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Product Details
See What's Inside
Product Description
By Nathalie Sinclair, David Pimm, Melanie Skelin, Rose Mary Zbiek
Why does it matter whether we state definitions carefully when we all know what particular geometric figures look like? What does it mean to say that a reflection is a transformation—a function? How does the study of transformations and matrices in high school connect with later work with vector spaces in linear algebra? How much do you know… and how much do you need to know?
Helping your students develop a robust understanding of geometry requires that you understand this mathematics deeply. But what does that mean?
This eBook focuses on essential knowledge for teachers about geometry. It is organized around four big ideas, supported by multiple smaller, interconnected ideas—essential understandings. Taking you beyond a simple introduction to geometry, the book will broaden and deepen your mathematical understanding of one of the most challenging topics for students—and teachers. It will help you engage your students, anticipate their perplexities, avoid pitfalls, and dispel misconceptions. You will also learn to develop appropriate tasks, techniques, and tools for assessing students' understanding of the topic. Focus on the ideas that you need to understand thoroughly to teach confidently.
Move beyond the mathematics you expect your students to learn. Students who fail to get a solid grounding in pivotal concepts struggle in subsequent work in mathematics and related disciplines. By bringing a deeper understanding to your teaching, you can help students who don't get it the first time by presenting the mathematics in multiple ways. The Essential Understanding Series addresses topics in school mathematics that are critical to the mathematical development of students but are often difficult to teach. Each book in the series gives an overview of the topic, highlights the differences between what teachers and students need to know, examines the big ideas and related essential understandings, reconsiders the ideas presented in light of connections with other mathematical ideas, and includes questions for readers' reflectionThis book focuses on essential knowledge for teachers about proof and the process of proving. It is organized around five big ideas, supported by multiple smaller, interconnected ideas—essential understandings.
Connect the process of problem solving with the content of the Common Core. The first of a series, this book will help mathematics educators illuminate a crucial link between problem solving and the Common Core State StandardsWhat tasks can you offer—what questions can you ask—to determine what your students know or don't know—and move them forward in their thinking?
This book focuses on the specialized pedagogical content knowledge that you need to teach fractions effectively in grades 3–5. The authors demonstrate how to use this multifaceted knowledge to address the big ideas and essential understandings that students must develop for success with fractions—not only in their current work, but also in higher-level mathematics and a myriad of real-world contexts.
The National Council of Teachers of Mathematics is the public voice of mathematics education, supporting teachers to ensure equitable mathematics learning of the highest quality for all students through vision, leadership, professional development, and research.
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Linear Algebra With Application Alt. Edition - 8th edition
Summary: Introductory courses in Linear Algebra can be taught in a variety of ways and the order of topics offered may vary based on the needs of the students. Linear Algebra with Applications, Alternate Eighth Edition provides instructors with an additional presentation of course material. In this edition earlier chapters cover systems of linear equations, matrices, and determinants. The more abstract material on vector spaces starts later, in Chapter 4, with the introduction of the vector s...show morepace R(n). This leads directly into general vector spaces and linear transformations. This alternate edition is especially appropriate for students preparing to apply linear equations and matrices in their own fields.Clear, concise, and comprehensive--the Alternate Eighth Edition continues to educate and enlighten students, leading to a mastery of the matehmatics and an understainding of how to apply it
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More About
This Textbook
Overview
Middle school teaching and learning has a distinct pedagogy and curriculum that is grounded in the concept of developmentally appropriate education. This text is designed to meet the very specific professional development needs of future teachers of mathematics in middle school environments.
Closely aligned with the NCTM Principles and Standards for School Mathematics, the reader-friendly, interactive format encourages readers to begin developing their own teaching style and making informed decisions about how to approach their future teaching career. A variety of examples establish a broad base of ideas intended to stimulate the formative development of concepts and models that can be employed in the classroom. Readers are encouraged and motivated to become teaching professionals who are lifelong learners.
The text offers a wealth of technology-related information and activities; reflective, thought-provoking questions; mathematical challenges; student life-based applications; TAG (tricks-activities-games) sections; and group discussion prompts to stimulate each future teacher's thinking. "Your Turn" sections ask readers to work with middle school students directly in field experience settings. This core text for middle school mathematics methods courses is also appropriate for elementary and secondary mathematics methods courses that address teaching in the middle school grades and as an excellent in-service resource for aspiring or practicing teachers of middle school mathematics as they update their knowledge base.
Topics covered in Teaching Middle School Mathematics:
*NCTM Principles for School Mathematics;
*Representation;
*Connections;
*Communication;
*Reasoning and Proof;
*Problem Solving;
*Number and Operations;
*Measurement;
*Data Analysis and Probability;
*Algebra in the Middle School Classroom; and
*Geometry in the Middle School Class
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Course information for Math 330
Linear Algebra
Instructor
Jennifer Johnson-Leung
303 Brink Hall
885-6258
jenfns@uidaho.edu
Class Times
MWF 1:30-2:20, REN 127 Office Hours
TBA
Text
Lay Linear Algebra and its Applications Third Edition
Course Description
Linear algebra is one of the most important parts of modern mathematics. In addition, its applications continue to spread more and more fields. The purpose of this course is to present the principal topics of linear algebra and to illustrate the power of this subject through a variety of applications.
This course is a transitional course, in which a mathematical argument and the construction of the proof are introduced. However, the focus is on student mastery of the techniques of linear algebra.
Grading Policies: Homework:
Homework exercises will be assigned in most classes, and you should complete the assignment before the next class period. Collaboration on homework is allowed and encouraged. A set of the homework exercises will be handed in for grading at the end of each week. Late homework will not be accepted except in extenuating circumstances.
Exams:
There will be three midterm exams and a final exam. The midterm exams are scheduled for 2/15, 3/12, and 4/16. If you have a conflict with any of the exam days, you should let me know immediately. Our final exam is scheduled for May 14 at 12:30 pm. Exception will be made for illness if a note from the health center (or attending physician) is provided.
Collaboration:
You are encouraged to collaborate on solving the problems given as homework. However, the solutions should be written on your own and in your own words. Clearly, there is no collaboration on exams. Blackboard and Electronic Submission of Assignments:
Our course webpage will be on Blackboard
If you have a documented need for accommodations such as extra time on exams, you should discuss this with me as soon as possible so that we can make the necessary arrangements. Point Distribution:
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Course Description
This online course will prepare you to take the Mathematics portion of the GED® (General Educational Development) test. The testing format will be explained and broken down to highlight the content this part of the exam will cover. This course will provide several resources from strategies, review of material, test taking tips, practice exercises and tests.
Topics covered:
-Arithmetic (different forms of numbers, signs and symbols, how to perform various operations, how to solve problems with Fractions, Percents and Decimals, and much more)
-Algebra (linear Equations and Inequalities, how to solve various word problems, how to simplify and factor numbers, and much more)
-Geometry (Geometric Concepts and Relationships, understanding measurements of congruence and similarity, the Pythagorean Theorem, and much more)
You cannot take the GED® test online. UniversalClass does not administer state-sanctioned GED® tests.
We recommend the following steps to pass the official, state-administered, GED® test:
Enroll in a GED® Preparation course
Purchase all recommended GED® Preparation books and materials
Study all course materials
Take all practice tests in the course and book materials
After you have mastered all material and passed all tests, you must sign up for the state-administered GED® test
Take the GED® test in person approved by both your state and the American Council on Education with a proctor
If you pass the state-administered GED® test, the state will issue you a high school equivalency diploma
*GED® is a registered trademark of the American Council on Education and may not be used without permission. The GED® and GED Testing Service®® brands are administered by GED Testing Service® LLC under license. THIS WORK IS NOT AUTHORIZED, ENDORSED, OR LICENSED BY AMERICAN COUNSEL ON EDUCATION OR GED Testing Service®, AND ANY REFERENCE TO "GED®" IN THE TITLE OR BODY OF THIS WORK IS IN NO WAY INTENDED TO IMPLY AN AFFILIATION WITH, OR SPONSORSHIP BY, GED Testing Service® OR ANY STATE OR ENTITY AUTHORIZED TO PROVIDE GED® BRANDED GOODS OR SERVICES.
Earn a final grade of 70% or higher to receive an online/downloadable CEU Certification documenting CEUs earned
Assessment Method:
Lesson assignments and review exams
Learning Outcomes
By successfully completing this course, students will be able to:
Demonstrate mastery of lesson content at levels of 70% or higher.
Lesson 1: Whole Numbers & Operations
Exam
30
Lesson 2: Number Sense
Exam
25
Lesson 3: Decimal Numbers
Exam
24
Post Test 1
Exam
81
Lesson 4: Fractions & Fraction Operations
Exam
27
Lesson 5: Number Relationships
Exam
24
Lesson 6: Statistics & Data Analysis
Exam
23
Post Test 2
Exam
75
Lesson 7: Percents
Exam
25
Lesson 8: Probability
Exam
22
Lesson 9: Data Analysis
Exam
24
Post Test 3
Exam
72
Lesson 10: Algebra
Exam
25
Lesson 11: Measurement
Exam
26
Lesson 12: Geometry
Exam
25
Post Test 4
Exam
76
Total Points:
604
Student Testimonials
"All of it was very helpful. It gave me the knowledge and confidence necessary for this section of the GED which, beforehand, I was very worried about. What was most helpful were the sections that most expanded on what I learned in school, particularly algebra, geometry and trigonometry." -- Kaleb H.
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Online statistics open textbook and additional resources to assist students in understanding of statistics. Topics covered...
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Online statistics open textbook and additional resources The applets in the simulations, demonstrations, an d caste studiesare in the public domain and can therefore be used without restriction. Simulations and demonstrations and source code are available for download. To view a video of the award winning author, go to View Rice Virtual Lab - Statistics Award Winner 2007 video
This is a free, online textbook that provides information on dimensions, from longtitude and latitude to the proof of a...
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This is a free, online textbook that provides information on dimensions, from longtitude and latitude to the proof of a theorem of geometry. There are 9 chapters, each 13 minutes long. The book contains a total of 117 minutes of video, but can also be read as an ordinary textbook.The film can be enjoyed by anyone, provided the chapters are well-chosen. There are 9 chapters, each 13 minutes long. Chapters 3-4, 5-6 and 7-8 are double chapters, but apart from that, they are more or less independent of each other.
A First Course in Linear Algebra is an introductory textbook aimed at college-level sophomores and juniors. Typically such a...
see more
A First Course in Linear Algebra is an introductory textbook aimed at college-level sophomores and juniors. Typically such a student will have taken calculus, but this is not a prerequisite. The book begins with systems of linear equations, then covers matrix algebra, before taking up finite-dimensional vector spaces in full generality. The final chapter covers matrix representations of linear transformations, through diagonalization, change of basis and Jordan canonical form.PDF versions are available to download for printing or on-screen viewing, an online version is available, and physical copies may be purchased from the print-on-demand service at Lulu.com. GNU Free Documentation License
The book provides a thorough introduction to "modern'' or "abstract'' algebra at a level suitable for upper-level...
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The book provides a thorough introduction to "modern'' or "abstract'' algebra at a level suitable for upper-level undergraduates and beginning graduate students. The book addresses the conventional topics: groups, rings, fields, and linear algebra, with symmetry as a unifying theme.
This introductory probability book, published by the American Mathematical Society, is available from AMS bookshop. We are...
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This introductory probability book, published by the American Mathematical Society, is available from AMS bookshop. We are pleased to announce that our book has now been made freely redistributable under the terms of the GNU Free Documentation License (FDL), as published by the Free Software Foundation. Briefly stated, the FDL permits you to do whatever you like with a work, as long as you don't prevent anyone else from doing what they like with it.The book emphasizes the use of computing to simulate experiments and make computations. We have prepared a set of programs to go with the book. We have Mathematica, Maple, and TrueBASIC versions of these programs
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MATH 135: Precalculus: Elementary Functions (3 credits)
Course Description
An analysis of elementary functions. A study of polynomial, rational, exponential and logarithmic functions. Topics also include graphing techniques, transformations, applications and related topics. Emphasis is placed on topics which will prove useful to students planning to take calculus and also to those who are interested in pursuing math-related careers. (3 hours lecture)
Pre-Requisite(s): Grade of "C" or better in MATH 103 or equivalent, satisfactory math placement test score, or consent of instructor.
Student Learning Outcomes
Upon completion of the course, the student will be able to:
Demonstrate proficiency in writing math expressions into different forms and finding the solutions to an equation and inequality using complex numbers where appropriate, by applying formal rules or algorithms.
Use appropriate symbolic techniques (such as algebraic techniques) to analyze and solve applied problems, and in the critical evaluation of evidence.
Interpret equations geometrically and use geometrical information to obtain the equation of lines and circles.
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The most helpful favourable review
The most helpful critical review
5.0 out of 5 starsVery useful
This book is so useful for beginning AS Maths, it bridges the gap from GCSE brilliantly. It was a really good buy, glad I got it.
Published on 2 Sep 2009 by E. Thompson
2 of 2 people found the following review helpful
2.0 out of 5 starsNot as I have hoped
Very short in material- could have been much better with important key component missing. I would not really recommend it and I am disappointed as our usual experience with CGP series of revision books is much better.
This is a useful guide for revising maths during the summer holidays. It recaps on things covered at GCSE as well as expanding on them to prepare for AS level maths. I found this book useful but thought that it lacked depth in some areas.
A Wonderful book, CGP really do know how to present books, I found it to be both enjoyable and resourceful, it's a must for students going onto AS Level Maths. Since there is a pretty big gap between the standard required from students, with pretty much no transition. A Must Buy for Student, Parents, Teachers and Tutors.
Very short in material- could have been much better with important key component missing. I would not really recommend it and I am disappointed as our usual experience with CGP series of revision books is much better.
The book is very slim and teaches you how to apply what you learnt from GCSE maths into A level maths. It helps GCSE students with the transition to A level. The only thing i did not like, was the fact that it's not as colourful as the Science cgp books and there aren't as many colourful pictures which makes the book quite bland and boring to read but it does get the job done in a simple effective way.
Bought this to start preparing for AS maths. Covers the harder GCSE topics and has practice questions; however, it's not in as much depth as the science Head Start to AS books which is a shame. Never-the-less, still helpful and solidifies some topics that will come up in AS.
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General Purpose Tools
General purpose tools are used to gain access to a wide range of problem-solving situations and mathematical content. These tools require strategies and skills highly applicable or transferable to other technology tools being used simultaneously or in the future.
The Spreadsheet tool displays and computes data stored in cells of a table; relates rows and columns by formulas; explores problems and models numerically including linear programming and "what-if" scenarios; and produces graphs for models not easily represented by algebraic models.
The National Council of Teachers of Mathematics is the public voice of mathematics education, supporting teachers to ensure equitable mathematics learning of the highest quality for all students through vision, leadership, professional development, and research.
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Principles Underlying the Ontario Mathematics Curriculum. This curriculum recognizes the diversity that exists ... This curriculum is designed to help students build the solid conceptual foundation in ...
Students work with visual and concrete examples of concepts. Students are assessed multiple pathways – Recognition Network: (a) ... Small group work to check for understanding. Differentiation/UDL. ...
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Survey of Mathmatics With Applications - Text Only - 9th edition
Summary: In a Liberal Arts Math course, a common question students ask is, ''Why do I have to know this?'' A Survey of Mathematics with Applicationscontinues to be a best-seller because it shows studentshowwe use mathematics in our daily lives andwhythis is important. The Ninth Edition further emphasizes this with the addition of new ''Why This Is Important'' sections throughout the text. Real-life and up-to-date examples motivate the topics throughout, and ...show morea wide range of exercises help students to develop their problem-solving and critical thinking skills.
Angel549011397.25 +$3.99 s/h
Good
A Book Company Lexington, KY
May contain some highlighting. Supplemental materials may not be included. We select best copy available. - 9th Edition - Hardcover - ISBN 9780321759665
$153159.24 +$3.99 s/h
Good
Dominion Bookstore Norfolk, VA
2012 Hardcover Good
$163.43 +$3.99 s/h
Good
FLORIDA BOOKSTORE Gainesville, FL
2012 Hardcover Good
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Differential Equations - With Student Solution Manual - 4th edition
Summary: Incorporating an innovative modeling approach, this book for a one-semester differential equations course emphasizes conceptual understanding to help users relate information taught in the classroom to real-world experiences. Certain models reappear throughout the book as running themes to synthesize different concepts from multiple angles, and a dynamical systems focus emphasizes predicting the long-term behavior of these recurring models. Users will discover how to...show more identify and harness the mathematics they will use in their careers, and apply it effectively outside the classroom
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Summary
The sixth edition of this best-selling text balances solid mathematical coverage with a comprehensive overview of mathematical ideas as they relate to varied disciplines. This text provides an appreciation of mathematics, highlighting mathematical history, applications of mathematics to the arts and sciences across cultures, and introduces students to the uses of technology in mathematics. It is an ideal book for students who require a general overview of mathematics, especially those majoring in the liberal arts, elementary education, the social sciences, business, nursing and allied health fields.
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Beginning Algebra
The Lial series has helped thousands of readers succeed in developmental mathematics through its approachable writing style, relevant real-world ...Show synopsisThe Lial series has helped thousands of readers succeed in developmental mathematics through its approachable writing style, relevant real-world examples, extensive exercise sets, and complete supplements package. The Real Number System; Linear Equations and Inequalities in One Variable; Linear Equations and Inequalities in Two Variables: Functions; Systems of Linear Equations and Inequalities; Exponents and Polynomials; Factoring and Applications; Rational Expressions and Applications; Roots and Radicals; Quadratic Equations For all readers interested in Beginning Algebra.Hide synopsis
Description:New in new dust jacket. Brand New as listed. ISBN 9780321702531....New in new dust jacket. Brand New as listed. ISBN 9780321702531. Clean! Out of sight Shipping & Customer Service! We process all orders same day! !
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0123850819
9780123850812
Matlab:Assuming no knowledge of programming, this book presents both programming concepts and MATLAB's built-in functions, providing a perfect platform for exploiting MATLAB's extensive capabilities for tackling engineering problems. It starts with programming concepts such as variables, assignments, input/output, and selection statements, moves onto loops and then solves problems using both the 'programming concept' and the 'power of MATLAB' side-by-side. In-depth coverage is given to input/output, a topic that is fundamental to many engineering applications. Ancillaries available with the text: Instructor solution manual (available Aug. 1st)electronic images from the text (available Aug 16th)m-files (available Aug 1st)
* Presents programming concepts and MATLAB built-in functions side-by-side, giving students the ability to program efficiently and exploit the power of MATLAB to solve problems. * In depth coverage of file input/output, a topic essential for many engineering applications * Systematic, step-by-step approach, building on concepts throughout the book, facilitating easier learning * Sections on 'common pitfalls' and 'programming guidelines' direct students towards best practice * New to this edition: More engineering applications help the reader learn Matlab in the context of solving technical problemsNew and revised end of chapter problemsStronger coverage of loops and vectorizing in a new chapter, chapter 5Updated to reflect current features and functions of the current release of Matlab
Back to top
Rent Matlab 2nd edition today, or search our site for Stormy textbooks. Every textbook comes with a 21-day "Any Reason" guarantee. Published by Butterworth-Heinemann.
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University – Mathematics Professor
Start Here for Free Information from Top Online Schools
Besides just teaching mathematics a university level math professor also participates in many activities on campus. These activities may include working with gifted or exceptional students, sitting on various committees and steering groups for the university, or even publishing research material on an ongoing basis. Most university mathematics professors also interact on a regular basis with colleagues and other mathematics professionals from around the world using various listserves and email programs. This constant networking may provide opportunities for a university mathematics professor to teach internationally or to become involved in mathematics research projects that are global in scope.
Mathematics professors use many different computer programs to verify mathematical research concepts that would simply be too time consuming to do with pencil and paper verification. A good understanding of databases, mathematical software programs and data entry protocols are required. In addition an ability to conceptualize and visualize even complex mathematic problems and break them down into small steps is critical for a professor.
Teaching students is one of the major responsibilities of a mathematics professor. Excellent communication skills and an ability to talk about abstract concepts in meaningful ways is truly important for effective teaching. The mathematics professor is also responsible for developing the curriculum they will use, evaluating student learning through projects and examinations and providing grades to students that reflect their understanding of the concepts. Mathematics professors may teach in various locations and may even travel internationally to present at conferences or to be a guest lecturer at a university in another location, state or country.
Common work activities include:
Developing and implementing a curriculum as approved by the mathematics department.
Teaching students based on mathematical principles in a manner that is within their ability to comprehend, assisting student learning, answering questions and working one-on-one with students during office hours.
Participating in research projects and keeping abreast of current changes in mathematical theory or teaching practices.
Traveling to various universities, conferences and professional development seminars.
Networking and interacting with other mathematics professors around the world.
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Schools- General Education
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I would like more information on the following: Pre-AlgebraAlgebra 1GeometryAlgebra 2PreCalculusCalculus
Getting Started
The initial assessment is the first step in documenting a student's academic performance.
This ensures that each student is placed at the appropriate entry level in the curriculum and helps identify conceptual holes in the student's understanding. Math-U-See's Placements Tests are designed to help you ensure proper initial placement.
For assistance in administering or interpreting these placement tests, please contact us.
Once your student is properly placed, you can determine which materials you need for instruction. Then read over our Four-Step Approach to understand how to get the most from the Math-U-See program in your homeschool.
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The basic homeschool set of math books for Saxon Math 87 comes with three books, the tests and worksheets book, the teacher's manual, and the solutions manual.
Like other Saxon Math books, the title of the book contains two numbers. In this case, the number is 87. The second number refers to the grade level when this book is generally taught, and lets parents know when to teach the curriculum to average or bright students in the 7th grade. The first number refers to when to teach this math curriculum to slower students.
The format of the Math 87 student textbook has a very consistent daily layout. The typical Saxon format gives students a few short pages explaining the lesson, including step-by-step example problems. Following the lesson, most of the day's work comes from the problem set, a set of 30 or so math problems of all different types. This is a cumulative review and could basically include any type of problem that has been introduced in the current book or in previous math books.
Topics and Methodology
In Saxon Math 87, students get into rigorous pre-algebra topics. Basic concepts from previous years are the subject of review and reinforcement; word problems practice problem solving techniques. New concepts and procedures, as well as vocabulary are introduced to prepare students for success in upper-level math. Some of the new topics covered include ratios, perimeter, circumference, area, and volume; scientific notation.
The characteristic incremental approach and spiral review of Saxon Math books assist the seventh grade student to learn and retain material. The new concepts are broken into small parts and each part is introduced in a separate lesson, gradually building up to the complete concept. Daily problem set review constantly practices all previous problem types.
Teacher Requirements
Teaching is easy with the way Saxon Math books are laid out. There are no special background requirements. The format encourages independent learning, with the student reading the lesson on his own, then working the problems. The solutions manual with its complete worked out problems is a great boon to a teacher who wishes to give partial credit on a students' work without having to write out the whole problem again.
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This popular Physical Chemistry text book is now available in electronic format. We have preserved much of the material of the former hard copy editions, making changes to improve understanding of the concepts in addition to including some of the recent discoveries in physical chemistry. Many chapters have new sections and the coverage of several chapters has been greatly expanded. The chapter on statistical mechanics, 15, has been completely rewritten.
The eBook has also been divided into smaller modules that are appropriate for specific courses in Physical Chemistry.
Easy to use Clebsch-Gordan coefficient solver for adding two angular momentums in Quantum Mechanics. This tool is created for my Quantum Mechanics II course offered by Dr. Thompson in Summer of 2007.
[Instruction]
Execute "GUI.m" script by invoking "GUI".
Inspired by a discussion with my father on how to solve sudokus, I decided to implement a GUI for MATLAB and play around with automatic solving. The result can be found here: You can use the GUI just for playing sudoku and having an online check or you may turn on the solving aids: Display tooltips showing all valid numbers so far, or have a semiautomatic or a automatic solver which evaluates the logical constraints. On top of that, a branching algorithm is implemented, which solves any arbitrary sudoku very fast.
Math Solver Free for Windows 8 is a handy tool for performing frequently used operations used for solving math problems. You can use this tool for solving quadratic equations or calculating the angles of a triangle.
The app also includes a unit converter and other useful tools for dealing with math problems by using your Windows 8 device.
Worksheet Generator for Chemistry is a handy and reliable software that helps you to easily and quickly create and customize your personal chemistry worksheets.
The application provides you with various exercise templates that allow you to adjust your worksheets. You are able to insert various chemistry exercises of different areas such as units and chemical formulae, thermochemistry, chemical kinetics, Redox reactions and organic chemistry
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More About
This Textbook
Overview
Prior to the nineteenth century, algebra meant the study of the solution of polynomial equations. By the twentieth century algebra came to encompass the study of abstract, axiomatic systems such as groups, rings, and fields. This presentation provides an account of the intellectual lineage behind many of the basic concepts, results, and theories of abstract algebra.
The development of abstract algebra was propelled by the need for new tools to address certain classical problems that appeared unsolvable by classical means. A major theme of the approach in this book is to show how abstract algebra has arisen in attempts to solve some of these classical problems, providing context from which the reader may gain a deeper appreciation of the mathematics involved.
Key features:
* Begins with an overview of classical algebra
* Contains separate chapters on aspects of the development of groups, rings, and fields
* Examines the evolution of linear algebra as it relates to other elements of abstract algebra
* Highlights the lives and works of six notables: Cayley, Dedekind, Galois, Gauss, Hamilton, and especially the pioneering work of Emmy Noether
* Offers suggestions to instructors on ways of integrating the history of abstract algebra into their teaching
* Each chapter concludes with extensive references to the relevant literature
Mathematics instructors, algebraists, and historians of science will find the work a valuable reference. The book may also serve as a supplemental text for courses in abstract algebra or the history of mathematics.
Editorial Reviews
From the Publisher
From the reviews:
"This concise history conveniently brings together topics in modern algebra that one might otherwise only find in scattered sources. … it reflects a deep attention to the mathematics and to how its history can be used to help understand the subject today. … The author provides an outline of his course in abstract algebra, a course that is intended for teachers of mathematics … ." (Albert C. Lewis, Mathematical Reviews, Issue 2008 g)
"This book gives an overview of the origin and development of the basic ideas of modern abstract algebra. … In each chapter, the author makes extensive references to relevant literature. The book can be recommended to mathematicians, teachers of mathematics (especially of algebra), historians of the sciences and students, who can find many useful references and ideas for their research, teaching or studies." (EMS Newsletter, September, 2008)
"This remarkable book presents both the history of algebra as well as selected detailed biographies of algebraists. … the origin of important ideas and concepts is presented very skillfully, even in a way such that the development of ideas can be used as a very good textbook for algebra. … This book combines in relatively few pages non-trivial algebra with detailed historical facts and ideas and should bring the reader a wealth of new insights." (G. Pilz, Internationle Mathematische Nachrichten, Issue 210
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Here is an algebra concept you already know. When most people think of algebra, they think of solving equations. That mysterious process that mathematicians use to manipulate equations and find an unknown value.
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viii Contents You are today where your knowledge brought you; you will be tomorrow where your knowledge takes you. — Anonymous
PrefaceScaffolding Reacting to criticism concerning the lack of motivation in his writings,Gauss remarked that architects of great cathedrals do not obscure the beautyof their work by leaving the scaffolding in place after the construction has beencompleted. His philosophy epitomized the formal presentation and teaching ofmathematics throughout the nineteenth and twentieth centuries, and it is stillcommonly found in mid-to-upper-level mathematics textbooks. The inherent ef-ficiency and natural beauty of mathematics are compromised by straying too farfrom Gauss's viewpoint. But, as with most things in life, appreciation is gen-erally preceded by some understanding seasoned with a bit of maturity, and inmathematics this comes from seeing some of the scaffolding.Purpose, Gap, and Challenge The purpose of this text is to present the contemporary theory and applica-tions of linear algebra to university students studying mathematics, engineering,or applied science at the postcalculus level. Because linear algebra is usually en-countered between basic problem solving courses such as calculus or differentialequations and more advanced courses that require students to cope with mathe-matical rigors, the challenge in teaching applied linear algebra is to expose someof the scaffolding while conditioning students to appreciate the utility and beautyof the subject. Effectively meeting this challenge and bridging the inherent gapsbetween basic and more advanced mathematics are primary goals of this book.Rigor and Formalism To reveal portions of the scaffolding, narratives, examples, and summariesare used in place of the formal definition–theorem–proof development. But whilewell-chosen examples can be more effective in promoting understanding thanrigorous proofs, and while precious classroom minutes cannot be squandered ontheoretical details, I believe that all scientifically oriented students should beexposed to some degree of mathematical thought, logic, and rigor. And if logicand rigor are to reside anywhere, they have to be in the textbook. So even whenlogic and rigor are not the primary thrust, they are always available. Formaldefinition–theorem–proof designations are not used, but definitions, theorems,and proofs nevertheless exist, and they become evident as a student's maturityincreases. A significant effort is made to present a linear development that avoidsforward references, circular arguments, and dependence on prior knowledge of thesubject. This results in some inefficiencies—e.g., the matrix 2-norm is presented
x Preface before eigenvalues or singular values are thoroughly discussed. To compensate, I try to provide enough "wiggle room" so that an instructor can temper the inefficiencies by tailoring the approach to the students' prior background. Comprehensiveness and Flexibility A rather comprehensive treatment of linear algebra and its applications is presented and, consequently, the book is not meant to be devoured cover-to-cover in a typical one-semester course. However, the presentation is structured to pro- vide flexibility in topic selection so that the text can be easily adapted to meet the demands of different course outlines without suffering breaks in continuity. Each section contains basic material paired with straightforward explanations, examples, and exercises. But every section also contains a degree of depth coupled with thought-provoking examples and exercises that can take interested students to a higher level. The exercises are formulated not only to make a student think about material from a current section, but they are designed also to pave the way for ideas in future sections in a smooth and often transparent manner. The text accommodates a variety of presentation levels by allowing instructors to select sections, discussions, examples, and exercises of appropriate sophistication. For example, traditional one-semester undergraduate courses can be taught from the basic material in Chapter 1 (Linear Equations); Chapter 2 (Rectangular Systems and Echelon Forms); Chapter 3 (Matrix Algebra); Chapter 4 (Vector Spaces); Chapter 5 (Norms, Inner Products, and Orthogonality); Chapter 6 (Determi- nants); and Chapter 7 (Eigenvalues and Eigenvectors). The level of the course and the degree of rigor are controlled by the selection and depth of coverage in the latter sections of Chapters 4, 5, and 7. An upper-level course might consist of a quick review of Chapters 1, 2, and 3 followed by a more in-depth treatment of Chapters 4, 5, and 7. For courses containing advanced undergraduate or grad- uate students, the focus can be on material in the latter sections of Chapters 4, 5, 7, and Chapter 8 (Perron–Frobenius Theory of Nonnegative Matrices). A rich two-semester course can be taught by using the text in its entirety. What Does "Applied" Mean? Most people agree that linear algebra is at the heart of applied science, but there are divergent views concerning what "applied linear algebra" really means; the academician's perspective is not always the same as that of the practitioner. In a poll conducted by SIAM in preparation for one of the triannual SIAM con- ferences on applied linear algebra, a diverse group of internationally recognized scientific corporations and government laboratories was asked how linear algebra finds application in their missions. The overwhelming response was that the pri- mary use of linear algebra in applied industrial and laboratory work involves the development, analysis, and implementation of numerical algorithms along with some discrete and statistical modeling. The applications in this book tend to reflect this realization. While most of the popular "academic" applications are included, and "applications" to other areas of mathematics are honestly treated,
Preface xi there is an emphasis on numerical issues designed to prepare students to use linear algebra in scientific environments outside the classroom. Computing Projects Computing projects help solidify concepts, and I include many exercises that can be incorporated into a laboratory setting. But my goal is to write a mathematics text that can last, so I don't muddy the development by marrying the material to a particular computer package or language. I am old enough to remember what happened to the FORTRAN- and APL-based calculus and linear algebra texts that came to market in the 1970s. I provide instructors with a flexible environment that allows for an ancillary computing laboratory in which any number of popular packages and lab manuals can be used in conjunction with the material in the text. History Finally, I believe that revealing only the scaffolding without teaching some- thing about the scientific architects who erected it deprives students of an im- portant part of their mathematical heritage. It also tends to dehumanize mathe- matics, which is the epitome of human endeavor. Consequently, I make an effort to say things (sometimes very human things that are not always complimentary) about the lives of the people who contributed to the development and applica- tions of linear algebra. But, as I came to realize, this is a perilous task because writing history is frequently an interpretation of facts rather than a statement of facts. I considered documenting the sources of the historical remarks to help mitigate the inevitable challenges, but it soon became apparent that the sheer volume required to do so would skew the direction and flavor of the text. I can only assure the reader that I made an effort to be as honest as possible, and I tried to corroborate "facts." Nevertheless, there were times when interpreta- tions had to be made, and these were no doubt influenced by my own views and experiences. Supplements Included with this text is a solutions manual and a CD-ROM. The solutions manual contains the solutions for each exercise given in the book. The solutions are constructed to be an integral part of the learning process. Rather than just providing answers, the solutions often contain details and discussions that are intended to stimulate thought and motivate material in the following sections. The CD, produced by Vickie Kearn and the people at SIAM, contains the entire book along with the solutions manual in PDF format. This electronic version of the text is completely searchable and linked. With a click of the mouse a student can jump to a referenced page, equation, theorem, definition, or proof, and then jump back to the sentence containing the reference, thereby making learning quite efficient. In addition, the CD contains material that extends his- torical remarks in the book and brings them to life with a large selection of
xii Preface portraits, pictures, attractive graphics, and additional anecdotes. The support- ing Internet site at MatrixAnalysis.com contains updates, errata, new material, and additional supplements as they become available. SIAM I thank the SIAM organization and the people who constitute it (the in- frastructure as well as the general membership) for allowing me the honor of publishing my book under their name. I am dedicated to the goals, philosophy, and ideals of SIAM, and there is no other company or organization in the world that I would rather have publish this book. In particular, I am most thankful to Vickie Kearn, publisher at SIAM, for the confidence, vision, and dedication she has continually provided, and I am grateful for her patience that allowed me to write the book that I wanted to write. The talented people on the SIAM staff went far above and beyond the call of ordinary duty to make this project special. This group includes Lois Sellers (art and cover design), Michelle Mont- gomery and Kathleen LeBlanc (promotion and marketing), Marianne Will and Deborah Poulson (copy for CD-ROM biographies), Laura Helfrich and David Comdico (design and layout of the CD-ROM), Kelly Cuomo (linking the CD- ROM), and Kelly Thomas (managing editor for the book). Special thanks goes to Jean Anderson for her eagle-sharp editor's eye. Acknowledgments This book evolved over a period of several years through many different courses populated by hundreds of undergraduate and graduate students. To all my students and colleagues who have offered suggestions, corrections, criticisms, or just moral support, I offer my heartfelt thanks, and I hope to see as many of you as possible at some point in the future so that I can convey my feelings to you in person. I am particularly indebted to Michele Benzi for conversations and suggestions that led to several improvements. All writers are influenced by people who have written before them, and for me these writers include (in no particular order) Gil Strang, Jim Ortega, Charlie Van Loan, Leonid Mirsky, Ben Noble, Pete Stewart, Gene Golub, Charlie Johnson, Roger Horn, Peter Lancaster, Paul Halmos, Franz Hohn, Nick Rose, and Richard Bellman—thanks for lighting the path. I want to offer particular thanks to Richard J. Painter and Franklin A. Graybill, two exceptionally fine teachers, for giving a rough Colorado farm boy a chance to pursue his dreams. Finally, neither this book nor anything else I have done in my career would have been possible without the love, help, and unwavering support from Bethany, my friend, partner, and wife. Her multiple readings of the manuscript and suggestions were invaluable. I dedicate this book to Bethany and our children, Martin and Holly, to our granddaughter, Margaret, and to the memory of my parents, Carl and Louise Meyer. Carl D. Meyer April 19, 2000
CHAPTER 1 Linear Equations1.1 INTRODUCTION A fundamental problem that surfaces in all mathematical sciences is that of analyzing and solving m algebraic equations in n unknowns. The study of a system of simultaneous linear equations is in a natural and indivisible alliance with the study of the rectangular array of numbers defined by the coefficients of the equations. This link seems to have been made at the outset. The earliest recorded analysis of simultaneous equations is found in the ancient Chinese book Chiu-chang Suan-shu (Nine Chapters on Arithmetic), es- timated to have been written some time around 200 B.C. In the beginning of Chapter VIII, there appears a problem of the following form. Three sheafs of a good crop, two sheafs of a mediocre crop, and one sheaf of a bad crop are sold for 39 dou. Two sheafs of good, three mediocre, and one bad are sold for 34 dou; and one good, two mediocre, and three bad are sold for 26 dou. What is the price received for each sheaf of a good crop, each sheaf of a mediocre crop, and each sheaf of a bad crop? Today, this problem would be formulated as three equations in three un- knowns by writing 3x + 2y + z = 39, 2x + 3y + z = 34, x + 2y + 3z = 26, where x, y, and z represent the price for one sheaf of a good, mediocre, and bad crop, respectively. The Chinese saw right to the heart of the matter. They placed the coefficients (represented by colored bamboo rods) of this system in
2 Chapter 1 Linear Equations a square array on a "counting board" and then manipulated the lines of the array according to prescribed rules of thumb. Their counting board techniques and rules of thumb found their way to Japan and eventually appeared in Europe with the colored rods having been replaced by numerals and the counting board replaced by pen and paper. In Europe, the technique became known as Gaussian 1 elimination in honor of the German mathematician Carl Gauss, whose extensive use of it popularized the method. Because this elimination technique is fundamental, we begin the study of our subject by learning how to apply this method in order to compute solutions for linear equations. After the computational aspects have been mastered, we will turn to the more theoretical facets surrounding linear systems. 1 Carl Friedrich Gauss (1777–1855) is considered by many to have been the greatest mathemati- cian who has ever lived, and his astounding career requires several volumes to document. He was referred to by his peers as the "prince of mathematicians." Upon Gauss's death one of them wrote that "His mind penetrated into the deepest secrets of numbers, space, and nature; He measured the course of the stars, the form and forces of the Earth; He carried within himself the evolution of mathematical sciences of a coming century." History has proven this remark to be true.
1.2 Gaussian Elimination and Matrices 31.2 GAUSSIAN ELIMINATION AND MATRICES The problem is to calculate, if possible, a common solution for a system of m linear algebraic the xi 's are the unknowns and the aij 's and the bi 's are known constants. The aij 's are called the coefficients of the system, and the set of bi 's is referred to as the right-hand side of the system. For any such system, there are exactly three possibilities for the set of solutions. Three Possibilities • UNIQUE SOLUTION: There is one and only one set of values for the xi 's that satisfies all equations simultaneously. • NO SOLUTION: There is no set of values for the xi 's that satisfies all equations simultaneously—the solution set is empty. • INFINITELY MANY SOLUTIONS: There are infinitely many different sets of values for the xi 's that satisfy all equations simultaneously. It is not difficult to prove that if a system has more than one solution, then it has infinitely many solutions. For example, it is impossible for a system to have exactly two different solutions. Part of the job in dealing with a linear system is to decide which one of these three possibilities is true. The other part of the task is to compute the solution if it is unique or to describe the set of all solutions if there are many solutions. Gaussian elimination is a tool that can be used to accomplish all of these goals. Gaussian elimination is a methodical process of systematically transform- ing one system into another simpler, but equivalent, system (two systems are called equivalent if they possess equal solution sets) by successively eliminating unknowns and eventually arriving at a system that is easily solvable. The elimi- nation process relies on three simple operations by which to transform one system to another equivalent system. To describe these operations, let Ek denote the k th equation Ek : ak1 x1 + ak2 x2 + · · · + akn xn = bk
1.2 Gaussian Elimination and Matrices 5 Providing explanations for why each of these operations cannot change the solution set is left as an exercise. The most common problem encountered in practice is the one in which there are n equations as well as n unknowns—called a square system—for which there is a unique solution. Since Gaussian elimination is straightforward for this case, we begin here and later discuss the other possibilities. What follows is a detailed description of Gaussian elimination as applied to the following simple (but typical) square system: 2x + y + z = 1, 6x + 2y + z = − 1, (1.2.4) −2x + 2y + z = 7. At each step, the strategy is to focus on one position, called the pivot po- sition, and to eliminate all terms below this position using the three elementary operations. The coefficient in the pivot position is called a pivotal element (or simply a pivot), while the equation in which the pivot lies is referred to as the pivotal equation. Only nonzero numbers are allowed to be pivots. If a coef- ficient in a pivot position is ever 0, then the pivotal equation is interchanged with an equation below the pivotal equation to produce a nonzero pivot. (This is always possible for square systems possessing a unique solution.) Unless it is 0, the first coefficient of the first equation is taken as the first pivot. For example, the circled 2 in the system below is the pivot for the first step: 2 x + y + z = 1, 6x + 2y + z = − 1, −2x + 2y + z = 7. Step 1. Eliminate all terms below the first pivot. • Subtract three times the first equation from the second so as to produce the equivalent system: 2 x + y + z = 1, − y − 2z = − 4 (E2 − 3E1 ), −2x + 2y + z = 7. • Add the first equation to the third equation to produce the equivalent system: 2 x + y + z = 1, − y − 2z = − 4, 3y + 2z = 8 (E3 + E1 ).
6 Chapter 1 Linear Equations Step 2. Select a new pivot. 2 • For the time being, select a new pivot by moving down and to the right. If this coefficient is not 0, then it is the next pivot. Otherwise, interchange with an equation below this position so as to bring a nonzero number into this pivotal position. In our example, −1 is the second pivot as identified below: 2x + y + z = 1, -1y − 2z = − 4, 3y + 2z = 8. Step 3. Eliminate all terms below the second pivot. • Add three times the second equation to the third equation so as to produce the equivalent system: 2x + y + z = 1, -1 y − 2z = − 4, (1.2.5) − 4z = − 4 (E3 + 3E2 ). • In general, at each step you move down and to the right to select the next pivot, then eliminate all terms below the pivot until you can no longer pro- ceed. In this example, the third pivot is −4, but since there is nothing below the third pivot to eliminate, the process is complete. At this point, we say that the system has been triangularized. A triangular system is easily solved by a simple method known as back substitution in which the last equation is solved for the value of the last unknown and then substituted back into the penultimate equation, which is in turn solved for the penultimate unknown, etc., until each unknown has been determined. For our example, solve the last equation in (1.2.5) to obtain z = 1. Substitute z = 1 back into the second equation in (1.2.5) and determine y = 4 − 2z = 4 − 2(1) = 2. 2 The strategy of selecting pivots in numerical computation is usually a bit more complicated than simply using the next coefficient that is down and to the right. Use the down-and-right strategy for now, and later more practical strategies will be discussed.
1.2 Gaussian Elimination and Matrices 7 Finally, substitute z = 1 and y = 2 back into the first equation in (1.2.5) to get 1 1 x = (1 − y − z) = (1 − 2 − 1) = −1, 2 2 which completes the solution. It should be clear that there is no reason to write down the symbols such as " x, " " y, " " z, " and " = " at each step since we are only manipulating the coefficients. If such symbols are discarded, then a system of linear equations reduces to a rectangular array of numbers in which each horizontal line represents one equation. For example, the system in (1.2.4) reduces to the following array: 2 1 1 1 6 2 1 −1 . (The line emphasizes where = appeared.) −2 2 1 7 The array of coefficients—the numbers on the left-hand side of the vertical line—is called the coefficient matrix for the system. The entire array—the coefficient matrix augmented by the numbers from the right-hand side of the system—is called the augmented matrix associated with the system. If the coefficient matrix is denoted by A and the right-hand side is denoted by b , then the augmented matrix associated with the system is denoted by [A|b]. Formally, a scalar is either a real number or a complex number, and a matrix is a rectangular array of scalars. It is common practice to use uppercase boldface letters to denote matrices and to use the corresponding lowercase letters with two subscripts to denote individual entries in a matrix. For example, a11 a12 ··· a1n a21 a22 ··· a2n A= . . . . .. . . . . . . . am1 am2 · · · amn The first subscript on an individual entry in a matrix designates the row (the horizontal line), and the second subscript denotes the column (the vertical line) that the entry occupies. For example, if 2 1 3 4 A= 8 6 5 −9 , then a11 = 2, a12 = 1, . . . , a34 = 7. (1.2.6) −3 8 3 7 A submatrix of a given matrix A is an array obtained by deleting any combination of rows and columns from A. For example, B = −3 4 is a 2 7 submatrix of the matrix A in (1.2.6) because B is the result of deleting the second row and the second and third columns of A.
10 Chapter 1 Linear Equations Algorithm for Back Substitution Determine the xi 's from (1.2.10) by first setting xn = cn /tnn and then recursively computing 1 xi = (ci − ti,i+1 xi+1 − ti,i+2 xi+2 − · · · − tin xn ) tii for i = n − 1, n − 2, . . . , 2, 1. One way to gauge the efficiency of an algorithm is to count the number of 3 arithmetical operations required. For a variety of reasons, no distinction is made between additions and subtractions, and no distinction is made between multipli- cations and divisions. Furthermore, multiplications/divisions are usually counted separately from additions/subtractions. Even if you do not work through the de- tails, it is important that you be aware of the operational counts for Gaussian elimination with back substitution so that you will have a basis for comparison when other algorithms are encountered. Gaussian Elimination Operation Counts Gaussian elimination with back substitution applied to an n × n system requires n3 n + n2 − multiplications/divisions 3 3 and n3 n2 5n + − additions/subtractions. 3 2 6 As n grows, the n3 /3 term dominates each of these expressions. There- fore, the important thing to remember is that Gaussian elimination with back substitution on an n × n system requires about n3 /3 multiplica- tions/divisions and about the same number of additions/subtractions. 3 Operation counts alone may no longer be as important as they once were in gauging the ef- ficiency of an algorithm. Older computers executed instructions sequentially, whereas some contemporary machines are capable of executing instructions in parallel so that different nu- merical tasks can be performed simultaneously. An algorithm that lends itself to parallelism may have a higher operational count but might nevertheless run faster on a parallel machine than an algorithm with a lesser operational count that cannot take advantage of parallelism.
1.2 Gaussian Elimination and Matrices 13 1.2.8. The following system has no solution: −x1 + 3x2 − 2x3 = 1, −x1 + 4x2 − 3x3 = 0, −x1 + 5x2 − 4x3 = 0. Attempt to solve this system using Gaussian elimination and explain what occurs to indicate that the system is impossible to solve. 1.2.9. Attempt to solve the system −x1 + 3x2 − 2x3 = 4, −x1 + 4x2 − 3x3 = 5, −x1 + 5x2 − 4x3 = 6, using Gaussian elimination and explain why this system must have in- finitely many solutions. 1.2.10. By solving a 3 × 3 system, find the coefficients in the equation of the parabola y = α+βx+γx2 that passes through the points (1, 1), (2, 2), and (3, 0). 1.2.11. Suppose that 100 insects are distributed in an enclosure consisting of four chambers with passageways between them as shown below. #3 #4 #2 #1 At the end of one minute, the insects have redistributed themselves. Assume that a minute is not enough time for an insect to visit more than one chamber and that at the end of a minute 40% of the insects in each chamber have not left the chamber they occupied at the beginning of the minute. The insects that leave a chamber disperse uniformly among the chambers that are directly accessible from the one they initially occupied—e.g., from #3, half move to #2 and half move to #4.
14 Chapter 1 Linear Equations (a) If at the end of one minute there are 12, 25, 26, and 37 insects in chambers #1, #2, #3, and #4, respectively, determine what the initial distribution had to be. (b) If the initial distribution is 20, 20, 20, 40, what is the distribution at the end of one minute? 1.2.12. Show that the three types of elementary row operations discussed on p. 8 are not independent by showing that the interchange operation (1.2.7) can be accomplished by a sequence of the other two types of row operations given in (1.2.8) and (1.2.9). 1.2.13. Suppose that [A|b] is the augmented matrix associated with a linear system. You know that performing row operations on [A|b] does not change the solution of the system. However, no mention of column oper- ations was ever made because column operations can alter the solution. (a) Describe the effect on the solution of a linear system when columns A∗j and A∗k are interchanged. (b) Describe the effect when column A∗j is replaced by αA∗j for α = 0. (c) Describe the effect when A∗j is replaced by A∗j + αA∗k . Hint: Experiment with a 2 × 2 or 3 × 3 system. 1.2.14. Consider the n × n Hilbert matrix defined by 1 1 2 1 3 ··· 1 n 1 2 1 1 ··· 1 3 4 n+1 H=1 1 1 ··· 1 . 3 4 5 n+2 . . . . . . . ··· . . . . . 1 n 1 n+1 1 n+2 ··· 1 2n−1 Express the individual entries hij in terms of i and j. 1.2.15. Verify that the operation counts given in the text for Gaussian elimi- nation with back substitution are correct for a general 3 × 3 system. If you are up to the challenge, try to verify these counts for a general n × n system. 1.2.16. Explain why a linear system can never have exactly two different solu- tions. Extend your argument to explain the fact that if a system has more than one solution, then it must have infinitely many different solutions.
1.3 Gauss–Jordan Method 151.3 GAUSS–JORDAN METHOD The purpose of this section is to introduce a variation of Gaussian elimination 4 that is known as the Gauss–Jordan method. The two features that dis- tinguish the Gauss–Jordan method from standard Gaussian elimination are as follows. • At each step, the pivot element is forced to be 1. • At each step, all terms above the pivot as well as all terms below the pivot are eliminated. In other words, if a11 a12 ··· a1n b1 a21 a22 ··· a2n b2 . . . . .. . . .. . . . . . an1 an2 · · · ann bn is the augmented matrix associated with a linear system, then elementary row operations are used to reduce this matrix to 1 0 ··· 0 s1 0 1 ··· 0 s2 . . . . . . . . . .. .. . . . 0 0 ··· 1 sn The solution then appears in the last column (i.e., xi = si ) so that this procedure circumvents the need to perform back substitution.Example 1.3.1 Problem: Apply the Gauss–Jordan method to solve the following system: 2x1 + 2x2 + 6x3 = 4, 2x1 + x2 + 7x3 = 6, −2x1 − 6x2 − 7x3 = − 1. 4 Although there has been some confusion as to which Jordan should receive credit for this algorithm, it now seems clear that the method was in fact introduced by a geodesist named Wilhelm Jordan (1842–1899) and not by the more well known mathematician Marie Ennemond Camille Jordan (1838–1922), whose name is often mistakenly associated with the technique, but who is otherwise correctly credited with other important topics in matrix analysis, the "Jordan canonical form" being the most notable. Wilhelm Jordan was born in southern Germany, educated in Stuttgart, and was a professor of geodesy at the technical college in Karlsruhe. He was a prolific writer, and he introduced his elimination scheme in the 1888 publication Handbuch der Vermessungskunde. Interestingly, a method similar to W. Jordan's variation of Gaussian elimination seems to have been discovered and described independently by an obscure Frenchman named Clasen, who appears to have published only one scientific article, which appeared in 1888—the same year as W. Jordan's Handbuch appeared.
1.3 Gauss–Jordan Method 17 number of additions/subtractions. Compare this with the n3 /2 factor required by the Gauss–Jordan method, and you can see that Gauss–Jordan requires about 50% more effort than Gaussian elimination with back substitution. For small sys- tems of the textbook variety (e.g., n = 3 ), these comparisons do not show a great deal of difference. However, in practical work, the systems that are encountered can be quite large, and the difference between Gauss–Jordan and Gaussian elim- ination with back substitution can be significant. For example, if n = 100, then n3 /3 is about 333,333, while n3 /2 is 500,000, which is a difference of 166,667 multiplications/divisions as well as that many additions/subtractions. Although the Gauss–Jordan method is not recommended for solving linear systems that arise in practical applications, it does have some theoretical advan- tages. Furthermore, it can be a useful technique for tasks other than computing solutions to linear systems. We will make use of the Gauss–Jordan procedure when matrix inversion is discussed—this is the primary reason for introducing Gauss–Jordan.Exercises for section 1.3 1.3.1. Use the Gauss–Jordan method to solve the following system: 4x2 − 3x3 = 3, −x1 + 7x2 − 5x3 = 4, −x1 + 8x2 − 6x3 = 5. 1.3.2. Apply the Gauss–Jordan method to the following system: x1 + x2 + x3 + x4 = 1, x1 + 2x2 + 2x3 + 2x4 = 0, x1 + 2x2 + 3x3 + 3x4 = 0, x1 + 2x2 + 3x3 + 4x4 = 0. 1.3.3. Use the Gauss–Jordan method to solve the following three systems at the same time. 2x1 − x2 = 1 0 0, −x1 + 2x2 − x3 = 0 1 0, −x2 + x3 = 0 0 1. 1.3.4. Verify that the operation counts given in the text for the Gauss–Jordan method are correct for a general 3 × 3 system. If you are up to the challenge, try to verify these counts for a general n × n system.
20 Chapter 1 Linear Equations Notice the pattern of the entries in the coefficient matrix in the above ex- ample. The nonzero elements occur only on the subdiagonal, main-diagonal, and superdiagonal lines—such a system (or matrix) is said to be tridiagonal. This is characteristic in the sense that when finite difference approximations are ap- plied to the general two-point boundary value problem, a tridiagonal system is the result. Tridiagonal systems are particularly nice in that they are inexpensive to solve. When Gaussian elimination is applied, only two multiplications/divisions are needed at each step of the triangularization process because there is at most only one nonzero entry below and to the right of each pivot. Furthermore, Gaus- sian elimination preserves all of the zero entries that were present in the original tridiagonal system. This makes the back substitution process cheap to execute because there are at most only two multiplications/divisions required at each substitution step. Exercise 3.10.6 contains more details.Exercises for section 1.4 1.4.1. Divide the interval [0, 1] into five equal subintervals, and apply the finite difference method in order to approximate the solution of the two-point boundary value problem y (t) = 125t, y(0) = y(1) = 0 at the four interior grid points. Compare your approximate values at the grid points with the exact solution at the grid points. Note: You should not expect very accurate approximations with only four interior grid points. 1.4.2. Divide [0, 1] into n+1 equal subintervals, and apply the finite difference approximation method to derive the linear system associated with the two-point boundary value problem y (t) − y (t) = f (t), y(0) = y(1) = 0. 1.4.3. Divide [0, 1] into five equal subintervals, and approximate the solution to y (t) − y (t) = 125t, y(0) = y(1) = 0 at the four interior grid points. Compare the approximations with the exact values at the grid points.
1.5 Making Gaussian Elimination Work 211.5 MAKING GAUSSIAN ELIMINATION WORK Now that you understand the basic Gaussian elimination technique, it's time to turn it into a practical algorithm that can be used for realistic applications. For pencil and paper computations where you are doing exact arithmetic, the strategy is to keep things as simple as possible (like avoiding messy fractions) in order to minimize those "stupid arithmetic errors" we are all prone to make. But very few problems in the real world are of the textbook variety, and practical applications involving linear systems usually demand the use of a computer. Computers don't care about messy fractions, and they don't introduce errors of the "stupid" variety. Computers produce a more predictable kind of error, called 6 roundoff error, and it's important to spend a little time up front to understand this kind of error and its effects on solving linear systems. Numerical computation in digital computers is performed by approximating the infinite set of real numbers with a finite set of numbers as described below. Floating-Point Numbers A t -digit, base-β floating-point number has the form f = ±.d1 d2 · · · dt × β with d1 = 0, where the base β, the exponent , and the digits 0 ≤ di ≤ β − 1 are integers. For internal machine representation, β = 2 (binary rep- resentation) is standard, but for pencil-and-paper examples it's more convenient to use β = 10. The value of t, called the precision, and the exponent can vary with the choice of hardware and software. Floating-point numbers are just adaptations of the familiar concept of sci- entific notation where β = 10, which will be the value used in our examples. For any fixed set of values for t, β, and , the corresponding set F of floating- point numbers is necessarily a finite set, so some real numbers can't be found in F. There is more than one way of approximating real numbers with floating- point numbers. For the remainder of this text, the following common rounding convention is adopted. Given a real number x, the floating-point approximation f l(x) is defined to be the nearest element in F to x, and in case of a tie we round away from 0. This means that for t-digit precision with β = 10, we need 6 The computer has been the single most important scientific and technological development of our century and has undoubtedly altered the course of science for all future time. The prospective young scientist or engineer who passes through a contemporary course in linear algebra and matrix theory and fails to learn at least the elementary aspects of what is involved in solving a practical linear system with a computer is missing a fundamental tool of applied mathematics.
22 Chapter 1 Linear Equations to look at digit dt+1 in x = .d1 d2 · · · dt dt+1 · · · × 10 (making sure d1 = 0) and then set .d1 d2 · · · dt × 10 if dt+1 < 5, f l(x) = −t ([.d1 d2 · · · dt ] + 10 ) × 10 if dt+1 ≥ 5. For example, in 2 -digit, base-10 floating-point arithmetic, f l (3/80) = f l(.0375) = f l(.375 × 10−1 ) = .38 × 10−1 = .038. By considering η = 21/2 and ξ = 11/2 with 2 -digit base-10 arithmetic, it's also easy to see that f l(η + ξ) = f l(η) + f l(ξ) and f l(ηξ) = f l(η)f l(ξ). Furthermore, several familiar rules of real arithmetic do not hold for floating- point arithmetic—associativity is one outstanding example. This, among other reasons, makes the analysis of floating-point computation difficult. It also means that you must be careful when working the examples and exercises in this text because although most calculators and computers can be instructed to display varying numbers of digits, most have a fixed internal precision with which all calculations are made before numbers are displayed, and this internal precision cannot be altered. The internal precision of your calculator is greater than the precision called for by the examples and exercises in this book, so each time you make a t-digit calculation with a calculator you should manually round the result to t significant digits and then manually reenter the rounded number in your calculator before proceeding to the next calculation. In other words, don't "chain" operations in your calculator or computer. To understand how to execute Gaussian elimination using floating-point arithmetic, let's compare the use of exact arithmetic with the use of 3-digit base-10 arithmetic to solve the following system: 47x + 28y = 19, 89x + 53y = 36. Using Gaussian elimination with exact arithmetic, we multiply the first equation by the multiplier m = 89/47 and subtract the result from the second equation to produce 47 28 19 . 0 −1/47 1/47 Back substitution yields the exact solution x=1 and y = −1. Using 3-digit arithmetic, the multiplier is 89 f l(m) = f l = .189 × 101 = 1.89. 47
1.5 Making Gaussian Elimination Work 23 Since f l f l(m)f l(47) = f l(1.89 × 47) = .888 × 102 = 88.8, f l f l(m)f l(28) = f l(1.89 × 28) = .529 × 102 = 52.9, f l f l(m)f l(19) = f l(1.89 × 19) = .359 × 102 = 35.9, the first step of 3-digit Gaussian elimination is as shown below: 47 28 19 f l(89 − 88.8) f l(53 − 52.9) f l(36 − 35.9) 47 28 19 = . .2 .1 .1 The goal is to triangularize the system—to produce a zero in the circled (2,1)-position—but this cannot be accomplished with 3-digit arithmetic. Unless the circled value .2 is replaced by 0, back substitution cannot be executed. Henceforth, we will agree simply to enter 0 in the position that we are trying to annihilate, regardless of the value of the floating-point number that might actually appear. The value of the position being annihilated is generally not even computed. For example, don't even bother computing f l 89 − f l f l(m)f l(47) = f l(89 − 88.8) = .2 in the above example. Hence the result of 3-digit Gaussian elimination for this example is 47 28 19 . 0 .1 .1 Apply 3-digit back substitution to obtain the 3-digit floating-point solution .1 y = fl = 1, .1 19 − 28 −9 x = fl = fl = −.191. 47 47 The vast discrepancy between the exact solution (1, −1) and the 3-digit solution (−.191, 1) illustrates some of the problems we can expect to encounter while trying to solve linear systems with floating-point arithmetic. Sometimes using a higher precision may help, but this is not always possible because on all machines there are natural limits that make extended precision arithmetic impractical past a certain point. Even if it is possible to increase the precision, it
24 Chapter 1 Linear Equations may not buy you very much because there are many cases for which an increase in precision does not produce a comparable decrease in the accumulated roundoff error. Given any particular precision (say, t ), it is not difficult to provide exam- ples of linear systems for which the computed t-digit solution is just as bad as the one in our 3-digit example above. Although the effects of rounding can almost never be eliminated, there are some simple techniques that can help to minimize these machine induced errors. Partial Pivoting At each step, search the positions on and below the pivotal position for the coefficient of maximum magnitude. If necessary perform the appro- priate row interchange to bring this maximal coefficient into the pivotal position. Illustrated below is the third step in a typical case: ∗ ∗ ∗ ∗ ∗ ∗ 0 ∗ ∗ ∗ ∗ ∗ 0 0 S ∗ ∗ ∗. 0 0 S ∗ ∗ ∗ 0 0 S ∗ ∗ ∗ Search the positions in the third column marked " S " for the coefficient of maximal magnitude and, if necessary, interchange rows to bring this coefficient into the circled pivotal position. Simply stated, the strategy is to maximize the magnitude of the pivot at each step by using only row interchanges. On the surface, it is probably not apparent why partial pivoting should make a difference. The following example not only shows that partial pivoting can indeed make a great deal of difference, but it also indicates what makes this strategy effective.Example 1.5.1 It is easy to verify that the exact solution to the system −10−4 x + y = 1, x + y = 2, is given by 1 1.0002 x= and y= . 1.0001 1.0001 If 3-digit arithmetic without partial pivoting is used, then the result is
1.5 Making Gaussian Elimination Work 25 −10−4 1 1 −10−4 1 1 −→ 1 1 2 R2 + 104 R1 0 104 104 because f l(1 + 104 ) = f l(.10001 × 105 ) = .100 × 105 = 104 (1.5.1) and f l(2 + 104 ) = f l(.10002 × 105 ) = .100 × 105 = 104 . (1.5.2) Back substitution now produces x=0 and y = 1. Although the computed solution for y is close to the exact solution for y, the computed solution for x is not very close to the exact solution for x —the computed solution for x is certainly not accurate to three significant figures as you might hope. If 3-digit arithmetic with partial pivoting is used, then the result is −10−4 1 1 1 1 2 −→ 1 1 2 −10−4 1 1 R2 + 10−4 R1 1 1 2 −→ 0 1 1 because f l(1 + 10−4 ) = f l(.10001 × 101 ) = .100 × 101 = 1 (1.5.3) and f l(1 + 2 × 10−4 ) = f l(.10002 × 101 ) = .100 × 101 = 1. (1.5.4) This time, back substitution produces the computed solution x=1 and y = 1, which is as close to the exact solution as one can reasonably expect—the com- puted solution agrees with the exact solution to three significant digits. Why did partial pivoting make a difference? The answer lies in comparing (1.5.1) and (1.5.2) with (1.5.3) and (1.5.4). Without partial pivoting the multiplier is 104 , and this is so large that it completely swamps the arithmetic involving the relatively smaller numbers 1 and 2 and prevents them from being taken into account. That is, the smaller numbers 1 and 2 are "blown away" as though they were never present so that our 3-digit computer produces the exact solution to another system, namely, −10−4 1 1 , 1 0 0
26 Chapter 1 Linear Equations which is quite different from the original system. With partial pivoting the mul- tiplier is 10−4 , and this is small enough so that it does not swamp the numbers 1 and 2. In this case, the 3-digit computer produces the exact solution to the 0 1 1 7 system 1 1 2 , which is close to the original system. In summary, the villain in Example 1.5.1 is the large multiplier that pre- vents some smaller numbers from being fully accounted for, thereby resulting in the exact solution of another system that is very different from the original system. By maximizing the magnitude of the pivot at each step, we minimize the magnitude of the associated multiplier thus helping to control the growth of numbers that emerge during the elimination process. This in turn helps cir- cumvent some of the effects of roundoff error. The problem of growth in the elimination procedure is more deeply analyzed on p. 348. When partial pivoting is used, no multiplier ever exceeds 1 in magnitude. To see that this is the case, consider the following two typical steps in an elimination procedure: ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 ∗ ∗ ∗ ∗ ∗ 0 ∗ ∗ ∗ ∗ ∗ 0 0 p ∗ ∗ ∗ −→ 0 0 p ∗ ∗ ∗. 0 0 q ∗ ∗ ∗ R4 − (q/p)R3 0 0 0 ∗ ∗ ∗ 0 0 r ∗ ∗ ∗ R5 − (r/p)R3 0 0 0 ∗ ∗ ∗ The pivot is p, while q/p and r/p are the multipliers. If partial pivoting has been employed, then |p| ≥ |q| and |p| ≥ |r| so that q r ≤1 and ≤ 1. p p By guaranteeing that no multiplier exceeds 1 in magnitude, the possibility of producing relatively large numbers that can swamp the significance of smaller numbers is much reduced, but not completely eliminated. To see that there is still more to be done, consider the following example.Example 1.5.2 The exact solution to the system −10x + 105 y = 105 , x+ y = 2, 7 Answering the question, "What system have I really solved (i.e., obtained the exact solution of), and how close is this system to the original system," is called backward error analysis, as opposed to forward analysis in which one tries to answer the question, "How close will a computed solution be to the exact solution?" Backward analysis has proven to be an effective way to analyze the numerical stability of algorithms.
1.5 Making Gaussian Elimination Work 27 is given by 1 1.0002 x= and y = . 1.0001 1.0001 Suppose that 3-digit arithmetic with partial pivoting is used. Since | − 10| > 1, no interchange is called for and we obtain −10 105 105 −10 105 105 −1 −→ 1 1 2 R2 + 10 R1 0 104 104 because f l(1 + 104 ) = f l(.10001 × 105 ) = .100 × 105 = 104 and f l(2 + 104 ) = f l(.10002 × 105 ) = .100 × 105 = 104 . Back substitution yields x=0 and y = 1, which must be considered to be very bad—the computed 3-digit solution for y is not too bad, but the computed 3-digit solution for x is terrible! What is the source of difficulty in Example 1.5.2? This time, the multi- plier cannot be blamed. The trouble stems from the fact that the first equation contains coefficients that are much larger than the coefficients in the second equation. That is, there is a problem of scale due to the fact that the coefficients are of different orders of magnitude. Therefore, we should somehow rescale the system before attempting to solve it. If the first equation in the above example is rescaled to insure that the coefficient of maximum magnitude is a 1, which is accomplished by multiplying the first equation by 10−5 , then the system given in Example 1.5.1 is obtained, and we know from that example that partial pivoting produces a very good approximation to the exact solution. This points to the fact that the success of partial pivoting can hinge on maintaining the proper scale among the coefficients. Therefore, the second re- finement needed to make Gaussian elimination practical is a reasonable scaling strategy. Unfortunately, there is no known scaling procedure that will produce optimum results for every possible system, so we must settle for a strategy that will work most of the time. The strategy is to combine row scaling—multiplying selected rows by nonzero multipliers—with column scaling—multiplying se- lected columns of the coefficient matrix A by nonzero multipliers. Row scaling doesn't alter the exact solution, but column scaling does—see Exercise 1.2.13(b). Column scaling is equivalent to changing the units of the k th unknown. For example, if the units of the k th unknown xk in [A|b] are millimeters, and if the k th column of A is multiplied by . 001, then the k th ˆ unknown in the scaled system [A | b] is xi = 1000xi , and thus the units of the ˆ scaled unknown xk become meters. ˆ
28 Chapter 1 Linear Equations Experience has shown that the following strategy for combining row scaling with column scaling usually works reasonably well. Practical Scaling Strategy 1. Choose units that are natural to the problem and do not dis- tort the relationships between the sizes of things. These natural units are usually self-evident, and further column scaling past this point is not ordinarily attempted. 2. Row scale the system [A|b] so that the coefficient of maximum magnitude in each row of A is equal to 1. That is, divide each equation by the coefficient of maximum magnitude. Partial pivoting together with the scaling strategy described above makes Gaussian elimination with back substitution an extremely effec- tive tool. Over the course of time, this technique has proven to be reliable for solving a majority of linear systems encountered in practical work. Although it is not extensively used, there is an extension of partial pivoting known as complete pivoting which, in some special cases, can be more effective than partial pivoting in helping to control the effects of roundoff error. Complete Pivoting If [A|b] is the augmented matrix at the k th step of Gaussian elimina- tion, then search the pivotal position together with every position in A that is below or to the right of the pivotal position for the coefficient of maximum magnitude. If necessary, perform the appropriate row and column interchanges to bring the coefficient of maximum magnitude into the pivotal position. Shown below is the third step in a typical situation: ∗ ∗ ∗ ∗ ∗ ∗ 0 ∗ ∗ ∗ ∗ ∗ 0 0 S S S ∗ 0 0 S S S ∗ 0 0 S S S ∗ Search the positions marked " S " for the coefficient of maximal magni- tude. If necessary, interchange rows and columns to bring this maximal coefficient into the circled pivotal position. Recall from Exercise 1.2.13 that the effect of a column interchange in A is equivalent to permuting (or renaming) the associated unknowns.
1.5 Making Gaussian Elimination Work 29 You should be able to see that complete pivoting should be at least as effec- tive as partial pivoting. Moreover, it is possible to construct specialized exam- ples where complete pivoting is superior to partial pivoting—a famous example is presented in Exercise 1.5.7. However, one rarely encounters systems of this nature in practice. A deeper comparison between no pivoting, partial pivoting, and complete pivoting is given on p. 348.Example 1.5.3 Problem: Use 3-digit arithmetic together with complete pivoting to solve the following system: x− y = −2, −9x + 10y = 12. Solution: Since 10 is the coefficient of maximal magnitude that lies in the search pattern, interchange the first and second rows and then interchange the first and second columns: 1 −1 −2 −9 10 12 −→ −9 10 12 1 −1 −2 10 −9 12 10 −9 12 −→ −→ . −1 1 −2 0 .1 −.8 The effect of the column interchange is to rename the unknowns to x and y , ˆ ˆ where x = y and y = x. Back substitution yields y = −8 and x = −6 so that ˆ ˆ ˆ ˆ x = y = −8 ˆ and y = x = −6. ˆ In this case, the 3-digit solution and the exact solution agree. If only partial pivoting is used, the 3-digit solution will not be as accurate. However, if scaled partial pivoting is used, the result is the same as when complete pivoting is used. If the cost of using complete pivoting was nearly the same as the cost of using partial pivoting, we would always use complete pivoting. However, it is not diffi- cult to show that complete pivoting approximately doubles the cost over straight Gaussian elimination, whereas partial pivoting adds only a negligible amount. Couple this with the fact that it is extremely rare to encounter a practical system where scaled partial pivoting is not adequate while complete pivoting is, and it is easy to understand why complete pivoting is seldom used in practice. Gaus- sian elimination with scaled partial pivoting is the preferred method for dense systems (i.e., not a lot of zeros) of moderate size.
30 Chapter 1 Linear EquationsExercises for section 1.5 1.5.1. Consider the following system: 10−3 x − y = 1, x + y = 0. (a) Use 3-digit arithmetic with no pivoting to solve this system. (b) Find a system that is exactly satisfied by your solution from part (a), and note how close this system is to the original system. (c) Now use partial pivoting and 3-digit arithmetic to solve the original system. (d) Find a system that is exactly satisfied by your solution from part (c), and note how close this system is to the original system. (e) Use exact arithmetic to obtain the solution to the original sys- tem, and compare the exact solution with the results of parts (a) and (c). (f) Round the exact solution to three significant digits, and compare the result with those of parts (a) and (c). 1.5.2. Consider the following system: x+ y = 3, −10x + 10 y = 105 . 5 (a) Use 4-digit arithmetic with partial pivoting and no scaling to compute a solution. (b) Use 4-digit arithmetic with complete pivoting and no scaling to compute a solution of the original system. (c) This time, row scale the original system first, and then apply partial pivoting with 4-digit arithmetic to compute a solution. (d) Now determine the exact solution, and compare it with the re- sults of parts (a), (b), and (c). 1.5.3. With no scaling, compute the 3-digit solution of −3x + y = −2, 10x − 3y = 7, without partial pivoting and with partial pivoting. Compare your results with the exact solution.
1.5 Making Gaussian Elimination Work 31 1.5.4. Consider the following system in which the coefficient matrix is the Hilbert matrix: 1 1 1 x+ y+ z= , 2 3 3 1 1 1 1 x+ y+ z= , 2 3 4 3 1 1 1 1 x+ y+ z= . 3 4 5 5 (a) First convert the coefficients to 3-digit floating-point numbers, and then use 3-digit arithmetic with partial pivoting but with no scaling to compute the solution. (b) Again use 3-digit arithmetic, but row scale the coefficients (after converting them to floating-point numbers), and then use partial pivoting to compute the solution. (c) Proceed as in part (b), but this time row scale the coefficients before each elimination step. (d) Now use exact arithmetic on the original system to determine the exact solution, and compare the result with those of parts (a), (b), and (c). 1.5.5. To see that changing units can affect a floating-point solution, consider a mining operation that extracts silica, iron, and gold from the earth. Capital (measured in dollars), operating time (in hours), and labor (in man-hours) are needed to operate the mine. To extract a pound of silica requires $.0055, .0011 hours of operating time, and .0093 man-hours of labor. For each pound of iron extracted, $.095, .01 operating hours, and .025 man-hours are required. For each pound of gold extracted, $960, 112 operating hours, and 560 man-hours are required. (a) Suppose that during 600 hours of operation, exactly $5000 and 3000 man-hours are used. Let x, y, and z denote the number of pounds of silica, iron, and gold, respectively, that are recov- ered during this period. Set up the linear system whose solution will yield the values for x, y, and z. (b) With no scaling, use 3-digit arithmetic and partial pivoting to compute a solution (˜, y , z ) of the system of part (a). Then x ˜ ˜ approximate the exact solution (x, y, z) by using your machine's (or calculator's) full precision with partial pivoting to solve the system in part (a), and compare this with your 3-digit solution by computing the relative error defined by (x − x)2 + (y − y )2 + (z − z )2 ˜ ˜ ˜ er = . x2 + y 2 + z 2
32 Chapter 1 Linear Equations (c) Using 3-digit arithmetic, column scale the coefficients by chang- ing units: convert pounds of silica to tons of silica, pounds of iron to half-tons of iron, and pounds of gold to troy ounces of gold (1 lb. = 12 troy oz.). (d) Use 3-digit arithmetic with partial pivoting to solve the column scaled system of part (c). Then approximate the exact solution by using your machine's (or calculator's) full precision with par- tial pivoting to solve the system in part (c), and compare this with your 3-digit solution by computing the relative error er as defined in part (b). 1.5.6. Consider the system given in Example 1.5.3. (a) Use 3-digit arithmetic with partial pivoting but with no scaling to solve the system. (b) Now use partial pivoting with scaling. Does complete pivoting provide an advantage over scaled partial pivoting in this case? 1.5.7. Consider the following well-scaled matrix: 1 0 0 ··· 0 01 −1 1 0 ··· 0 01 .. −1 −1 1 . 0 0 1 . . .. .. . . Wn = . . . . . . .. . . .. . . −1 −1 −1 . . . 1 0 1 −1 −1 −1 · · · −1 1 1 −1 −1 −1 · · · −1 −1 1 (a) Reduce Wn to an upper-triangular form using Gaussian elimi- nation with partial pivoting, and determine the element of max- imal magnitude that emerges during the elimination procedure. (b) Now use complete pivoting and repeat part (a). (c) Formulate a statement comparing the results of partial pivoting with those of complete pivoting for Wn , and describe the effect this would have in determining the t -digit solution for a system whose augmented matrix is [Wn | b]. 1.5.8. Suppose that A is an n × n matrix of real numbers that has been scaled so that each entry satisfies |aij | ≤ 1, and consider reducing A to tri- angular form using Gaussian elimination with partial pivoting. Demon- strate that after k steps of the process, no entry can have a magnitude that exceeds 2k . Note: The previous exercise shows that there are cases where it is possible for some elements to actually attain the maximum magnitude of 2k after k steps.
1.6 Ill-Conditioned Systems 331.6 ILL-CONDITIONED SYSTEMS Gaussian elimination with partial pivoting on a properly scaled system is perhaps the most fundamental algorithm in the practical use of linear algebra. However, it is not a universal algorithm nor can it be used blindly. The purpose of this section is to make the point that when solving a linear system some discretion must always be exercised because there are some systems that are so inordinately sensitive to small perturbations that no numerical technique can be used with confidence.Example 1.6.1 Consider the system .835x + .667y = .168, .333x + .266y = .067, for which the exact solution is x=1 and y = −1. If b2 = .067 is only slightly perturbed to become ˆ2 = .066, then the exact b solution changes dramatically to become x = −666 ˆ and y = 834. ˆ This is an example of a system whose solution is extremely sensitive to a small perturbation. This sensitivity is intrinsic to the system itself and is not a result of any numerical procedure. Therefore, you cannot expect some "numerical trick" to remove the sensitivity. If the exact solution is sensitive to small perturbations, then any computed solution cannot be less so, regardless of the algorithm used. Ill-Conditioned Linear Systems A system of linear equations is said to be ill-conditioned when some small perturbation in the system can produce relatively large changes in the exact solution. Otherwise, the system is said to be well- conditioned. It is easy to visualize what causes a 2 × 2 system to be ill-conditioned. Geometrically, two equations in two unknowns represent two straight lines, and the point of intersection is the solution for the system. An ill-conditioned system represents two straight lines that are almost parallel.
34 Chapter 1 Linear Equations If two straight lines are almost parallel and if one of the lines is tilted only slightly, then the point of intersection (i.e., the solution of the associated 2 × 2 linear system) is drastically altered. L L Perturbed Solution Original Solution Figure 1.6.1 This is illustrated in Figure 1.6.1 in which line L is slightly perturbed to become line L . Notice how this small perturbation results in a large change in the point of intersection. This was exactly the situation for the system given in Example 1.6.1. In general, ill-conditioned systems are those that represent almost parallel lines, almost parallel planes, and generalizations of these notions. Because roundoff errors can be viewed as perturbations to the original coeffi- cients of the system, employing even a generally good numerical technique—short of exact arithmetic—on an ill-conditioned system carries the risk of producing nonsensical results. In dealing with an ill-conditioned system, the engineer or scientist is often confronted with a much more basic (and sometimes more disturbing) problem than that of simply trying to solve the system. Even if a minor miracle could be performed so that the exact solution could be extracted, the scientist or engineer might still have a nonsensical solution that could lead to totally incorrect conclusions. The problem stems from the fact that the coefficients are often empirically obtained and are therefore known only within certain tolerances. For an ill-conditioned system, a small uncertainty in any of the coefficients can mean an extremely large uncertainty may exist in the solution. This large uncertainty can render even the exact solution totally useless.Example 1.6.2 Suppose that for the system .835x + .667y = b1 .333x + .266y = b2 the numbers b1 and b2 are the results of an experiment and must be read from the dial of a test instrument. Suppose that the dial can be read to within a
1.6 Ill-Conditioned Systems 35 tolerance of ±.001, and assume that values for b1 and b2 are read as . 168 and . 067, respectively. This produces the ill-conditioned system of Example 1.6.1, and it was seen in that example that the exact solution of the system is (x, y) = (1, −1). (1.6.1) However, due to the small uncertainty in reading the dial, we have that .167 ≤ b1 ≤ .169 and .066 ≤ b2 ≤ .068. (1.6.2) For example, this means that the solution associated with the reading (b1 , b2 ) = (.168, .067) is just as valid as the solution associated with the reading (b1 , b2 ) = (.167, .068), or the reading (b1 , b2 ) = (.169, .066), or any other reading falling in the range (1.6.2). For the reading (b1 , b2 ) = (.167, .068), the exact solution is (x, y) = (934, −1169), (1.6.3) while for the other reading (b1 , b2 ) = (.169, .066), the exact solution is (x, y) = (−932, 1167). (1.6.4) Would you be willing to be the first to fly in the plane or drive across the bridge whose design incorporated a solution to this problem? I wouldn't! There is just too much uncertainty. Since no one of the solutions (1.6.1), (1.6.3), or (1.6.4) can be preferred over any of the others, it is conceivable that totally different designs might be implemented depending on how the technician reads the last significant digit on the dial. Due to the ill-conditioned nature of an associated linear system, the successful design of the plane or bridge may depend on blind luck rather than on scientific principles. Rather than trying to extract accurate solutions from ill-conditioned sys- tems, engineers and scientists are usually better off investing their time and re- sources in trying to redesign the associated experiments or their data collection methods so as to avoid producing ill-conditioned systems. There is one other discomforting aspect of ill-conditioned systems. It con- cerns what students refer to as "checking the answer" by substituting a computed solution back into the left-hand side of the original system of equations to see how close it comes to satisfying the system—that is, producing the right-hand side. More formally, if xc = ( ξ1 ξ2 · · · ξn ) is a computed solution for a system a11 x1 + a12 x2 + · · · + a1n xn = b1 , a21 x1 + a22 x2 + · · · + a2n xn = b2 , . . . an1 x1 + an2 x2 + · · · + ann xn = bn ,
36 Chapter 1 Linear Equations then the numbers ri = ai1 ξ1 + ai2 ξ2 + · · · + ain ξn − bi for i = 1, 2, . . . , n are called the residuals. Suppose that you compute a solution xc and substitute it back to find that all the residuals are relatively small. Does this guarantee that xc is close to the exact solution? Surprisingly, the answer is a resounding "no!" whenever the system is ill-conditioned.Example 1.6.3 For the ill-conditioned system given in Example 1.6.1, suppose that somehow you compute a solution to be ξ1 = −666 and ξ2 = 834. If you attempt to "check the error" in this computed solution by substituting it back into the original system, then you find—using exact arithmetic—that the residuals are r1 = .835ξ1 + .667ξ2 − .168 = 0, r2 = .333ξ1 + .266ξ2 − .067 = −.001. That is, the computed solution (−666, 834) exactly satisfies the first equation and comes very close to satisfying the second. On the surface, this might seem to suggest that the computed solution should be very close to the exact solution. In fact a naive person could probably be seduced into believing that the computed solution is within ±.001 of the exact solution. Obviously, this is nowhere close to being true since the exact solution is x=1 and y = −1. This is always a shock to a student seeing this illustrated for the first time because it is counter to a novice's intuition. Unfortunately, many students leave school believing that they can always "check" the accuracy of their computations by simply substituting them back into the original equations—it is good to know that you're not among them. This raises the question, "How can I check a computed solution for accu- racy?" Fortunately, if the system is well-conditioned, then the residuals do indeed provide a more effective measure of accuracy (a rigorous proof along with more insight appears in Example 5.12.2 on p. 416). But this means that you must be able to answer some additional questions. For example, how can one tell before- hand if a given system is ill-conditioned? How can one measure the extent of ill-conditioning in a linear system? One technique to determine the extent of ill-conditioning might be to exper- iment by slightly perturbing selected coefficients and observing how the solution
1.6 Ill-Conditioned Systems 37 changes. If a radical change in the solution is observed for a small perturbation to some set of coefficients, then you have uncovered an ill-conditioned situation. If a given perturbation does not produce a large change in the solution, then nothing can be concluded—perhaps you perturbed the wrong set of coefficients. By performing several such experiments using different sets of coefficients, a feel (but not a guarantee) for the extent of ill-conditioning can be obtained. This is expensive and not very satisfying. But before more can be said, more sophisti- cated tools need to be developed—the topics of sensitivity and conditioning are revisited on p. 127 and in Example 5.12.1 on p. 414.Exercises for section 1.6 1.6.1. Consider the ill-conditioned system of Example 1.6.1: .835x + .667y = .168, .333x + .266y = .067. (a) Describe the outcome when you attempt to solve the system using 5-digit arithmetic with no scaling. (b) Again using 5-digit arithmetic, first row scale the system before attempting to solve it. Describe to what extent this helps. (c) Now use 6-digit arithmetic with no scaling. Compare the results with the exact solution. (d) Using 6-digit arithmetic, compute the residuals for your solution of part (c), and interpret the results. (e) For the same solution obtained in part (c), again compute the residuals, but use 7-digit arithmetic this time, and interpret the results. (f) Formulate a concluding statement that summarizes the points made in parts (a)–(e). 1.6.2. Perturb the ill-conditioned system given in Exercise 1.6.1 above so as to form the following system: .835x + .667y = .1669995, .333x + .266y = .066601. (a) Determine the exact solution, and compare it with the exact solution of the system in Exercise 1.6.1. (b) On the basis of the results of part (a), formulate a statement concerning the necessity for the solution of an ill-conditioned system to undergo a radical change for every perturbation of the original system.
38 Chapter 1 Linear Equations 1.6.3. Consider the two straight lines determined by the graphs of the following two equations: .835x + .667y = .168, .333x + .266y = .067. (a) Use 5-digit arithmetic to compute the slopes of each of the lines, and then use 6-digit arithmetic to do the same. In each case, sketch the graphs on a coordinate system. (b) Show by diagram why a small perturbation in either of these lines can result in a large change in the solution. (c) Describe in geometrical terms the situation that must exist in order for a system to be optimally well-conditioned. 1.6.4. Using geometric considerations, rank the following three systems accord- ing to their condition. 1.001x − y = .235, 1.001x − y = .235, (a) (b) x + .0001y = .765. x + .9999y = .765. 1.001x + y = .235, (c) x + .9999y = .765. 1.6.5. Determine the exact solution of the following system: 8x + 5y + 2z = 15, 21x + 19y + 16z = 56, 39x + 48y + 53z = 140. Now change 15 to 14 in the first equation and again solve the system with exact arithmetic. Is the system ill-conditioned? 1.6.6. Show that the system v − w − x − y − z = 0, w − x − y − z = 0, x − y − z = 0, y − z = 0, z = 1,
40 Chapter 1 Linear Equations To isolate mathematics from the practical demands of the sciences is to invite the sterility of a cow shut away from the bulls. — Pafnuty Lvovich Chebyshev (1821–1894)
CHAPTER 2 Rectangular Systems and Echelon Forms2.1 ROW ECHELON FORM AND RANK We are now ready to analyze more general linear systems consisting of m linear equations involving m may be different from n. If we do not know for sure that m and n are the same, then the system is said to be rectangular. The case m = n is still allowed in the discussion—statements concerning rectangular systems also are valid for the special case of square systems. The first goal is to extend the Gaussian elimination technique from square systems to completely general rectangular systems. Recall that for a square sys- tem with a unique solution, the pivotal positions are always located along the main diagonal—the diagonal line from the upper-left-hand corner to the lower- right-hand corner—in the coefficient matrix A so that Gaussian elimination results in a reduction of A to a triangular matrix, such as that illustrated below for the case n = 4: * ∗ ∗ ∗ 0 * ∗ ∗ T= . 0 0 * ∗ 0 0 0 *
42 Chapter 2 Rectangular Systems and Echelon Forms Remember that a pivot must always be a nonzero number. For square sys- tems possessing a unique solution, it is a fact (proven later) that one can al- ways bring a nonzero number into each pivotal position along the main diag- 8 onal. However, in the case of a general rectangular system, it is not always possible to have the pivotal positions lying on a straight diagonal line in the coefficient matrix. This means that the final result of Gaussian elimination will not be triangular in form. For example, consider the following system: x1 + 2x2 + x3 + 3x4 + 3x5 = 5, 2x1 + 4x2 + 4x4 + 4x5 = 6, x1 + 2x2 + 3x3 + 5x4 + 5x5 = 9, 2x1 + 4x2 + 4x4 + 7x5 = 9. Focus your attention on the coefficient matrix 1 2 1 3 3 2 4 0 4 4 A= , (2.1.1) 1 2 3 5 5 2 4 0 4 7 and ignore the right-hand side for a moment. Applying Gaussian elimination to A yields the following result: 1 2 1 3 3 1 2 1 3 3 2 4 0 4 4 0 0 −2 −2 −2 −→ . 1 2 3 5 5 0 0 2 2 2 2 4 0 4 7 0 0 −2 −2 1 In the basic elimination process, the strategy is to move down and to the right to the next pivotal position. If a zero occurs in this position, an interchange with a row below the pivotal row is executed so as to bring a nonzero number into the pivotal position. However, in this example, it is clearly impossible to bring a nonzero number into the (2, 2) -position by interchanging the second row with a lower row. In order to handle this situation, the elimination process is modified as follows. 8 This discussion is for exact arithmetic. If floating-point arithmetic is used, this may no longer be true. Part (a) of Exercise 1.6.1 is one such example.
44 Chapter 2 Rectangular Systems and Echelon Forms Notice that the final result of applying Gaussian elimination in the above example is not a purely triangular form but rather a jagged or "stair-step" type of triangular form. Hereafter, a matrix that exhibits this stair-step structure will be said to be in row echelon form. Row Echelon Form An m × n matrix E with rows Ei∗ and columns E∗j is said to be in row echelon form provided the following two conditions hold. • If Ei∗ consists entirely of zeros, then all rows below Ei∗ are also entirely zero; i.e., all zero rows are at the bottom. • If the first nonzero entry in Ei∗ lies in the j th position, then all entries below the ith position in columns E∗1 , E∗2 , . . . , E∗j are zero. These two conditions say that the nonzero entries in an echelon form must lie on or above a stair-step line that emanates from the upper- left-hand corner and slopes down and to the right. The pivots are the first nonzero entries in each row. A typical structure for a matrix in row echelon form is illustrated below with the pivots circled. * ∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 0 * ∗ ∗ ∗ ∗ ∗ 0 0 0 * ∗ ∗ ∗ ∗ 0 0 0 0 0 0 * ∗ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Because of the flexibility in choosing row operations to reduce a matrix A to a row echelon form E, the entries in E are not uniquely determined by A. Nevertheless, it can be proven that the "form" of E is unique in the sense that the positions of the pivots in E (and A) are uniquely determined by the entries 9 in A . Because the pivotal positions are unique, it follows that the number of pivots, which is the same as the number of nonzero rows in E, is also uniquely 10 determined by the entries in A . This number is called the rank of A, and it 9 The fact that the pivotal positions are unique should be intuitively evident. If it isn't, take the matrix given in (2.1.1) and try to force some different pivotal positions by a different sequence of row operations. 10 The word "rank" was introduced in 1879 by the German mathematician Ferdinand Georg Frobenius (p. 662), who thought of it as the size of the largest nonzero minor determinant in A. But the concept had been used as early as 1851 by the English mathematician James J. Sylvester (1814–1897).
2.1 Row Echelon Form and Rank 45 is extremely important in the development of our subject. Rank of a Matrix Suppose Am×n is reduced by row operations to an echelon form E. The rank of A is defined to be the number rank (A) = number of pivots = number of nonzero rows in E = number of basic columns in A, where the basic columns of A are defined to be those columns in A that contain the pivotal positions.Example 2.1.2 Problem: Determine the rank, and identify the basic columns in 1 2 1 1 A = 2 4 2 2. 3 6 3 4 Solution: Reduce A to row echelon form as shown below: 1 2 1 1 1 2 1 1 1 2 1 1 A= 2 4 2 2 −→ 0 0 0 0 −→ 0 0 0 1 = E. 3 6 3 4 0 0 0 1 0 0 0 0 Consequently, rank (A) = 2. The pivotal positions lie in the first and fourth columns so that the basic columns of A are A∗1 and A∗4 . That is, 1 1 Basic Columns = 2 , 2 . 3 4 Pay particular attention to the fact that the basic columns are extracted from A and not from the row echelon form E .
48 Chapter 2 Rectangular Systems and Echelon Forms Compare the results of this example with the results of Example 2.1.1, and notice that the "form" of the final matrix is the same in both examples, which indeed must be the case because of the uniqueness of "form" mentioned in the previous section. The only difference is in the numerical value of some of the entries. By the nature of Gauss–Jordan elimination, each pivot is 1 and all entries above and below each pivot are 0. Consequently, the row echelon form produced by the Gauss–Jordan method contains a reduced number of nonzero entries, so 11 it seems only natural to refer to this as a reduced row echelon form. Reduced Row Echelon Form A matrix Em×n is said to be in reduced row echelon form provided that the following three conditions hold. • E is in row echelon form. • The first nonzero entry in each row (i.e., each pivot) is 1. • All entries above each pivot are 0. A typical structure for a matrix in reduced row echelon form is illustrated below, where entries marked * can be either zero or nonzero numbers: 1 ∗ 0 0 ∗ ∗ 0 ∗ 0 0 1 0 ∗ ∗ 0 ∗ 0 0 0 1 ∗ ∗ 0 ∗ . 0 0 0 0 0 0 1 ∗ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 As previously stated, if matrix A is transformed to a row echelon form by row operations, then the "form" is uniquely determined by A, but the in- dividual entries in the form are not unique. However, if A is transformed by 12 row operations to a reduced row echelon form EA , then it can be shown that both the "form" as well as the individual entries in EA are uniquely determined by A. In other words, the reduced row echelon form EA produced from A is independent of whatever elimination scheme is used. Producing an unreduced form is computationally more efficient, but the uniqueness of EA makes it more useful for theoretical purposes. 11 In some of the older books this is called the Hermite normal form in honor of the French mathematician Charles Hermite (1822–1901), who, around 1851, investigated reducing matrices by row operations. 12 A formal uniqueness proof must wait until Example 3.9.2, but you can make this intuitively clear right now with some experiments. Try to produce two different reduced row echelon forms from the same matrix.
50 Chapter 2 Rectangular Systems and Echelon Forms general matrix A are usually obscure, but the relationships among the columns in EA are absolutely transparent. For example, notice that the multipliers used in the relationships (2.2.1) and (2.2.2) appear explicitly in the two nonbasic columns in EA —they are just the nonzero entries in these nonbasic columns. This is important because it means that EA can be used as a "map" or "key" to discover or unlock the hidden relationships among the columns of A . Finally, observe from Example 2.2.2 that only the basic columns to the left of a given nonbasic column are needed in order to express the nonbasic column as a combination of basic columns—e.g., representing A∗2 requires only A∗1 and not A∗3 or A∗5 , while representing A∗4 requires only A∗1 and A∗3 . This too is typical. For the time being, we accept the following statements to be true. A rigorous proof is given later on p. 136. Column Relationships in A and EA • Each nonbasic column E∗k in EA is a combination (a sum of mul- tiples) of the basic columns in EA to the left of E∗k . That is, E∗k = µ1 E∗b1 + µ2 E∗b2 + · · · + µj E∗bj µ 1 0 0 1 0 1 0 µ2 . . . . . . . . , . . . . = µ1 + µ2 + · · · + µj = 0 0 1 µj . . . . . . . . . . . . 0 0 0 0 where the E∗bi's are the basic columns to the left of E∗k and where the multipliers µi are the first j entries in E∗k . • The relationships that exist among the columns of A are exactly the same as the relationships that exist among the columns of EA . In particular, if A∗k is a nonbasic column in A , then A∗k = µ1 A∗b1 + µ2 A∗b2 + · · · + µj A∗bj , (2.2.3) where the A∗bi's are the basic columns to the left of A∗k , and where the multipliers µi are as described above—the first j entries in E∗k .
52 Chapter 2 Rectangular Systems and Echelon Forms 2.2.2. Construct a matrix A whose reduced row echelon form is 1 2 0 −3 0 0 0 0 0 1 −4 0 1 0 0 0 0 0 1 0 0 EA = . 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Is A unique? 2.2.3. Suppose that A is an m × n matrix. Give a short explanation of why rank (A) < n whenever one column in A is a combination of other columns in A . 2.2.4. Consider the following matrix: .1 .2 .3 A = .4 .5 .6 . .7 .8 .901 (a) Use exact arithmetic to determine EA . (b) Now use 3-digit floating-point arithmetic (without partial piv- oting or scaling) to determine EA and formulate a statement concerning "near relationships" between the columns of A . 2.2.5. Consider the matrix 1 0 −1 E = 0 1 2. 0 0 0 You already know that E∗3 can be expressed in terms of E∗1 and E∗2 . However, this is not the only way to represent the column dependencies in E . Show how to write E∗1 in terms of E∗2 and E∗3 and then express E∗2 as a combination of E∗1 and E∗3 . Note: This exercise illustrates that the set of pivotal columns is not the only set that can play the role of "basic columns." Taking the basic columns to be the ones containing the pivots is a matter of convenience because everything becomes automatic that way.
2.3 Consistency of Linear Systems 532.3 CONSISTENCY OF LINEAR SYSTEMS A system of m linear equations in n unknowns is said to be a consistent sys- tem if it possesses at least one solution. If there are no solutions, then the system is called inconsistent. The purpose of this section is to determine conditions under which a given system will be consistent. Stating conditions for consistency of systems involving only two or three unknowns is easy. A linear equation in two unknowns represents a line in 2-space, and a linear equation in three unknowns is a plane in 3-space. Consequently, a linear system of m equations in two unknowns is consistent if and only if the m lines defined by the m equations have at least one common point of intersection. Similarly, a system of m equations in three unknowns is consistent if and only if the associated m planes have at least one common point of intersection. However, when m is large, these geometric conditions may not be easy to verify visually, and when n > 3, the generalizations of intersecting lines or planes are impossible to visualize with the eye. Rather than depending on geometry to establish consistency, we use Gaus- sian elimination. If the associated augmented matrix [A|b] is reduced by row operations to a matrix [E|c] that is in row echelon form, then consistency—or lack of it—becomes evident. Suppose that somewhere in the process of reduc- ing [A|b] to [E|c] a situation arises in which the only nonzero entry in a row appears on the right-hand side, as illustrated below: ∗ ∗ ∗ ∗ ∗ ∗ ∗ 0 0 0 ∗ ∗ ∗ ∗ 0 0 0 0 ∗ ∗ ∗ Row i −→ 0 0 0 0 0 0 α ←− α = 0. • • • • • • • • • • • • • • If this occurs in the ith row, then the ith equation of the associated system is 0x1 + 0x2 + · · · + 0xn = α. For α = 0, this equation has no solution, and hence the original system must also be inconsistent (because row operations don't alter the solution set). The converse also holds. That is, if a system is inconsistent, then somewhere in the elimination process a row of the form (0 0 ··· 0 | α), α=0 (2.3.1) must appear. Otherwise, the back substitution process can be completed and a solution is produced. There is no inconsistency indicated when a row of the form (0 0 · · · 0 | 0) is encountered. This simply says that 0 = 0, and although
54 Chapter 2 Rectangular Systems and Echelon Forms this is no help in determining the value of any unknown, it is nevertheless a true statement, so it doesn't indicate inconsistency in the system. There are some other ways to characterize the consistency (or inconsistency) of a system. One of these is to observe that if the last column b in the augmented matrix [A|b] is a nonbasic column, then no pivot can exist in the last column, and hence the system is consistent because the situation (2.3.1) cannot occur. Conversely, if the system is consistent, then the situation (2.3.1) never occurs during Gaussian elimination and consequently the last column cannot be basic. In other words, [A|b] is consistent if and only if b is a nonbasic column. Saying that b is a nonbasic column in [A|b] is equivalent to saying that all basic columns in [A|b] lie in the coefficient matrix A . Since the number of basic columns in a matrix is the rank, consistency may also be characterized by stating that a system is consistent if and only if rank[A|b] = rank (A). Recall from the previous section the fact that each nonbasic column in [A|b] must be expressible in terms of the basic columns. Because a consistent system is characterized by the fact that the right-hand side b is a nonbasic column, it follows that a system is consistent if and only if the right-hand side b is a combination of columns from the coefficient matrix A. 13 Each of the equivalent ways of saying that a system is consistent is sum- marized below. Consistency Each of the following is equivalent to saying that [A|b] is consistent. • In row reducing [A|b], a row of the following form never appears: (0 0 ··· 0 | α), where α = 0. (2.3.2) • b is a nonbasic column in [A|b]. (2.3.3) • rank[A|b] = rank (A). (2.3.4) • b is a combination of the basic columns in A. (2.3.5)Example 2.3.1 Problem: Determine if the following system is consistent:. 13 Statements P and Q are said to be equivalent when (P implies Q) as well as its converse (Q implies P ) are true statements. This is also the meaning of the phrase "P if and only if Q."
56 Chapter 2 Rectangular Systems and Echelon Forms 2.3.4. Consider two consistent systems whose augmented matrices are of the form [A|b] and [A|c]. That is, they differ only on the right-hand side. Is the system associated with [A | b + c] also consistent? Explain why. 2.3.5. Is it possible for a parabola whose equation has the form y = α+βx+γx2 to pass through the four points (0, 1), (1, 3), (2, 15), and (3, 37)? Why? 2.3.6. Consider using floating-point arithmetic (without scaling) to solve the following system: .835x + .667y = .168, .333x + .266y = .067. (a) Is the system consistent when 5-digit arithmetic is used? (b) What happens when 6-digit arithmetic is used? 2.3.7 say brand X , brand Y , and brand Z . Determine whether or not it is possible to meet exactly the recommendation by applying some combination of the three brands of fertilizer. 2.3.8. Suppose that an augmented matrix [A|b] is reduced by means of Gaus- sian elimination to a row echelon form [E|c]. If a row of the form (0 0 ··· 0 | α), α=0 does not appear in [E|c], is it possible that rows of this form could have appeared at earlier stages in the reduction process? Why?
2.4 Homogeneous Systems 572.4 HOMOGENEOUS SYSTEMS A system of m linear equations in n unknowns a11 x1 + a12 x2 + · · · + a1n xn = 0, a21 x1 + a22 x2 + · · · + a2n xn = 0, . . . am1 x1 + am2 x2 + · · · + amn xn = 0, in which the right-hand side consists entirely of 0's is said to be a homogeneous system. If there is at least one nonzero number on the right-hand side, then the system is called nonhomogeneous. The purpose of this section is to examine some of the elementary aspects concerning homogeneous systems. Consistency is never an issue when dealing with homogeneous systems be- cause the zero solution x1 = x2 = · · · = xn = 0 is always one solution regardless of the values of the coefficients. Hereafter, the solution consisting of all zeros is referred to as the trivial solution. The only question is, "Are there solutions other than the trivial solution, and if so, how can we best describe them?" As before, Gaussian elimination provides the answer. While reducing the augmented matrix [A|0] of a homogeneous system to a row echelon form using Gaussian elimination, the zero column on the right- hand side can never be altered by any of the three elementary row operations. That is, any row echelon form derived from [A|0] by means of row operations must also have the form [E|0]. This means that the last column of 0's is just excess baggage that is not necessary to carry along at each step. Just reduce the coefficient matrix A to a row echelon form E, and remember that the right- hand side is entirely zero when you execute back substitution. The process is best understood by considering a typical example. In order to examine the solutions of the homogeneous system x1 + 2x2 + 2x3 + 3x4 = 0, 2x1 + 4x2 + x3 + 3x4 = 0, (2.4.1) 3x1 + 6x2 + x3 + 4x4 = 0, reduce the coefficient matrix to a row echelon form. 1 2 2 3 1 2 2 3 1 2 2 3 A = 2 4 1 3 −→ 0 0 −3 −3 −→ 0 0 −3 −3 = E. 3 6 1 4 0 0 −5 −5 0 0 0 0 Therefore, the original homogeneous system is equivalent to the following reduced homogeneous system: x1 + 2x2 + 2x3 + 3x4 = 0, (2.4.2) − 3x3 − 3x4 = 0.
58 Chapter 2 Rectangular Systems and Echelon Forms Since there are four unknowns but only two equations in this reduced system, it is impossible to extract a unique solution for each unknown. The best we can do is to pick two "basic" unknowns—which will be called the basic variables and solve for these in terms of the other two unknowns—whose values must remain arbitrary or "free," and consequently they will be referred to as the free variables. Although there are several possibilities for selecting a set of basic variables, the convention is to always solve for the unknowns corresponding to the pivotal positions—or, equivalently, the unknowns corresponding to the basic columns. In this example, the pivots (as well as the basic columns) lie in the first and third positions, so the strategy is to apply back substitution to solve the reduced system (2.4.2) for the basic variables x1 and x3 in terms of the free variables x2 and x4 . The second equation in (2.4.2) yields x3 = −x4 and substitution back into the first equation produces x1 = −2x2 − 2x3 − 3x4 , = −2x2 − 2(−x4 ) − 3x4 , = −2x2 − x4 . Therefore, all solutions of the original homogeneous system can be described by saying x1 = −2x2 − x4 , x2 is "free," (2.4.3) x3 = −x4 , x4 is "free." As the free variables x2 and x4 range over all possible values, the above ex- pressions describe all possible solutions. For example, when x2 and x4 assume the values x2 = 1 and x4 = −2, then the particular solution x1 = 0, x2 = 1, x3 = 2, x4 = −2 √ is produced. When x2 = π and x4 = 2, then another particular solution √ √ √ x1 = −2π − 2, x2 = π, x3 = − 2, x4 = 2 is generated. Rather than describing the solution set as illustrated in (2.4.3), future de- velopments will make it more convenient to express the solution set by writing x1 −2x2 − x4 −2 −1 x2 x2 1 0 = = x2 + x4 (2.4.4) x3 −x4 0 −1 x4 x4 0 1
2.4 Homogeneous Systems 59 with the understanding that x2 and x4 are free variables that can range over all possible numbers. This representation will be called the general solution of the homogeneous system. This expression for the general solution emphasizes that every solution is some combination of the two particular solutions −2 −1 1 0 h1 = and h2 = . 0 −1 0 1 The fact that h1 and h2 are each solutions is clear because h1 is produced when the free variables assume the values x2 = 1 and x4 = 0, whereas the solution h2 is generated when x2 = 0 and x4 = 1. Now consider a general homogeneous system [A|0] of m linear equations in n unknowns. If the coefficient matrix is such that rank (A) = r, then it should be apparent from the preceding discussion that there will be exactly r basic variables—corresponding to the positions of the basic columns in A —and exactly n − r free variables—corresponding to the positions of the nonbasic columns in A . Reducing A to a row echelon form using Gaussian elimination and then using back substitution to solve for the basic variables in terms of the free variables produces the general solution, which has the form x = xf1 h1 + xf2 h2 + · · · + xfn−r hn−r , (2.4.5) where xf1 , xf2 , . . . , xfn−r are the free variables and where h1 , h2 , . . . , hn−r are n × 1 columns that represent particular solutions of the system. As the free variables xfi range over all possible values, the general solution generates all possible solutions. The general solution does not depend on which row echelon form is used in the sense that using back substitution to solve for the basic variables in terms of the nonbasic variables generates a unique set of particular solutions {h1 , h2 , . . . , hn−r }, regardless of which row echelon form is used. Without going into great detail, one can argue that this is true because using back substitution in any row echelon form to solve for the basic variables must produce exactly the same result as that obtained by completely reducing A to EA and then solving the reduced homogeneous system for the basic variables. Uniqueness of EA guarantees the uniqueness of the hi 's. For example, if the coefficient matrix A associated with the system (2.4.1) is completely reduced by the Gauss–Jordan procedure to EA 1 2 2 3 1 2 0 1 A = 2 4 1 3 −→ 0 0 1 1 = EA , 3 6 1 4 0 0 0 0
60 Chapter 2 Rectangular Systems and Echelon Forms then we obtain the following reduced system: x1 + 2x2 + x4 = 0, x3 + x4 = 0. Solving for the basic variables x1 and x3 in terms of x2 and x4 produces exactly the same result as given in (2.4.3) and hence generates exactly the same general solution as shown in (2.4.4). Because it avoids the back substitution process, you may find it more con- venient to use the Gauss–Jordan procedure to reduce A completely to EA and then construct the general solution directly from the entries in EA . This approach usually will be adopted in the examples and exercises. As was previously observed, all homogeneous systems are consistent because the trivial solution consisting of all zeros is always one solution. The natural question is, "When is the trivial solution the only solution?" In other words, we wish to know when a homogeneous system possesses a unique solution. The form of the general solution (2.4.5) makes the answer transparent. As long as there is at least one free variable, then it is clear from (2.4.5) that there will be an infinite number of solutions. Consequently, the trivial solution is the only solution if and only if there are no free variables. Because the number of free variables is given by n − r, where r = rank (A), the previous statement can be reformulated to say that a homogeneous system possesses a unique solution—the trivial solution—if and only if rank (A) = n.Example 2.4.1 The homogeneous system x1 + 2x2 + 2x3 = 0, 2x1 + 5x2 + 7x3 = 0, 3x1 + 6x2 + 8x3 = 0, has only the trivial solution because 1 2 2 1 2 2 A = 2 5 7 −→ 0 1 3 = E 3 6 8 0 0 2 shows that rank (A) = n = 3. Indeed, it is also obvious from E that applying back substitution in the system [E|0] yields only the trivial solution.Example 2.4.2 Problem: Explain why the following homogeneous system has infinitely many solutions, and exhibit the general solution: x1 + 2x2 + 2x3 = 0, 2x1 + 5x2 + 7x3 = 0, 3x1 + 6x2 + 6x3 = 0.
2.4 Homogeneous Systems 61 Solution: 1 2 2 1 2 2 A = 2 5 7 −→ 0 1 3 = E 3 6 6 0 0 0 shows that rank (A) = 2 < n = 3. Since the basic columns lie in positions one and two, x1 and x2 are the basic variables while x3 is free. Using back substitution on [E|0] to solve for the basic variables in terms of the free variable produces x2 = −3x3 and x1 = −2x2 − 2x3 = 4x3 , so the general solution is x1 4 x2 = x3 −3 , where x3 is free. x3 1 4 That is, every solution is a multiple of the one particular solution h1 = −3 . 1 Summary Let Am×n be the coefficient matrix for a homogeneous system of m linear equations in n unknowns, and suppose rank (A) = r. • The unknowns that correspond to the positions of the basic columns (i.e., the pivotal positions) are called the basic variables, and the unknowns corresponding to the positions of the nonbasic columns are called the free variables. • There are exactly r basic variables and n − r free variables. • To describe all solutions, reduce A to a row echelon form using Gaussian elimination, and then use back substitution to solve for the basic variables in terms of the free variables. This produces the general solution that has the form x = xf1 h1 + xf2 h2 + · · · + xfn−r hn−r , where the terms xf1 , xf2 , . . . , xfn−r are the free variables and where h1 , h2 , . . . , hn−r are n × 1 columns that represent particular solu- tions of the homogeneous system. The hi 's are independent of which row echelon form is used in the back substitution process. As the free variables xfi range over all possible values, the general solution gen- erates all possible solutions. • A homogeneous system possesses a unique solution (the trivial solu- tion) if and only if rank (A) = n —i.e., if and only if there are no free variables.
2.4 Homogeneous Systems 63 2.4.5. Suppose that A is the coefficient matrix for a homogeneous system of four equations in six unknowns and suppose that A has at least one nonzero row. (a) Determine the fewest number of free variables that are possible. (b) Determine the maximum number of free variables that are pos- sible. 2.4.6. Explain why a homogeneous system of m equations in n unknowns where m < n must always possess an infinite number of solutions. 2.4.7. Construct a homogeneous system of three equations in four unknowns that has −2 −3 1 0 x2 + x4 0 2 0 1 as its general solution. 2.4.8. If c1 and c2 are columns that represent two particular solutions of the same homogeneous system, explain why the sum c1 + c2 must also represent a solution of this system.
64 Chapter 2 Rectangular Systems and Echelon Forms2.5 NONHOMOGENEOUS SYSTEMS Recall that a system of m linear is said to be nonhomogeneous whenever bi = 0 for at least one i. Unlike homogeneous systems, a nonhomogeneous system may be inconsistent and the techniques of §2.3 must be applied in order to determine if solutions do indeed exist. Unless otherwise stated, it is assumed that all systems in this section are consistent. To describe the set of all possible solutions of a consistent nonhomogeneous system, construct a general solution by exactly the same method used for homo- geneous systems as follows. • Use Gaussian elimination to reduce the associated augmented matrix [A|b] to a row echelon form [E|c]. • Identify the basic variables and the free variables in the same manner de- scribed in §2.4. • Apply back substitution to [E|c] and solve for the basic variables in terms of the free variables. • Write the result in the form x = p + xf1 h1 + xf2 h2 + · · · + xfn−r hn−r , (2.5.1) where xf1 , xf2 , . . . , xfn−r are the free variables and p, h1 , h2 , . . . , hn−r are n × 1 columns. This is the general solution of the nonhomogeneous system. As the free variables xfi range over all possible values, the general solu- tion (2.5.1) generates all possible solutions of the system [A|b]. Just as in the homogeneous case, the columns hi and p are independent of which row eche- lon form [E|c] is used. Therefore, [A|b] may be completely reduced to E[A|b] by using the Gauss–Jordan method thereby avoiding the need to perform back substitution. We will use this approach whenever it is convenient. The difference between the general solution of a nonhomogeneous system and the general solution of a homogeneous system is the column p that appears
66 Chapter 2 Rectangular Systems and Echelon Forms Furthermore, recall from (2.4.4) that the general solution of the associated homogeneous system x1 + 2x2 + 2x3 + 3x4 = 0, 2x1 + 4x2 + x3 + 3x4 = 0, (2.5.4) 3x1 + 6x2 + x3 + 4x4 = 0, is given by −2x2 − x4 −2 −1 x2 1 0 = x2 + x4 . −x4 0 −1 x4 0 1 That is, the general solution of the associated homogeneous system (2.5.4) is a part of the general solution of the original nonhomogeneous system (2.5.2). These two observations can be combined by saying that the general solution of the nonhomogeneous system is given by a particular solution plus the general 14 solution of the associated homogeneous system. To see that the previous statement is always true, suppose [A|b] represents a general m × n consistent system where rank (A) = r. Consistency guarantees that b is a nonbasic column in [A|b], and hence the basic columns in [A|b] are in the same positions as the basic columns in [A|0] so that the nonhomogeneous system and the associated homogeneous system have exactly the same set of basic variables as well as free variables. Furthermore, it is not difficult to see that E[A|0] = [EA |0] and E[A|b] = [EA |c], ξ1 . . . ξr where c is some column of the form c = . This means that if you solve 0 . . . 0 the ith equation in the reduced homogeneous system for the ith basic variable xbi in terms of the free variables xfi , xfi+1 , . . . , xfn−r to produce xbi = αi xfi + αi+1 xfi+1 + · · · + αn−r xfn−r , (2.5.5) then the solution for the ith basic variable in the reduced nonhomogeneous system must have the form xbi = ξi + αi xfi + αi+1 xfi+1 + · · · + αn−r xfn−r . (2.5.6) 14 For those students who have studied differential equations, this statement should have a familiar ring. Exactly the same situation holds for the general solution to a linear differential equation. This is no accident—it is due to the inherent linearity in both problems. More will be said about this issue later in the text.
2.5 Nonhomogeneous Systems 69 Now turn to the question, "When does a consistent system have a unique solution?" It is known from (2.5.7) that the general solution of a consistent m × n nonhomogeneous system [A|b] with rank (A) = r is given by x = p + xf1 h1 + xf2 h2 + · · · + xfn−r hn−r , where xf1 h1 + xf2 h2 + · · · + xfn−r hn−r is the general solution of the associated homogeneous system. Consequently, it is evident that the nonhomogeneous system [A|b] will have a unique solution (namely, p ) if and only if there are no free variables—i.e., if and only if r = n (= number of unknowns)—this is equivalent to saying that the associated ho- mogeneous system [A|0] has only the trivial solution.Example 2.5.2 Consider the following nonhomogeneous system: 2x1 + 4x2 + 6x3 = 2, x1 + 2x2 + 3x3 = 1, x1 + x3 = −3, 2x1 + 4x2 = 8. Reducing [A|b] to E[A|b] yields 2 4 6 2 1 0 0 −2 1 2 3 1 0 1 0 3 [A|b] = −→ = E[A|b] . 1 0 1 −3 0 0 1 −1 2 4 0 8 0 0 0 0 The system is consistent because the last column is nonbasic. There are several ways to see that the system has a unique solution. Notice that rank (A) = 3 = number of unknowns, which is the same as observing that there are no free variables. Furthermore, the associated homogeneous system clearly has only the trivial solution. Finally, because we completely reduced [A|b] to E[A|b] , is obvious that there is only it −2 one solution possible and that it is given by p = 3 . −1
2.5 Nonhomogeneous Systems 71 2.5.2. Among the solutions that satisfy the set of linear equations, find all those that also satisfy the following two constraints: (x1 − x2 )2 − 4x2 = 0, 5 x2 − x2 = 0. 3 5 2.5.3say brand X , brand Y, and brand Z. (a) Take into account the obvious fact that a negative number of pounds of any brand can never be applied, and suppose that because of the way fertilizer is sold only an integral number of pounds of each brand will be applied. Under these constraints, determine all possible combinations of the three brands that can be applied to satisfy the recommendations exactly. (b) Suppose that brand X costs $1 per pound, brand Y costs $6 per pound, and brand Z costs $3 per pound. Determine the least expensive solution that will satisfy the recommendations exactly as well as the constraints of part (a). 2.5.4. Consider the following system: 2x + 2y + 3z = 0, 4x + 8y + 12z = −4, 6x + 2y + αz = 4. (a) Determine all values of α for which the system is consistent. (b) Determine all values of α for which there is a unique solution, and compute the solution for these cases. (c) Determine all values of α for which there are infinitely many different solutions, and give the general solution for these cases.
72 Chapter 2 Rectangular Systems and Echelon Forms 2.5.5. If columns s1 and s2 are particular solutions of the same nonhomo- geneous system, must it be the case that the sum s1 + s2 is also a solution? 2.5.6. Suppose that [A|b] is the augmented matrix for a consistent system of m equations in n unknowns where m ≥ n. What must EA look like when the system possesses a unique solution? 2.5.7. Construct a nonhomogeneous system of three equations in four un- knowns that has 1 −2 −3 0 1 0 + x2 + x4 1 0 2 0 0 1 as its general solution. 2.5.8. Consider using floating-point arithmetic (without partial pivoting or scaling) to solve the system represented by the following augmented matrix: .835 .667 .5 .168 .333 .266 .1994 .067 . 1.67 1.334 1.1 .436 (a) Determine the 4-digit general solution. (b) Determine the 5-digit general solution. (c) Determine the 6-digit general solution.
2.6 Electrical Circuits 732.6 ELECTRICAL CIRCUITS The theory of electrical circuits is an important application that naturally gives rise to rectangular systems of linear equations. Because the underlying mathe- matics depends on several of the concepts discussed in the preceding sections, you may find it interesting and worthwhile to make a small excursion into the elementary mathematical analysis of electrical circuits. However, the continuity of the text is not compromised by omitting this section. In a direct current circuit containing resistances and sources of electromo- tive force (abbreviated EMF) such as batteries, a point at which three or more conductors are joined is called a node or branch point of the circuit, and a closed conduction path is called a loop. Any part of a circuit between two ad- joining nodes is called a branch of the circuit. The circuit shown in Figure 2.6.1 is a typical example that contains four nodes, seven loops, and six branches. E1 E2 R1 1 R2 I1 I2 A R5 B E3 R3 I5 R6 2 4 I3 3 I6 C I4 R4 E4 Figure 2.6.1 The problem is to relate the currents Ik in each branch to the resistances Rk 15 and the EMFs Ek . This is accomplished by using Ohm's law in conjunction with Kirchhoff 's rules to produce a system of linear equations. Ohm's Law Ohm's law states that for a current of I amps, the voltage drop (in volts) across a resistance of R ohms is given by V = IR. Kirchhoff's rules—formally stated below—are the two fundamental laws that govern the study of electrical circuits. 15 For an EMF source of magnitude E and a current I, there is always a small internal resistance in the source, and the voltage drop across it is V = E −I ×(internal resistance). But internal source resistance is usually negligible, so the voltage drop across the source can be taken as V = E. When internal resistance cannot be ignored, its effects may be incorporated into existing external resistances, or it can be treated as a separate external resistance.
74 Chapter 2 Rectangular Systems and Echelon Forms Kirchhoff's Rules NODE RULE: The algebraic sum of currents toward each node is zero. That is, the total incoming current must equal the total outgoing current. This is simply a statement of conser- vation of charge. LOOP RULE: The algebraic sum of the EMFs around each loop must equal the algebraic sum of the IR products in the same loop. That is, assuming internal source resistances have been accounted for, the algebraic sum of the voltage drops over the sources equals the algebraic sum of the voltage drops over the resistances in each loop. This is a statement of conservation of energy. Kirchhoff's rules may be used without knowing the directions of the currents and EMFs in advance. You may arbitrarily assign directions. If negative values emerge in the final solution, then the actual direction is opposite to that assumed. To apply the node rule, consider a current to be positive if its direction is toward the node—otherwise, consider the current to be negative. It should be clear that the node rule will always generate a homogeneous system. For example, applying the node rule to the circuit in Figure 2.6.1 yields four homogeneous equations in six unknowns—the unknowns are the Ik 's: Node 1: I1 − I2 − I5 = 0, Node 2: − I1 − I3 + I4 = 0, Node 3: I3 + I5 + I6 = 0, Node 4: I2 − I4 − I6 = 0. To apply the loop rule, some direction (clockwise or counterclockwise) must be chosen as the positive direction, and all EMFs and currents in that direction are considered positive and those in the opposite direction are negative. It is possible for a current to be considered positive for the node rule but considered negative when it is used in the loop rule. If the positive direction is considered to be clockwise in each case, then applying the loop rule to the three indicated loops A, B, and C in the circuit shown in Figure 2.6.1 produces the three non- homogeneous equations in six unknowns—the Ik 's are treated as the unknowns, while the Rk 's and Ek 's are assumed to be known. Loop A: I1 R1 − I3 R3 + I5 R5 = E1 − E3 , Loop B: I2 R2 − I5 R5 + I6 R6 = E2 , Loop C: I3 R3 + I4 R4 − I6 R6 = E3 + E4 .
2.6 Electrical Circuits 75 There are 4 additional loops that also produce loop equations thereby mak- ing a total of 11 equations (4 nodal equations and 7 loop equations) in 6 un- knowns. Although this appears to be a rather general 11 × 6 system of equations, it really is not. If the circuit is in a state of equilibrium, then the physics of the situation dictates that for each set of EMFs Ek , the corresponding currents Ik must be uniquely determined. In other words, physics guarantees that the 11 × 6 system produced by applying the two Kirchhoff rules must be consistent and possess a unique solution. Suppose that [A|b] represents the augmented matrix for the 11 × 6 system generated by Kirchhoff's rules. From the results in §2.5, we know that the system has a unique solution if and only if rank (A) = number of unknowns = 6. Furthermore, it was demonstrated in §2.3 that the system is consistent if and only if rank[A|b] = rank (A). Combining these two facts allows us to conclude that rank[A|b] = 6 so that when [A|b] is reduced to E[A|b] , there will be exactly 6 nonzero rows and 5 zero rows. Therefore, 5 of the original 11 equations are redundant in the sense that they can be "zeroed out" by forming combinations of some particular set of 6 "independent" equations. It is desirable to know beforehand which of the 11 equations will be redundant and which can act as the "independent" set. Notice that in using the node rule, the equation corresponding to node 4 is simply the negative sum of the equations for nodes 1, 2, and 3, and that the first three equations are independent in the sense that no one of the three can be written as a combination of any other two. This situation is typical. For a general circuit with n nodes, it can be demonstrated that the equations for the first n − 1 nodes are independent, and the equation for the last node is redundant. The loop rule also can generate redundant equations. Only simple loops— loops not containing smaller loops—give rise to independent equations. For ex- ample, consider the loop consisting of the three exterior branches in the circuit shown in Figure 2.6.1. Applying the loop rule to this large loop will produce no new information because the large loop can be constructed by "adding" the three simple loops A, B, and C contained within. The equation associated with the large outside loop is I1 R1 + I2 R2 + I4 R4 = E1 + E2 + E4 , which is precisely the sum of the equations that correspond to the three compo- nent loops A, B, and C. This phenomenon will hold in general so that only the simple loops need to be considered when using the loop rule.
76 Chapter 2 Rectangular Systems and Echelon Forms The point of this discussion is to conclude that the more general 11 × 6 rectangular system can be replaced by an equivalent 6 × 6 square system that has a unique solution by dropping the last nodal equation and using only the simple loop equations. This is characteristic of practical work in general. The physics of a problem together with natural constraints can usually be employed to replace a general rectangular system with one that is square and possesses a unique solution. One of the goals in our study is to understand more clearly the notion of "independence" that emerged in this application. So far, independence has been an intuitive idea, but this example helps make it clear that independence is a fundamentally important concept that deserves to be nailed down more firmly. This is done in §4.3, and the general theory for obtaining independent equations from electrical circuits is developed in Examples 4.4.6 and 4.4.7.Exercises for section 2.6 2.6.1. Suppose that Ri = i ohms and Ei = i volts in the circuit shown in Figure 2.6.1. (a) Determine the six indicated currents. (b) Select node number 1 to use as a reference point and fix its potential to be 0 volts. With respect to this reference, calculate the potentials at the other three nodes. Check your answer by verifying the loop rule for each loop in the circuit. 2.6.2. Determine the three currents indicated in the following circuit. 5Ω 8Ω I2 I1 12 volts 1Ω 1Ω 9 volts 10Ω I3 2.6.3. Determine the two unknown EMFs in the following circuit. 20 volts 6Ω 1 amp E1 4Ω 2 amps E2 2Ω
2.6 Electrical Circuits 77 2.6.4. Consider the circuit shown below and answer the following questions. R2 R3 R5 R1 R4 R6 I E (a) How many nodes does the circuit contain? (b) How many branches does the circuit contain? (c) Determine the total number of loops and then determine the number of simple loops. (d) Demonstrate that the simple loop equations form an "indepen- dent" system of equations in the sense that there are no redun- dant equations. (e) Verify that any three of the nodal equations constitute an "in- dependent" system of equations. (f) Verify that the loop equation associated with the loop containing R1 , R2 , R3 , and R4 can be expressed as the sum of the two equations associated with the two simple loops contained in the larger loop. (g) Determine the indicated current I if R1 = R2 = R3 = R4 = 1 ohm, R5 = R6 = 5 ohms, and E = 5 volts.
78 Chapter 2 Rectangular Systems and Echelon Forms Life is good for only two things, discovering mathematics and teaching mathematics. — Sim´on D. Poisson (1781–1840) e
CHAPTER 3 Matrix Algebra3.1 FROM ANCIENT CHINA TO ARTHUR CAYLEY The ancient Chinese appreciated the advantages of array manipulation in dealing with systems of linear equations, and they possessed the seed that might have germinated into a genuine theory of matrices. Unfortunately, in the year 213 B.C., emperor Shih Hoang-ti ordered that "all books be burned and all scholars be buried." It is presumed that the emperor wanted all knowledge and written records to begin with him and his regime. The edict was carried out, and it will never be known how much knowledge was lost. The book Chiu-chang Suan-shu (Nine Chapters on Arithmetic), mentioned in the introduction to Chapter 1, was compiled on the basis of remnants that survived. More than a millennium passed before further progress was documented. The Chinese counting board with its colored rods and its applications involving array manipulation to solve linear systems eventually found its way to Japan. Seki Kowa (1642–1708), whom many Japanese consider to be one of the greatest mathematicians that their country has produced, carried forward the Chinese principles involving "rule of thumb" elimination methods on arrays of numbers. His understanding of the elementary operations used in the Chinese elimination process led him to formulate the concept of what we now call the determinant. While formulating his ideas concerning the solution of linear systems, Seki Kowa anticipated the fundamental concepts of array operations that today form the basis for matrix algebra. However, there is no evidence that he developed his array operations to actually construct an algebra for matrices. From the middle 1600s to the middle 1800s, while Europe was flowering in mathematical development, the study of array manipulation was exclusively
80 Chapter 3 Matrix Algebra dedicated to the theory of determinants. Curiously, matrix algebra did not evolve along with the study of determinants. It was not until the work of the British mathematician Arthur Cayley (1821– 1895) that the matrix was singled out as a separate entity, distinct from the notion of a determinant, and algebraic operations between matrices were defined. In an 1855 paper, Cayley first introduced his basic ideas that were presented mainly to simplify notation. Finally, in 1857, Cayley expanded on his original ideas and wrote A Memoir on the Theory of Matrices. This laid the foundations for the modern theory and is generally credited for being the birth of the subjects of matrix analysis and linear algebra. Arthur Cayley began his career by studying literature at Trinity College, Cambridge (1838–1842), but developed a side interest in mathematics, which he studied in his spare time. This "hobby" resulted in his first mathematical paper in 1841 when he was only 20 years old. To make a living, he entered the legal profession and practiced law for 14 years. However, his main interest was still mathematics. During the legal years alone, Cayley published almost 300 papers in mathematics. In 1850 Cayley crossed paths with James J. Sylvester, and between the two of them matrix theory was born and nurtured. The two have been referred to as the "invariant twins." Although Cayley and Sylvester shared many mathe- matical interests, they were quite different people, especially in their approach to mathematics. Cayley had an insatiable hunger for the subject, and he read everything that he could lay his hands on. Sylvester, on the other hand, could not stand the sight of papers written by others. Cayley never forgot anything he had read or seen—he became a living encyclopedia. Sylvester, so it is said, would frequently fail to remember even his own theorems. In 1863, Cayley was given a chair in mathematics at Cambridge University, and thereafter his mathematical output was enormous. Only Cauchy and Euler were as prolific. Cayley often said, "I really love my subject," and all indica- tions substantiate that this was indeed the way he felt. He remained a working mathematician until his death at age 74. Because the idea of the determinant preceded concepts of matrix algebra by at least two centuries, Morris Kline says in his book Mathematical Thought from Ancient to Modern Times that "the subject of matrix theory was well developed before it was created." This must have indeed been the case because immediately after the publication of Cayley's memoir, the subjects of matrix theory and linear algebra virtually exploded and quickly evolved into a discipline that now occupies a central position in applied mathematics.
3.2 Addition and Transposition 813.2 ADDITION AND TRANSPOSITION In the previous chapters, matrix language and notation were used simply to for- mulate some of the elementary concepts surrounding linear systems. The purpose 16 now is to turn this language into a mathematical theory. Unless otherwise stated, a scalar is a complex number. Real numbers are a subset of the complex numbers, and hence real numbers are also scalar quan- tities. In the early stages, there is little harm in thinking only in terms of real scalars. Later on, however, the necessity for dealing with complex numbers will be unavoidable. Throughout the text, will denote the set of real numbers, and C will denote the complex numbers. The set of all n -tuples of real numbers will be denoted by n , and the set of all complex n -tuples will be denoted by C n . For example, 2 is the set of all ordered pairs of real numbers (i.e., the standard cartesian plane), and 3 is ordinary 3-space. Analogously, m×n and C m×n denote the m × n matrices containing real numbers and complex numbers, respectively. Matrices A = [aij ] and B = [bij ] are defined to be equal matrices when A and B have the same shape and corresponding entries are equal. That is, aij = bij for each i = 1, 2, . . . , m andj = 1, 2, . . . , n. In particular, this 1 definition applies to arrays such as u = 2 and v = ( 1 2 3 ) . Even 3 though u and v describe exactly the same point in 3-space, we cannot consider them to be equal matrices because they have different shapes. An array (or matrix) consisting of a single column, such as u, is called a column vector, while an array consisting of a single row, such as v, is called a row vector. Addition of Matrices If A and B are m × n matrices, the sum of A and B is defined to be the m × n matrix A + B obtained by adding corresponding entries. That is, [A + B]ij = [A]ij + [B]ij for each i and j. For example, −2 x 3 2 1 − x −2 0 1 1 + = . z + 3 4 −y −3 4 + x 4 + y z 8+x 4 16 The great French mathematician Pierre-Simon Laplace (1749–1827) said that, "Such is the ad- vantage of a well-constructed language that its simplified notation often becomes the source of profound theories." The theory of matrices is a testament to the validity of Laplace's statement.
82 Chapter 3 Matrix Algebra The symbol "+" is used two different ways—it denotes addition between scalars in some places and addition between matrices at other places. Although these are two distinct algebraic operations, no ambiguities will arise if the context in which "+" appears is observed. Also note that the requirement that A and B have the same shape prevents adding a row to a column, even though the two may contain the same number of entries. The matrix (−A), called the additive inverse of A, is defined to be the matrix obtained by negating each entry of A. That is, if A = [aij ], then −A = [−aij ]. This allows matrix subtraction to be defined in the natural way. For two matrices of the same shape, the difference A − B is defined to be the matrix A − B = A + (−B) so that [A − B]ij = [A]ij − [B]ij for each i and j. Since matrix addition is defined in terms of scalar addition, the familiar algebraic properties of scalar addition are inherited by matrix addition as detailed below. Properties of Matrix Addition For m × n matrices A, B, and C, the following properties hold. Closure property: A + B is again an m × n matrix. Associative property: (A + B) + C = A + (B + C). Commutative property: A + B = B + A. Additive identity: The m × n matrix 0 consisting of all ze- ros has the property that A + 0 = A. Additive inverse: The m × n matrix (−A) has the property that A + (−A) = 0. Another simple operation that is derived from scalar arithmetic is as follows. Scalar Multiplication The product of a scalar α times a matrix A, denoted by αA, is defined to be the matrix obtained by multiplying each entry of A by α. That is, [αA]ij = α[A]ij for each i and j. For example, 1 2 3 2 4 6 1 2 2 4 1 20 1 2 = 0 2 4 and 3 4 = 6 8. 2 1 4 2 2 8 4 0 1 0 2 The rules for combining addition and scalar multiplication are what you might suspect they should be. Some of the important ones are listed below.
3.2 Addition and Transposition 83 Properties of Scalar Multiplication For m × n matrices A and B and for scalars α and β, the following properties hold. Closure property: αA is again an m × n matrix. Associative property: (αβ)A = α(βA). Distributive property: α(A + B) = αA + αB. Scalar multiplica- tion is distributed over matrix addition. Distributive property: (α + β)A = αA + βA. Scalar multiplica- tion is distributed over scalar addition. Identity property: 1A = A. The number 1 is an identity el- ement under scalar multiplication. Other properties such as αA = Aα could have been listed, but the prop- erties singled out pave the way for the definition of a vector space on p. 160. A matrix operation that's not derived from scalar arithmetic is transposition as defined below. Transpose The transpose of Am×n is defined to be the n × m matrix AT ob- tained by interchanging rows and columns in A. More precisely, if A = [aij ], then [AT ]ij = aji . For example, T 1 2 3 4 = 1 3 5 . 2 4 6 5 6 T It should be evident that for all matrices, AT = A. Whenever a matrix contains complex entries, the operation of complex con- jugation almost always accompanies the transpose operation. (Recall that the complex conjugate of z = a + ib is defined to be z = a − ib.)
84 Chapter 3 Matrix Algebra Conjugate Transpose For A = [aij ], the conjugate matrix is defined to be A = [aij ] , and ¯ the conjugate transpose of A is defined to be AT = AT . From now on, AT will be denoted by A∗ , so [A∗ ]ij = aji . For example, ¯ ∗ 1 + 4i 3 1 − 4i i 2 = −i 2 − i. 3 2+i 0 2 0 ∗ (A∗ ) = A for all matrices, and A∗ = AT whenever A contains only real entries. Sometimes the matrix A∗ is called the adjoint of A. The transpose (and conjugate transpose) operation is easily combined with matrix addition and scalar multiplication. The basic rules are given below. Properties of the Transpose If A and B are two matrices of the same shape, and if α is a scalar, then each of the following statements is true. ∗ (A + B) = A∗ + B∗ . T (A + B) = AT + BT and (3.2.1) ∗ (αA) = αA∗ . T (αA) = αAT and (3.2.2) 17 Proof. We will prove that (3.2.1) and (3.2.2) hold for the transpose operation. The proofs of the statements involving conjugate transposes are similar and are left as exercises. For each i and j, it is true that T [(A + B) ]ij = [A + B]ji = [A]ji + [B]ji = [AT ]ij + [BT ]ij = [AT + BT ]ij . 17 Computers can outperform people in many respects in that they do arithmetic much faster and more accurately than we can, and they are now rather adept at symbolic computation and mechanical manipulation of formulas. But computers can't do mathematics—people still hold the monopoly. Mathematics emanates from the uniquely human capacity to reason abstractly in a creative and logical manner, and learning mathematics goes hand-in-hand with learning how to reason abstractly and create logical arguments. This is true regardless of whether your orientation is applied or theoretical. For this reason, formal proofs will appear more frequently as the text evolves, and it is expected that your level of comprehension as well as your ability to create proofs will grow as you proceed.
3.2 Addition and Transposition 85 T This proves that corresponding entries in (A + B) and AT + BT are equal, T so it must be the case that (A + B) = AT + BT . Similarly, for each i and j, T [(αA)T ]ij = [αA]ji = α[A]ji = α[AT ]ij =⇒ (αA) = αAT . Sometimes transposition doesn't change anything. For example, if 1 2 3 A = 2 4 5 , then AT = A. 3 5 6 This is because the entries in A are symmetrically located about the main di- agonal—the line from the upper-left-hand corner to the lower-right-hand corner. λ1 0 · · · 0 0 λ2 ··· 0 Matrices of the form D = . . . . .. . . are called diagonal matrices, . . . . 0 0 ··· λn and they are clearly symmetric in the sense that D = DT . This is one of several kinds of symmetries described below. Symmetries Let A = [aij ] be a square matrix. • A is said to be a symmetric matrix whenever A = AT , i.e., whenever aij = aji . • A is said to be a skew-symmetric matrix whenever A = −AT , i.e., whenever aij = −aji . • A is said to be a hermitian matrix whenever A = A∗ , i.e., whenever aij = aji . This is the complex analog of symmetry. • A is said to be a skew-hermitian matrix when A = −A∗ , i.e., whenever aij = −aji . This is the complex analog of skew symmetry. For example, consider 1 2 + 4i 1 − 3i 1 2 + 4i 1 − 3i A = 2 − 4i 3 8 + 6i and B = 2 + 4i 3 8 + 6i . 1 + 3i 8 − 6i 5 1 − 3i 8 + 6i 5 Can you see that A is hermitian but not symmetric, while B is symmetric but not hermitian? Nature abounds with symmetry, and very often physical symmetry manifests itself as a symmetric matrix in a mathematical model. The following example is an illustration of this principle.
86 Chapter 3 Matrix AlgebraExample 3.2.1 Consider two springs that are connected as shown in Figure 3.2.1. Node 1 k1 Node 2 k2 Node 3 x1 x2 x3 F1 -F1 -F3 F3 Figure 3.2.1 The springs at the top represent the "no tension" position in which no force is being exerted on any of the nodes. Suppose that the springs are stretched or compressed so that the nodes are displaced as indicated in the lower portion of Figure 3.2.1. Stretching or compressing the springs creates a force on each 18 node according to Hooke's law that says that the force exerted by a spring is F = kx, where x is the distance the spring is stretched or compressed and where k is a stiffness constant inherent to the spring. Suppose our springs have stiffness constants k1 and k2 , and let Fi be the force on node i when the springs are stretched or compressed. Let's agree that a displacement to the left is positive, while a displacement to the right is negative, and consider a force directed to the right to be positive while one directed to the left is negative. If node 1 is displaced x1 units, and if node 2 is displaced x2 units, then the left-hand spring is stretched (or compressed) by a total amount of x1 − x2 units, so the force on node 1 is F1 = k1 (x1 − x2 ). Similarly, if node 2 is displaced x2 units, and if node 3 is displaced x3 units, then the right-hand spring is stretched by a total amount of x2 − x3 units, so the force on node 3 is F3 = −k2 (x2 − x3 ). The minus sign indicates the force is directed to the left. The force on the left- hand side of node 2 is the opposite of the force on node 1, while the force on the right-hand side of node 2 must be the opposite of the force on node 3. That is, F2 = −F1 − F3 . 18 Hooke's law is named for Robert Hooke (1635–1703), an English physicist, but it was generally known to several people (including Newton) before Hooke's 1678 claim to it was made. Hooke was a creative person who is credited with several inventions, including the wheel barometer, but he was reputed to be a man of "terrible character." This characteristic virtually destroyed his scientific career as well as his personal life. It is said that he lacked mathematical sophis- tication and that he left much of his work in incomplete form, but he bitterly resented people who built on his ideas by expressing them in terms of elegant mathematical formulations.
88 Chapter 3 Matrix Algebra 3.2.4. Explain why the set of all n × n symmetric matrices is closed under matrix addition. That is, explain why the sum of two n × n symmetric matrices is again an n × n symmetric matrix. Is the set of all n × n skew-symmetric matrices closed under matrix addition? 3.2.5. Prove that each of the following statements is true. (a) If A = [aij ] is skew symmetric, then ajj = 0 for each j. (b) If A = [aij ] is skew hermitian, then each ajj is a pure imagi- nary number—i.e., a multiple of the imaginary unit i. (c) If A is real and symmetric, then B = iA is skew hermitian. 3.2.6. Let A be any square matrix. (a) Show that A+AT is symmetric and A−AT is skew symmetric. (b) Prove that there is one and only one way to write A as the sum of a symmetric matrix and a skew-symmetric matrix. 3.2.7. If A and B are two matrices of the same shape, prove that each of the following statements is true. ∗ (a) (A + B) = A∗ + B∗ . ∗ (b) (αA) = αA∗ . 3.2.8. Using the conventions given in Example 3.2.1, determine the stiffness matrix for a system of n identical springs, with stiffness constant k, connected in a line similar to that shown in Figure 3.2.1.
3.3 Linearity 893.3 LINEARITY The concept of linearity is the underlying theme of our subject. In elementary mathematics the term "linear function" refers to straight lines, but in higher mathematics linearity means something much more general. Recall that a func- tion f is simply a rule for associating points in one set D —called the domain of f —to points in another set R —the range of f. A linear function is a particular type of function that is characterized by the following two properties. Linear Functions Suppose that D and R are sets that possess an addition operation as well as a scalar multiplication operation—i.e., a multiplication between scalars and set members. A function f that maps points in D to points in R is said to be a linear function whenever f satisfies the conditions that f (x + y) = f (x) + f (y) (3.3.1) and f (αx) = αf (x) (3.3.2) for every x and y in D and for all scalars α. These two conditions may be combined by saying that f is a linear function whenever f (αx + y) = αf (x) + f (y) (3.3.3) for all scalars α and for all x, y ∈ D. One of the simplest linear functions is f (x) = αx, whose graph in 2 is a straight line through the origin. You should convince yourself that f is indeed a linear function according to the above definition. However, f (x) = αx + β does not qualify for the title "linear function"—it is a linear function that has been translated by a constant β. Translations of linear functions are referred to as affine functions. Virtually all information concerning affine functions can be derived from an understanding of linear functions, and consequently we will focus only on issues of linearity. In 3 , the surface described by a function of the form f (x1 , x2 ) = α1 x1 + α2 x2 is a plane through the origin, and it is easy to verify that f is a linear function. For β = 0, the graph of f (x1 , x2 ) = α1 x1 + α2 x2 + β is a plane not passing through the origin, and f is no longer a linear function—it is an affine function.
3.4 Why Do It This Way 933.4 WHY DO IT THIS WAY If you were given the task of formulating a definition for composing two ma- trices A and B in some sort of "natural" multiplicative fashion, your first attempt would probably be to compose A and B by multiplying correspond- ing entries—much the same way matrix addition is defined. Asked then to defend the usefulness of such a definition, you might be hard pressed to provide a truly satisfying response. Unless a person is in the right frame of mind, the issue of deciding how to best define matrix multiplication is not at all transparent, es- pecially if it is insisted that the definition be both "natural" and "useful." The world had to wait for Arthur Cayley to come to this proper frame of mind. As mentioned in §3.1, matrix algebra appeared late in the game. Manipula- tion on arrays and the theory of determinants existed long before Cayley and his theory of matrices. Perhaps this can be attributed to the fact that the "correct" way to multiply two matrices eluded discovery for such a long time. 19 Around 1855, Cayley became interested in composing linear functions. In particular, he was investigating linear functions of the type discussed in Example 3.3.2. Typical examples of two such functions are x1 ax1 + bx2 x1 Ax1 + Bx2 f (x) = f = and g(x) = g = . x2 cx1 + dx2 x2 Cx1 + Dx2 Consider, as Cayley did, composing f and g to create another linear function Ax1 + Bx2 (aA + bC)x1 + (aB + bD)x2 h(x) = f g(x) = f = . Cx1 + Dx2 (cA + dC)x1 + (cB + dD)x2 It was Cayley's idea to use matrices of coefficients to represent these linear functions. That is, f, g, and h are represented by a b A B aA + bC aB + bD F= , G= , and H= . c d C D cA + dC cB + dD After making this association, it was only natural for Cayley to call H the composition (or product) of F and G, and to write a b A B aA + bC aB + bD = . (3.4.1) c d C D cA + dC cB + dD In other words, the product of two matrices represents the composition of the two associated linear functions. By means of this observation, Cayley brought to life the subjects of matrix analysis and linear algebra. 19 Cayley was not the first to compose linear functions. In fact, Gauss used these compositions as early as 1801, but not in the form of an array of coefficients. Cayley was the first to make the connection between composition of linear functions and the composition of the associated matrices. Cayley's work from 1855 to 1857 is regarded as being the birth of our subject.
94 Chapter 3 Matrix AlgebraExercises for section 3.4 Each problem in this section concerns the following three linear transformations in 2 . f(p) Rotation: Rotate points counterclockwise through an angle θ. θ p Reflection: Reflect points about the x -axis. p f(p) x = y p Projection: Project points onto the line y = x in a perpendicular f(p) manner. 3.4.1. Determine the matrix associated with each of these linear functions. That is, determine the aij 's such that x1 a11 x1 + a12 x2 f (p) = f = . x2 a21 x1 + a22 x2 3.4.2. By using matrix multiplication, determine the linear function obtained by performing a rotation followed by a reflection. 3.4.3. By using matrix multiplication, determine the linear function obtained by first performing a reflection, then a rotation, and finally a projection.
3.5 Matrix Multiplication 97 For example, 1 3 −3 2 2 1 −4 8 3 −7 4 A= , B= 2 5 −1 8 =⇒ AB = . −3 0 5 −8 1 9 4 −1 2 0 2 Notice that in spite of the fact that the product AB exists, the product BA is not defined—matrix B is 3 × 4 and A is 2 × 3, and the inside dimensions don't match in this order. Even when the products AB and BA each exist and have the same shape, they need not be equal. For example, 1 −1 1 1 0 0 2 −2 A= , B= =⇒ AB = , BA = . (3.5.1) 1 −1 1 1 0 0 2 −2 This disturbing feature is a primary difference between scalar and matrix algebra. Matrix Multiplication Is Not Commutative Matrix multiplication is a noncommutative operation—i.e., it is possible for AB = BA, even when both products exist and have the same shape. There are other major differences between multiplication of matrices and multiplication of scalars. For scalars, αβ = 0 implies α = 0 or β = 0. (3.5.2) However, the analogous statement for matrices does not hold—the matrices given in (3.5.1) show that it is possible for AB = 0 with A = 0 and B = 0. Related to this issue is a rule sometimes known as the cancellation law. For scalars, this law says that αβ = αγ and α=0 implies β = γ. (3.5.3) This is true because we invoke (3.5.2) to deduce that α(β − γ) = 0 implies β − γ = 0. Since (3.5.2) does not hold for matrices, we cannot expect (3.5.3) to hold for matrices.Example 3.5.1 The cancellation law (3.5.3) fails for matrix multiplication. If 1 1 2 2 3 1 A= , B= , and C= , 1 1 2 2 1 3 then 4 4 AB = = AC but B = C 4 4 in spite of the fact that A = 0.
100 Chapter 3 Matrix Algebra For example, a very concise proof of the fact (2.3.5) stating that a system of equations Am×n xn×1 = bm×1 is consistent if and only if b is a linear combination of the columns in A is obtained by noting that the system is consistent if and only if there exists a column s that satisfies s1 s2 b = As = ( A∗1 A∗2 · · · A∗n ) . = A∗1 s1 + A∗2 s2 + · · · + A∗n sn . . . sn The following example illustrates a common situation in which matrix mul- tiplication arises naturally.Example 3.5.2 An airline serves five cities, say, A, B, C, D, and H, in which H is the "hub city." The various routes between the cities are indicated in Figure 3.5.1. A B H C D Figure 3.5.1 Suppose you wish to travel from city A to city B so that at least two connecting flights are required to make the trip. Flights (A → H) and (H → B) provide the minimal number of connections. However, if space on either of these two flights is not available, you will have to make at least three flights. Several questions arise. How many routes from city A to city B require exactly three connecting flights? How many routes require no more than four flights—and so forth? Since this particular network is small, these questions can be answered by "eyeballing" the diagram, but the "eyeball method" won't get you very far with the large networks that occur in more practical situations. Let's see how matrix algebra can be applied. Begin by creating a connectivity matrix C = [cij ] (also known as an adjacency matrix) in which 1 if there is a flight from city i to city j, cij = 0 otherwise.
3.5 Matrix Multiplication 103 3.5.5. Suppose that A and B are m × n matrices. If Ax = Bx holds for all n × 1 columns x, prove that A = B. Hint: What happens when x is a unit column? 1/2 α 3.5.6. For A = , determine limn→∞ An . Hint: Compute a few 0 1/2 powers of A and try to deduce the general form of An . 3.5.7. If Cm×1 and R1×n are matrices consisting of a single column and a single row, respectively, then the matrix product Pm×n = CR is sometimes called the outer product of C with R. For conformable matrices A and B, explain how to write the product AB as a sum of outer products involving the columns of A and the rows of B. 3.5.8. A square matrix U = [uij ] is said to be upper triangular whenever uij = 0 for i > j —i.e., all entries below the main diagonal are 0. (a) If A and B are two n × n upper-triangular matrices, explain why the product AB must also be upper triangular. (b) If An×n and Bn×n are upper triangular, what are the diagonal entries of AB? (c) L is lower triangular when ij = 0 for i < j. Is it true that the product of two n × n lower-triangular matrices is again lower triangular? 3.5.9. If A = [aij (t)] is a matrix whose entries are functions of a variable t, the derivative of A with respect to t is defined to be the matrix of derivatives. That is, dA daij = . dt dt Derive the product rule for differentiation d(AB) dA dB = B+A . dt dt dt 3.5.10. Let Cn×n be the connectivity matrix associated with a network of n nodes such as that described in Example 3.5.2, and let e be the n × 1 column of all 1's. In terms of the network, describe the entries in each of the following products. (a) Interpret the product Ce. (b) Interpret the product eT C.
104 Chapter 3 Matrix Algebra 3.5.11. Consider three tanks each containing V gallons of brine. The tanks are connected as shown in Figure 3.5.2, and all spigots are opened at once. As fresh water at the rate of r gal/sec is pumped into the top of the first tank, r gal/sec leaves from the bottom and flows into the next tank, and so on down the line—there are r gal/sec entering at the top and leaving through the bottom of each tank. r gal / sec r gal / sec r gal / sec r gal / sec Figure 3.5.2 Let xi (t) denote the number of pounds of salt in tank i at time t, and let x1 (t) dx1 /dt dx x = x2 (t) and = dx2 /dt . dt x3 (t) dx3 /dt Assuming that complete mixing occurs in each tank on a continuous basis, show that −1 0 0 dx r = Ax, where A = 1 −1 0. dt V 0 1 −1 Hint: Use the fact that dxi lbs lbs = rate of change = coming in − going out. dt sec sec
3.6 Properties of Matrix Multiplication 1053.6 PROPERTIES OF MATRIX MULTIPLICATION We saw in the previous section that there are some differences between scalar and matrix algebra—most notable is the fact that matrix multiplication is not commutative, and there is no cancellation law. But there are also some important similarities, and the purpose of this section is to look deeper into these issues. Although we can adjust to not having the commutative property, the situa- tion would be unbearable if the distributive and associative properties were not available. Fortunately, both of these properties hold for matrix multiplication. Distributive and Associative Laws For conformable matrices each of the following is true. • A(B + C) = AB + AC (left-hand distributive law). • (D + E)F = DF + EF (right-hand distributive law). • A(BC) = (AB)C (associative law). Proof. To prove the left-hand distributive property, demonstrate the corre- sponding entries in the matrices A(B + C) and AB + AC are equal. To this end, use the definition of matrix multiplication to write [A(B + C)]ij = Ai∗ (B + C)∗j = [A]ik [B + C]kj = [A]ik ([B]kj + [C]kj ) k k = ([A]ik [B]kj + [A]ik [C]kj ) = [A]ik [B]kj + [A]ik [C]kj k k k = Ai∗ B∗j + Ai∗ C∗j = [AB]ij + [AC]ij = [AB + AC]ij . Since this is true for each i and j, it follows that A(B + C) = AB + AC. The proof of the right-hand distributive property is similar and is omitted. To prove the associative law, suppose that B is p × q and C is q × n, and recall from (3.5.7) that the j th column of BC is a linear combination of the columns in B. That is, q [BC]∗j = B∗1 c1j + B∗2 c2j + · · · + B∗q cqj = B∗k ckj . k=1
106 Chapter 3 Matrix Algebra Use this along with the left-hand distributive property to write q q [A(BC)]ij = Ai∗ [BC]∗j = Ai∗ B∗k ckj = Ai∗ B∗k ckj k=1 k=1 q = [AB]ik ckj = [AB]i∗ C∗j = [(AB)C]ij . k=1Example 3.6.1 Linearity of Matrix Multiplication. Let A be an m × n matrix, and f be the function defined by matrix multiplication f (Xn×p ) = AX. The left-hand distributive property guarantees that f is a linear function be- cause for all scalars α and for all n × p matrices X and Y, f (αX + Y) = A(αX + Y) = A(αX) + AY = αAX + AY = αf (X) + f (Y). Of course, the linearity of matrix multiplication is no surprise because it was the consideration of linear functions that motivated the definition of the matrix product at the outset. For scalars, the number 1 is the identity element for multiplication because it has the property that it reproduces whatever it is multiplied by. For matrices, there is an identity element with similar properties. Identity Matrix The n × n matrix with 1's on the main diagonal and 0's elsewhere 1 0 ··· 0 0 1 ··· 0 In = . . . . .. . . . . . . 0 0 ··· 1 is called the identity matrix of order n. For every m × n matrix A, AIn = A and Im A = A. The subscript on In is neglected whenever the size is obvious from the context.
3.6 Properties of Matrix Multiplication 107 Proof. Notice that I∗j has a 1 in the j th position and 0's elsewhere. Recall from Exercise 3.5.4 that such columns were called unit columns, and they have the property that for any conformable matrix A, AI∗j = A∗j . Using this together with the fact that [AI]∗j = AI∗j produces AI = ( AI∗1 AI∗2 ··· AI∗n ) = ( A∗1 A∗2 ··· A∗n ) = A. A similar argument holds when I appears on the left-hand side of A. Analogous to scalar algebra, we define the 0th power of a square matrix to be the identity matrix of corresponding size. That is, if A is n × n, then A 0 = In . Positive powers of A are also defined in the natural way. That is, An = AA· · ·A . n times The associative law guarantees that it makes no difference how matrices are grouped for powering. For example, AA2 is the same as A2 A, so that A3 = AAA = AA2 = A2 A. Also, the usual laws of exponents hold. For nonnegative integers r and s, s Ar As = Ar+s and (Ar ) = Ars . We are not yet in a position to define negative or fractional powers, and due to the lack of conformability, powers of nonsquare matrices are never defined.Example 3.6.2 2 A Pitfall. For two n × n matrices, what is (A + B) ? Be careful! Because matrix multiplication is not commutative, the familiar formula from scalar alge- bra is not valid for matrices. The distributive properties must be used to write 2 (A + B) = (A + B)(A + B) = (A + B) A + (A + B) B = A2 + BA + AB + B2 , and this is as far as you can go. The familiar form A2 +2AB+B2 is obtained only k in those rare cases where AB = BA. To evaluate (A + B) , the distributive rules must be applied repeatedly, and the results are a bit more complicated—try it for k = 3.
108 Chapter 3 Matrix AlgebraExample 3.6.3 Suppose that the population migration between two geographical regions—say, the North and the South—is as follows. Each year, 50% of the population in the North migrates to the South, while only 25% of the population in the South moves to the North. This situation is depicted by drawing a transition diagram such as that shown in Figure 3.6.1. .5 .5 N S .75 .25 Figure 3.6.1 Problem: If this migration pattern continues, will the population in the North continually shrink until the entire population is eventually in the South, or will the population distribution somehow stabilize before the North is completely deserted? Solution: Let nk and sk denote the respective proportions of the total popula- tion living in the North and South at the end of year k and assume nk + sk = 1. The migration pattern dictates that the fractions of the population in each region at the end of year k + 1 are nk+1 = nk (.5) + sk (.25), (3.6.1) sk+1 = nk (.5) + sk (.75). If pT = (nk , sk ) and pT = (nk+1 , sk+1 ) denote the respective population k k+1 distributions at the end of years k and k + 1, and if N S N .5 .5 T= S .25 .75 is the associated transition matrix, then (3.6.1) assumes the matrix form pT = pT T. Inducting on pT = pT T, pT = pT T = pT T2 , pT = pT T = k+1 k 1 0 2 1 0 3 2 pT T3 , etc., leads to 0 pT = pT Tk . k 0 (3.6.2) Determining the long-run behavior involves evaluating limk→∞ pT , and it's clear k from (3.6.2) that this boils down to analyzing limk→∞ Tk . Later, in Example
3.6 Properties of Matrix Multiplication 109 7.3.5, a more sophisticated approach is discussed, but for now we will use the "brute force" method of successively powering P until a pattern emerges. The first several powers of P are shown below with three significant digits displayed. .375 .625 .344 .656 .328 .672 P2 = P3 = P4 = .312 .687 .328 .672 .332 .668 .334 .666 .333 .667 .333 .667 P5 = P6 = P7 = .333 .667 .333 .667 .333 .667 This sequence appears to be converging to a limiting matrix of the form 1/3 2/3 P∞ = lim Pk = , k→∞ 1/3 2/3 so the limiting population distribution is 1/3 2/3 pT = lim pT = lim pT Tk = pT lim Tk = ( n0 ∞ k 0 0 s0 ) k→∞ k→∞ k→∞ 1/3 2/3 n0 + s0 2(n0 + s0 ) = = ( 1/3 2/3 ) . 3 3 Therefore, if the migration pattern continues to hold, then the population dis- tribution will eventually stabilize with 1/3 of the population being in the North and 2/3 of the population in the South. And this is independent of the initial distribution! The powers of P indicate that the population distribution will be practically stable in no more than 6 years—individuals may continue to move, but the proportions in each region are essentially constant by the sixth year. The operation of transposition has an interesting effect upon a matrix product—a reversal of order occurs. Reverse Order Law for Transposition For conformable matrices A and B, T (AB) = BT AT . The case of conjugate transposition is similar. That is, ∗ (AB) = B∗ A∗ .
110 Chapter 3 Matrix Algebra Proof. By definition, T (AB)ij = [AB]ji = Aj∗ B∗i . Consider the (i, j)-entry of the matrix BT AT and write BT AT ij = BT i∗ AT ∗j = BT ik AT kj k = [B]ki [A]jk = [A]jk [B]ki k k = Aj∗ B∗i . T T Therefore, (AB)ij= BT AT ij for all i and j, and thus (AB) = BT A T . The proof for the conjugate transpose case is similar.Example 3.6.4 For every matrix Am×n , the products AT A and AAT are symmetric matrices because T T T T AT A = AT AT = AT A and AAT = AT AT = AAT .Example 3.6.5 Trace of a Product. Recall from Example 3.3.1 that the trace of a square matrix is the sum of its main diagonal entries. Although matrix multiplication is not commutative, the trace function is one of the few cases where the order of the matrices can be changed without affecting the results. Problem: For matrices Am×n and Bn×m , prove that trace (AB) = trace (BA). Solution: trace (AB) = [AB]ii = Ai∗ B∗i = aik bki = bki aik i i i k i k = bki aik = Bk∗ A∗k = [BA]kk = trace (BA). k i k k Note: This is true in spite of the fact that AB is m × m while BA is n × n. Furthermore, this result can be extended to say that any product of conformable matrices can be permuted cyclically without altering the trace of the product. For example, trace (ABC) = trace (BCA) = trace (CAB). However, a noncyclical permutation may not preserve the trace. For example, trace (ABC) = trace (BAC).
112 Chapter 3 Matrix AlgebraExample 3.6.7 Reducibility. Suppose that Tn×n x = b represents a system of linear equa- tions in which the coefficient matrix is block triangular. That is, T can be partitioned as A B T= , where A is r × r and C is n − r × n − r. (3.6.3) 0 C If x and b are similarly partitioned as x = x1 and b = b1 , then block x2 b2 multiplication shows that Tx = b reduces to two smaller systems Ax1 + Bx2 = b1 , Cx2 = b2 , so if all systems are consistent, a block version of back substitution is possible— i.e., solve Cx2 = b2 for x2 , and substituted this back into Ax1 = b1 − Bx2 , which is then solved for x1 . For obvious reasons, block-triangular systems of this type are sometimes referred to as reducible systems, and T is said to be a reducible matrix. Recall that applying Gaussian elimination with back substitution to an n × n system requires about n3 /3 multiplications/divisions and about n3 /3 additions/subtractions. This means that it's more efficient to solve two smaller subsystems than to solve one large main system. For exam- ple, suppose the matrix T in (3.6.3) is 100 × 100 while A and C are each 50 × 50. If Tx = b is solved without taking advantage of its reducibility, then about 106 /3 multiplications/divisions are needed. But by taking advantage of the reducibility, only about (250 × 103 )/3 multiplications/divisions are needed to solve both 50 × 50 subsystems. Another advantage of reducibility is realized when a computer's main memory capacity is not large enough to store the entire coefficient matrix but is large enough to hold the submatrices.Exercises for section 3.6 3.6.1. For the partitioned matrices −1 −1 1 0 0 3 3 3 0 0 1 0 0 3 3 3 0 0 A= and B= , −1 −2 1 2 2 0 0 0 −1 −2 −1 −2 use block multiplication with the indicated partitions to form the prod- uct AB.
3.6 Properties of Matrix Multiplication 113 3.6.2. For all matrices An×k and Bk×n , show that the block matrix I − BA B L= 2A − ABA AB − I has the property L2 = I. Matrices with this property are said to be involutory, and they occur in the science of cryptography. 3.6.3. For the matrix 1 0 0 1/3 1/3 1/3 0 1 0 1/3 1/3 1/3 0 0 1 1/3 1/3 1/3 A= , 0 0 0 1/3 1/3 1/3 0 0 0 1/3 1/3 1/3 0 0 0 1/3 1/3 1/3 determine A300 . Hint: A square matrix C is said to be idempotent when it has the property that C2 = C. Make use of idempotent sub- matrices in A. 3.6.4. For every matrix Am×n , demonstrate that the products A∗ A and AA∗ are hermitian matrices. 3.6.5. If A and B are symmetric matrices that commute, prove that the product AB is also symmetric. If AB = BA, is AB necessarily sym- metric? 3.6.6. Prove that the right-hand distributive property is true. 3.6.7. For each matrix An×n , explain why it is impossible to find a solution for Xn×n in the matrix equation AX − XA = I. Hint: Consider the trace function. 3.6.8. Let y1×m be a row of unknowns, and let Am×n and bT T 1×n be known matrices. (a) Explain why the matrix equation yT A = bT represents a sys- tem of n linear equations in m unknowns. (b) How are the solutions for yT in yT A = bT related to the solutions for x in AT x = b?
114 Chapter 3 Matrix Algebra 3.6.9. A particular electronic device consists of a collection of switching circuits that can be either in an ON state or an OFF state. These electronic switches are allowed to change state at regular time intervals called clock cycles. Suppose that at the end of each clock cycle, 30% of the switches currently in the OFF state change to ON, while 90% of those in the ON state revert to the OFF state. (a) Show that the device approaches an equilibrium in the sense that the proportion of switches in each state eventually becomes constant, and determine these equilibrium proportions. (b) Independent of the initial proportions, about how many clock cycles does it take for the device to become essentially stable? 3.6.10. Write the following system in the form Tn×n x = b, where T is block triangular, and then obtain the solution by solving two small systems as described in Example 3.6.7. x1 + x2 + 3x3 + 4x4 = − 1, 2x3 + 3x4 = 3, x1 + 2x2 + 5x3 + 6x4 = − 2, x3 + 2x4 = 4. 3.6.11. Prove that each of the following statements is true for conformable ma- trices. (a) trace (ABC) = trace (BCA) = trace (CAB). (b) trace (ABC) can be different from trace (BAC). (c) trace AT B = trace ABT . 3.6.12. Suppose that Am×n and xn×1 have real entries. (a) Prove that xT x = 0 if and only if x = 0. (b) Prove that trace AT A = 0 if and only if A = 0.
3.7 Matrix Inversion 1153.7 MATRIX INVERSION If α is a nonzero scalar, then for each number β the equation αx = β has a unique solution given by x = α−1 β. To prove that α−1 β is a solution, write α(α−1 β) = (αα−1 )β = (1)β = β. (3.7.1) Uniqueness follows because if x1 and x2 are two solutions, then αx1 = β = αx2 =⇒ α−1 (αx1 ) = α−1 (αx2 ) =⇒ (α−1 α)x1 = (α−1 α)x2 (3.7.2) =⇒ (1)x1 = (1)x2 =⇒ x1 = x2 . These observations seem pedantic, but they are important in order to see how to make the transition from scalar equations to matrix equations. In particular, these arguments show that in addition to associativity, the properties αα−1 = 1 and α−1 α = 1 (3.7.3) are the key ingredients, so if we want to solve matrix equations in the same fashion as we solve scalar equations, then a matrix analogue of (3.7.3) is needed. Matrix Inversion For a given square matrix An×n , the matrix Bn×n that satisfies the conditions AB = In and BA = In is called the inverse of A and is denoted by B = A−1 . Not all square matrices are invertible—the zero matrix is a trivial example, but there are also many nonzero matrices that are not invertible. An invertible matrix is said to be nonsingular, and a square matrix with no inverse is called a singular matrix. Notice that matrix inversion is defined for square matrices only—the con- dition AA−1 = A−1 A rules out inverses of nonsquare matrices.Example 3.7.1 If a b A= , where δ = ad − bc = 0, c d then 1 d −b A−1 = δ −c a because it can be verified that AA−1 = A−1 A = I2 .
116 Chapter 3 Matrix Algebra Although not all matrices are invertible, when an inverse exists, it is unique. To see this, suppose that X1 and X2 are both inverses for a nonsingular matrix A. Then X1 = X1 I = X1 (AX2 ) = (X1 A)X2 = IX2 = X2 , which implies that only one inverse is possible. Since matrix inversion was defined analogously to scalar inversion, and since matrix multiplication is associative, exactly the same reasoning used in (3.7.1) and (3.7.2) can be applied to a matrix equation AX = B, so we have the following statements. Matrix Equations • If A is a nonsingular matrix, then there is a unique solution for X in the matrix equation An×n Xn×p = Bn×p , and the solution is X = A−1 B. (3.7.4) • A system of n linear equations in n unknowns can be written as a single matrix equation An×n xn×1 = bn×1 (see p. 99), so it follows from (3.7.4) that when A is nonsingular, the system has a unique solution given by x = A−1 b. However, it must be stressed that the representation of the solution as x = A−1 b is mostly a notational or theoretical convenience. In practice, a nonsingular system Ax = b is almost never solved by first computing A−1 and then the product x = A−1 b. The reason will be apparent when we learn how much work is involved in computing A−1 . Since not all square matrices are invertible, methods are needed to distin- guish between nonsingular and singular matrices. There is a variety of ways to describe the class of nonsingular matrices, but those listed below are among the most important. Existence of an Inverse For an n × n matrix A, the following statements are equivalent. • A−1 exists (A is nonsingular). (3.7.5) • rank (A) = n. (3.7.6) Gauss–Jordan • A − − − −→ I. − − −− (3.7.7) • Ax = 0 implies that x = 0. (3.7.8)
3.7 Matrix Inversion 117 Proof. The fact that (3.7.6) ⇐⇒ (3.7.7) is a direct consequence of the defi- nition of rank, and (3.7.6) ⇐⇒ (3.7.8) was established in §2.4. Consequently, statements (3.7.6), (3.7.7), and (3.7.8) are equivalent, so if we establish that (3.7.5) ⇐⇒ (3.7.6), then the proof will be complete. Proof of (3.7.5) =⇒ (3.7.6). Begin by observing that (3.5.5) guarantees that a matrix X = [X∗1 | X∗2 | · · · | X∗n ] satisfies the equation AX = I if and only if X∗j is a solution of the linear system Ax = I∗j . If A is nonsingular, then we know from (3.7.4) that there exists a unique solution to AX = I, and hence each linear system Ax = I∗j has a unique solution. But in §2.5 we learned that a linear system has a unique solution if and only if the rank of the coefficient matrix equals the number of unknowns, so rank (A) = n. Proof of (3.7.6) =⇒ (3.7.5). If rank (A) = n, then (2.3.4) insures that each system Ax = I∗j is consistent because rank[A | I∗j ] = n = rank (A). Furthermore, the results of §2.5 guarantee that each system Ax = I∗j has a unique solution, and hence there is a unique solution to the matrix equation AX = I. We would like to say that X = A−1 , but we cannot jump to this conclusion without first arguing that XA = I. Suppose this is not true—i.e., suppose that XA − I = 0. Since A(XA − I) = (AX)A − A = IA − A = 0, it follows from (3.5.5) that any nonzero column of XA−I is a nontrivial solution of the homogeneous system Ax = 0. But this is a contradiction of the fact that (3.7.6) ⇐⇒ (3.7.8). Therefore, the supposition that XA − I = 0 must be false, and thus AX = I = XA, which means A is nonsingular. The definition of matrix inversion says that in order to compute A−1 , it is necessary to solve both of the matrix equations AX = I and XA = I. These two equations are necessary to rule out the possibility of nonsquare inverses. But when only square matrices are involved, then any one of the two equations will suffice—the following example elaborates.Example 3.7.2 Problem: If A and X are square matrices, explain why AX = I =⇒ XA = I. (3.7.9) −1 In other words, if A and X are square and AX = I, then X = A . Solution: Notice first that AX = I implies X is nonsingular because if X is singular, then, by (3.7.8), there is a column vector x = 0 such that Xx = 0, which is contrary to the fact that x = Ix = AXx = 0. Now that we know X−1 exists, we can establish (3.7.9) by writing AX = I =⇒ AXX−1 = X−1 =⇒ A = X−1 =⇒ XA = I. Caution! The argument above is not valid for nonsquare matrices. When m = n, it's possible that Am×n Xn×m = Im , but XA = In .
118 Chapter 3 Matrix Algebra Although we usually try to avoid computing the inverse of a matrix, there are times when an inverse must be found. To construct an algorithm that will yield A−1 when An×n is nonsingular, recall from Example 3.7.2 that determining A−1 is equivalent to solving the single matrix equation AX = I, and due to (3.5.5), this in turn is equivalent to solving the n linear systems defined by Ax = I∗j for j = 1, 2, . . . , n. (3.7.10) In other words, if X∗1 , X∗2 , . . . , X∗n are the respective solutions to (3.7.10), then X = [X∗1 | X∗2 | · · · | X∗n ] solves the equation AX = I, and hence X = A−1 . If A is nonsingular, then we know from (3.7.7) that the Gauss–Jordan method reduces the augmented matrix [A | I∗j ] to [I | X∗j ], and the results of §1.3 insure that X∗j is the unique solution to Ax = I∗j . That is, Gauss–Jordan [A | I∗j ] − − − −→ I [A−1 ]∗j . − − −− But rather than solving each system Ax = I∗j separately, we can solve them simultaneously by taking advantage of the fact that they all have the same coefficient matrix. In other words, applying the Gauss–Jordan method to the larger augmented array [A | I∗1 | I∗2 | · · · | I∗n ] produces Gauss–Jordan [A | I∗1 | I∗2 | · · · | I∗n ] − − − −→ I [A−1 ]∗1 [A−1 ]∗2 · · · [A−1 ]∗n , − − −− or more compactly, Gauss–Jordan [A | I] − − − −→ [I | A−1 ]. − − −− (3.7.11) What happens if we try to invert a singular matrix using this procedure? The fact that (3.7.5) ⇐⇒ (3.7.6) ⇐⇒ (3.7.7) guarantees that a singular matrix A cannot be reduced to I by Gauss–Jordan elimination because a zero row will have to emerge in the left-hand side of the augmented array at some point during the process. This means that we do not need to know at the outset whether A is nonsingular or singular—it becomes self-evident depending on whether or not the reduction (3.7.11) can be completed. A summary is given below. Computing an Inverse Gauss–Jordan elimination can be used to invert A by the reduction Gauss–Jordan [A | I] − − − −→ [I | A−1 ]. − − −− (3.7.12) The only way for this reduction to fail is for a row of zeros to emerge in the left-hand side of the augmented array, and this occurs if and only if A is a singular matrix. A different (and somewhat more practical) algorithm is given Example 3.10.3 on p. 148.
120 Chapter 3 Matrix Algebra Solving a nonsingular system Ax = b by first computing A−1 and then forming the product x = A−1 b requires n3 + n2 multiplications/divisions and n3 − n2 additions/subtractions. Recall from §1.5 that Gaussian elimination with back substitution requires only about n3 /3 multiplications/divisions and about n3 /3 additions/subtractions. In other words, using A−1 to solve a nonsingular system Ax = b requires about three times the effort as does Gaussian elimina- tion with back substitution. To put things in perspective, consider standard matrix multiplication be- tween two n × n matrices. It is not difficult to verify that n3 multiplications and n3 −n2 additions are required. Remarkably, it takes almost exactly as much effort to perform one matrix multiplication as to perform one matrix inversion. This fact always seems to be counter to a novice's intuition—it "feels" like ma- trix inversion should be a more difficult task than matrix multiplication, but this is not the case. The remainder of this section is devoted to a discussion of some of the important properties of matrix inversion. We begin with the four basic facts listed below. Properties of Matrix Inversion For nonsingular matrices A and B, the following properties hold. −1 • A−1 = A. (3.7.13) • The product AB is also nonsingular. (3.7.14) −1 −1 −1 • (AB) =B A (the reverse order law for inversion). (3.7.15) −1 T T −1 ∗ −1 • A = A and A−1 = (A∗ ) . (3.7.16) Proof. Property (3.7.13) follows directly from the definition of inversion. To prove (3.7.14) and (3.7.15), let X = B−1 A−1 and verify that (AB)X = I by writing (AB)X = (AB)B−1 A−1 = A(BB−1 )A−1 = A(I)A−1 = AA−1 = I. According to the discussion in Example 3.7.2, we are now guaranteed that X(AB) = I, and we need not bother to verify it. To prove property (3.7.16), let T X = A−1 and verify that AT X = I. Make use of the reverse order law for transposition to write T T AT X = AT A−1 = A−1 A = IT = I. −1 T Therefore, AT = X = A−1 . The proof of the conjugate transpose case is similar.
122 Chapter 3 Matrix Algebra 3.7.3. For a square matrix A, explain why each of the following statements must be true. (a) If A contains a zero row or a zero column, then A is singular. (b) If A contains two identical rows or two identical columns, then A is singular. (c) If one row (or column) is a multiple of another row (or column), then A must be singular. 3.7.4. Answer each of the following questions. (a) Under what conditions is a diagonal matrix nonsingular? De- scribe the structure of the inverse of a diagonal matrix. (b) Under what conditions is a triangular matrix nonsingular? De- scribe the structure of the inverse of a triangular matrix. 3.7.5. If A is nonsingular and symmetric, prove that A−1 is symmetric. 3.7.6. If A is a square matrix such that I − A is nonsingular, prove that A(I − A)−1 = (I − A)−1 A. 3.7.7. Prove that if A is m × n and B is n × m such that AB = Im and BA = In , then m = n. 3.7.8. If A, B, and A + B are each nonsingular, prove that −1 A(A + B)−1 B = B(A + B)−1 A = A−1 + B−1 . 3.7.9. Let S be a skew-symmetric matrix with real entries. (a) Prove that I − S is nonsingular. Hint: xT x = 0 =⇒ x = 0. (b) If A = (I + S)(I − S)−1 , show that A−1 = AT . 3.7.10. For matrices Ar×r , Bs×s , and Cr×s such that A and B are nonsin- gular, verify that each of the following is true. −1 A 0 A−1 0 (a) = 0 B 0 B−1 −1 A C A−1 −A−1 CB−1 (b) = 0 B 0 B−1
3.7 Matrix Inversion 123 Ar×r Cr×s 3.7.11. Consider the block matrix . When the indicated in- Rs×r Bs×s verses exist, the matrices defined by S = B − RA−1 C and T = A − CB−1 R 20 are called the Schur complements of A and B, respectively. (a) If A and S are both nonsingular, verify that −1 A C A−1 + A−1 CS−1 RA−1 −A−1 CS−1 = . R B −S−1 RA−1 S−1 (b) If B and T are nonsingular, verify that −1 A C T−1 −T−1 CB−1 = . R B −B RT−1 −1 B −1 + B−1 RT−1 CB−1 3.7.12. Suppose that A, B, C, and D are n × n matrices such that ABT and CDT are each symmetric and ADT − BCT = I. Prove that AT D − CT B = I. 20 This is named in honor of the German mathematician Issai Schur (1875–1941), who first studied matrices of this type. Schur was a student and collaborator of Ferdinand Georg Frobenius (p. 662). Schur and Frobenius were among the first to study matrix theory as a discipline unto itself, and each made great contributions to the subject. It was Emilie V. Haynsworth (1916–1987)—a mathematical granddaughter of Schur—who introduced the phrase "Schur complement" and developed several important aspects of the concept.
124 Chapter 3 Matrix Algebra3.8 INVERSES OF SUMS AND SENSITIVITY The reverse order law for inversion makes the inverse of a product easy to deal with, but the inverse of a sum is much more difficult. To begin with, (A + B)−1 may not exist even if A−1 and B−1 each exist. Moreover, if (A + B)−1 exists, then, with rare exceptions, (A + B)−1 = A−1 + B−1 . This doesn't even hold for scalars (i.e., 1 × 1 matrices), so it has no chance of holding in general. There is no useful general formula for (A+B)−1 , but there are some special sums for which something can be said. One of the most easily inverted sums is I + cdT in which c and d are n × 1 nonzero columns such that 1 + dT c = 0. It's straightforward to verify by direct multiplication that −1 cdT I + cdT =I− . (3.8.1) 1 + dT c If I is replaced by a nonsingular matrix A satisfying 1 + dT A−1 c = 0, then the reverse order law for inversion in conjunction with (3.8.1) yields −1 (A + cdT )−1 = A(I + A−1 cdT ) = (I + A−1 cdT )−1 A−1 A−1 cdT A−1 cdT A−1 = I− A−1 = A−1 − . 1 + dT A−1 c 1 + dT A−1 c 21 This is often called the Sherman–Morrison rank-one update formula because it can be shown (Exercise 3.9.9, p. 140) that rank (cdT ) = 1 when c = 0 = d. Sherman–Morrison Formula • If An×n is nonsingular and if c and d are n × 1 columns such that 1 + dT A−1 c = 0, then the sum A + cdT is nonsingular, and −1 A−1 cdT A−1 A + cdT = A−1 − . (3.8.2) 1 + dT A−1 c • The Sherman–Morrison–Woodbury formula is a generalization. If C and D are n × k such that (I + DT A−1 C)−1 exists, then (A + CDT )−1 = A−1 − A−1 C(I + DT A−1 C)−1 DT A−1 . (3.8.3) 21 This result appeared in the 1949–1950 work of American statisticians J. Sherman and W. J. Morrison, but they were not the first to discover it. The formula was independently presented by the English mathematician W. J. Duncan in 1944 and by American statisticians L. Guttman (1946), Max Woodbury (1950), and M. S. Bartlett (1951). Since its derivation is so natural, it almost certainly was discovered by many others along the way. Recognition and fame are often not afforded simply for introducing an idea, but rather for applying the idea to a useful end.
126 Chapter 3 Matrix Algebra Another sum that often requires inversion is I − A, but we have to be careful because (I − A)−1 need not always exist. However, we are safe when the entries in A are sufficiently small. In particular, if the entries in A are small enough in magnitude to insure that limn→∞ An = 0, then, analogous to scalar algebra, (I − A)(I + A + A2 + · · · + An−1 ) = I − An → I as n → ∞, so we have the following matrix version of a geometric series. Neumann Series If limn→∞ An = 0, then I − A is nonsingular and ∞ (I − A)−1 = I + A + A2 + · · · = Ak . (3.8.5) k=0 This is the Neumann series. It provides approximations of (I − A)−1 when A has entries of small magnitude. For example, a first-order ap- proximation is (I − A)−1 ≈ I+A. More on the Neumann series appears in Example 7.3.1, p. 527, and the complete statement is developed on p. 618. While there is no useful formula for (A + B)−1 in general, the Neumann series allows us to say something when B has small entries relative to A, or vice versa. For example, if A−1 exists, and if the entries in B are small enough n in magnitude to insure that limn→∞ A−1 B = 0, then −1 −1 (A + B)−1 = A I − −A−1 B = I − −A−1 B A−1 ∞ k = −A−1 B A−1 , k=0 and a first-order approximation is (A + B)−1 ≈ A−1 − A−1 BA−1 . (3.8.6) Consequently, if A is perturbed by a small matrix B, possibly resulting from errors due to inexact measurements or perhaps from roundoff error, then the resulting change in A−1 is about A−1 BA−1 . In other words, the effect of a small perturbation (or error) B is magnified by multiplication (on both sides) with A−1 , so if A−1 has large entries, small perturbations (or errors) in A can produce large perturbations (or errors) in the resulting inverse. You can reach
3.8 Inverses of Sums and Sensitivity 127 essentially the same conclusion from (3.8.4) when only a single entry is perturbed and from Exercise 3.8.2 when a single column is perturbed. This discussion resolves, at least in part, an issue raised in §1.6—namely, "What mechanism determines the extent to which a nonsingular system Ax = b is ill-conditioned?" To see how, an aggregate measure of the magnitude of the entries in A is needed, and one common measure is A = max |aij | = the maximum absolute row sum. (3.8.7) i j This is one example of a matrix norm, a detailed discussion of which is given in §5.1. Theoretical properties specific to (3.8.7) are developed on pp. 280 and 283, and one property established there is the fact that XY ≤ X Y for all conformable matrices X and Y. But let's keep things on an intuitive level for the time being and defer the details. Using the norm (3.8.7), the approximation (3.8.6) insures that if B is sufficiently small, then A−1 − (A + B)−1 ≈ A−1 BA−1 ≤ A−1 B A−1 , < so, if we interpret x ∼ y to mean that x is bounded above by something not far from y, we can write A−1 − (A + B)−1 < −1 B ∼ A B = A−1 A . A−1 A The term on the left is the relative change in the inverse, and B / A is the relative change in A. The number κ = A−1 A is therefore the "magnifi- cation factor" that dictates how much the relative change in A is magnified. This magnification factor κ is called a condition number for A. In other words, if κ is small relative to 1 (i.e., if A is well conditioned), then a small relative change (or error) in A cannot produce a large relative change (or error) in the inverse, but if κ is large (i.e., if A is ill conditioned), then a small rela- tive change (or error) in A can possibly (but not necessarily) result in a large relative change (or error) in the inverse. The situation for linear systems is similar. If the coefficients in a nonsingular system Ax = b are slightly perturbed to produce the system (A + B)˜ = b, x then x = A−1 b and x = (A + B)−1 b so that (3.8.6) implies ˜ x − x = A−1 b − (A + B)−1 b ≈ A−1 b − A−1 − A−1 BA−1 b = A−1 Bx. ˜ For column vectors, (3.8.7) reduces to x = maxi |xi |, and we have x − x ∼ A−1 ˜ < B x ,
128 Chapter 3 Matrix Algebra so the relative change in the solution is x−x < ˜ −1 B B ∼ A B = A−1 A =κ . (3.8.8) x A A Again, the condition number κ is pivotal because when κ is small, a small relative change in A cannot produce a large relative change in x, but for larger values of κ, a small relative change in A can possibly result in a large relative change in x. Below is a summary of these observations. Sensitivity and Conditioning • A nonsingular matrix A is said to be ill conditioned if a small relative change in A can cause a large relative change in A−1 . The degree of ill-conditioning is gauged by a condition number κ = A A−1 , where is a matrix norm. • The sensitivity of the solution of Ax = b to perturbations (or errors) in A is measured by the extent to which A is an ill- conditioned matrix. More is said in Example 5.12.1 on p. 414.Example 3.8.2 It was demonstrated in Example 1.6.1 that the system .835x + .667y = .168, .333x + .266y = .067, is sensitive to small perturbations. We can understand this in the current context by examining the condition number of the coefficient matrix. If the matrix norm (3.8.7) is employed with .835 .667 −266000 667000 A= and A−1 = , .333 .266 333000 −835000 then the condition number for A is κ=κ= A A−1 = (1.502)(1168000) = 1, 754, 336 ≈ 1.7 × 106 . Since the right-hand side of (3.8.8) is only an estimate of the relative error in the solution, the exact value of κ is not as important as its order of magnitude. Because κ is of order 106 , (3.8.8) holds the possibility that the relative change (or error) in the solution can be about a million times larger than the relative
3.8 Inverses of Sums and Sensitivity 129 change (or error) in A. Therefore, we must consider A and the associated linear system to be ill conditioned. A Rule of Thumb. If Gaussian elimination with partial pivoting is used to solve a well-scaled nonsingular system Ax = b using t -digit floating-point arithmetic, then, assuming no other source of error exists, it can be argued that when κ is of order 10p , the computed solution is expected to be accurate to at least t − p significant digits, more or less. In other words, one expects to lose roughly p significant figures. For example, if Gaussian elimination with 8- digit arithmetic is used to solve the 2 × 2 system given above, then only about t − p = 8 − 6 = 2 significant figures of accuracy should be expected. This doesn't preclude the possibility of getting lucky and attaining a higher degree of accuracy—it just says that you shouldn't bet the farm on it. The complete story of conditioning has not yet been told. As pointed out ear- lier, it's about three times more costly to compute A−1 than to solve Ax = b, so it doesn't make sense to compute A−1 just to estimate the condition of A. Questions concerning condition estimation without explicitly computing an in- verse still need to be addressed. Furthermore, liberties allowed by using the ≈ < and ∼ symbols produce results that are intuitively correct but not rigorous. Rigor will eventually be attained—see Example 5.12.1on p. 414.Exercises for section 3.8 3.8.1. Suppose you are given that 2 0 −1 1 0 1 A = −1 1 1 and A−1 = 0 1 −1 . −1 0 1 1 0 2 (a) Use the Sherman–Morrison formula to determine the inverse of the matrix B that is obtained by changing the (3, 2)-entry in A from 0 to 2. (b) Let C be the matrix that agrees with A except that c32 = 2 and c33 = 2. Use the Sherman–Morrison formula to find C−1 . 3.8.2. Suppose A and B are nonsingular matrices in which B is obtained from A by replacing A∗j with another column b. Use the Sherman– Morrison formula to derive the fact that A−1 b − ej [A−1 ]j∗ B−1 = A−1 − . [A−1 ]j∗ b
130 Chapter 3 Matrix Algebra 3.8.3. Suppose the coefficient matrix of a nonsingular system Ax = b is up- dated to produce another nonsingular system (A + cdT )z = b, where b, c, d ∈ n×1 , and let y be the solution of Ay = c. Show that z = x − ydT x/(1 + dT y). 3.8.4. (a) Use the Sherman–Morrison formula to prove that if A is non- singular, then A + αei eT is nonsingular for a sufficiently small j α. (b) Use part (a) to prove that I + E is nonsingular when all ij 's are sufficiently small in magnitude. This is an alternative to using the Neumann series argument. 3.8.5. For given matrices A and B, where A is nonsingular, explain why A + B is also nonsingular when the real number is constrained to a sufficiently small interval about the origin. In other words, prove that small perturbations of nonsingular matrices are also nonsingular. 3.8.6. Derive the Sherman–Morrison–Woodbury formula. Hint: Recall Exer- cise 3.7.11, and consider the product 0 C I I A DT −I C I DT I 0 . 3.8.7. Using the norm (3.8.7), rank the following matrices according to their degree of ill-conditioning: 100 0 −100 1 8 −1 A= 0 100 −100 , B = −9 −71 11 , −100 −100 300 1 17 18 1 22 −42 C= 0 1 −45 . −45 −948 1 3.8.8. Suppose that the entries in A(t), x(t), and b(t) are differentiable functions of a real variable t such that A(t)x(t) = b(t). (a) Assuming that A(t)−1 exists, explain why dA(t)−1 = −A(t)−1 A (t)A(t)−1 . dt (b) Derive the equation x (t) = A(t)−1 b (t) − A(t)−1 A (t)x(t). This shows that A−1 magnifies both the change in A and the change in b, and thus it confirms the observation derived from (3.8.8) saying that the sensitivity of a nonsingular system to small perturbations is directly related to the magnitude of the entries in A−1 .
3.9 Elementary Matrices and Equivalence 1313.9 ELEMENTARY MATRICES AND EQUIVALENCE A common theme in mathematics is to break complicated objects into more elementary components, such as factoring large polynomials into products of smaller polynomials. The purpose of this section is to lay the groundwork for similar ideas in matrix algebra by considering how a general matrix might be factored into a product of more "elementary" matrices. Elementary Matrices Matrices of the form I − uvT , where u and v are n × 1 columns such that vT u = 1 are called elementary matrices, and we know from (3.8.1) that all such matrices are nonsingular and −1 uvT I − uvT =I− . (3.9.1) vT u − 1 Notice that inverses of elementary matrices are elementary matrices. We are primarily interested in the elementary matrices associated with the three elementary row (or column) operations hereafter referred to as follows. • Type I is interchanging rows (columns) i and j. • Type II is multiplying row (column) i by α = 0. • Type III is adding a multiple of row (column) i to row (column) j. An elementary matrix of Type I, II, or III is created by performing an elementary operation of Type I, II, or III to an identity matrix. For example, the matrices 0 1 0 1 0 0 1 0 0 E1 = 1 0 0 , E2 = 0 α 0 , and E3 = 0 1 0 (3.9.2) 0 0 1 0 0 1 α 0 1 are elementary matrices of Types I, II, and III, respectively, because E1 arises by interchanging rows 1 and 2 in I3 , whereas E2 is generated by multiplying row 2 in I3 by α, and E3 is constructed by multiplying row 1 in I3 by α and adding the result to row 3. The matrices in (3.9.2) also can be generated by column operations. For example, E3 can be obtained by adding α times the third column of I3 to the first column. The fact that E1 , E2 , and E3 are of the form (3.9.1) follows by using the unit columns ei to write E1 = I−uuT , where u = e1 −e2 , E2 = I−(1−α)e2 eT , 2 and E3 = I+αe3 eT . 1
132 Chapter 3 Matrix Algebra These observations generalize to matrices of arbitrary size. One of our objectives is to remove the arrows from Gaussian elimination because the inability to do "arrow algebra" limits the theoretical analysis. For example, while it makes sense to add two equations together, there is no mean- ingful analog for arrows—reducing A → B and C → D by row operations does not guarantee that A + C → B + D is possible. The following properties are the mechanisms needed to remove the arrows from elimination processes. Properties of Elementary Matrices • When used as a left-hand multiplier, an elementary matrix of Type I, II, or III executes the corresponding row operation. • When used as a right-hand multiplier, an elementary matrix of Type I, II, or III executes the corresponding column operation. Proof. A proof for Type III operations is given—the other two cases are left to the reader. Using I + αej eT as a left-hand multiplier on an arbitrary matrix A i produces 0 0 ··· 0 . . . . . . . . . a · · · ain ← j th row . I + αej eT A = A + αej Ai∗ = A + α i1 ai2 i . . . . . . . . . 0 0 ··· 0 This is exactly the matrix produced by a Type III row operation in which the ith row of A is multiplied by α and added to the j th row. When I + αej eT i is used as a right-hand multiplier on A, the result is ith col ↓ 0 ··· a1j ··· 0 0 ··· a2j ··· 0 A I+ αej eT i =A+ αA∗j eT i =A+α . . . . . . . . . . 0 ··· anj ··· 0 This is the result of a Type III column operation in which the j th column of A is multiplied by α and then added to the ith column.
134 Chapter 3 Matrix Algebra Equivalence • Whenever B can be derived from A by a combination of elementary row and column operations, we write A ∼ B, and we say that A and B are equivalent matrices. Since elementary row and column operations are left-hand and right-hand multiplication by elementary matrices, respectively, and in view of (3.9.3), we can say that A ∼ B ⇐⇒ PAQ = B for nonsingular P and Q. • Whenever B can be obtained from A by performing a sequence row of elementary row operations only, we write A ∼ B, and we say that A and B are row equivalent. In other words, row A ∼ B ⇐⇒ PA = B for a nonsingular P. • Whenever B can be obtained from A by performing a sequence of col column operations only, we write A ∼ B, and we say that A and B are column equivalent. In other words, col A ∼ B ⇐⇒ AQ = B for a nonsingular Q. If it's possible to go from A to B by elementary row and column oper- ations, then clearly it's possible to start with B and get back to A because elementary operations are reversible—i.e., PAQ = B =⇒ P−1 BQ−1 = A. It therefore makes sense to talk about the equivalence of a pair of matrices without regard to order. In other words, A ∼ B ⇐⇒ B ∼ A. Furthermore, it's not difficult to see that each type of equivalence is transitive in the sense that A∼B and B ∼ C =⇒ A ∼ C. In §2.2 it was stated that each matrix A possesses a unique reduced row echelon form EA , and we accepted this fact because it is intuitively evident. However, we are now in a position to understand a rigorous proof.Example 3.9.2 Problem: Prove that EA is uniquely determined by A. Solution: Without loss of generality, we may assume that A is square— otherwise the appropriate number of zero rows or columns can be adjoined to A row row without affecting the results. Suppose that A ∼ E1 and A ∼ E2 , where E1 row and E2 are both in reduced row echelon form. Consequently, E1 ∼ E2 , and hence there is a nonsingular matrix P such that PE1 = E2 . (3.9.4)
3.9 Elementary Matrices and Equivalence 135 Furthermore, by permuting the rows of E1 and E2 to force the pivotal 1's to occupy the diagonal positions, we see that row row E1 ∼ T1 and E2 ∼ T2 , (3.9.5) where T1 and T2 are upper-triangular matrices in which the basic columns in each Ti occupy the same positions as the basic columns in Ei . For example, if 1 2 0 1 2 0 E = 0 0 1 , then T = 0 0 0 . 0 0 0 0 0 1 Each Ti has the property that T2 = Ti because there is a permutation i matrix Qi (a product of elementary interchange matrices of Type I) such that Ir i Ji Ir i Ji Qi Ti QT = i or, equivalently, Ti = QT i Qi , 0 0 0 0 and QT = Q−1 (see Exercise 3.9.4) implies T2 = Ti . It follows from (3.9.5) i i i row that T1 ∼ T2 , so there is a nonsingular matrix R such that RT1 = T2 . Thus T2 = RT1 = RT1 T1 = T2 T1 and T1 = R−1 T2 = R−1 T2 T2 = T1 T2 . Because T1 and T2 are both upper triangular, T1 T2 and T2 T1 have the same diagonal entries, and hence T1 and T2 have the same diagonal. Therefore, the positions of the basic columns (i.e., the pivotal positions) in T1 agree with those in T2 , and hence E1 and E2 have basic columns in exactly the same positions. This means there is a permutation matrix Q such that Ir J1 Ir J2 E1 Q = and E2 Q = . 0 0 0 0 Using (3.9.4) yields PE1 Q = E2 Q, or P11 P12 Ir J1 Ir J2 = , P21 P22 0 0 0 0 which in turn implies that P11 = Ir and P11 J1 = J2 . Consequently, J1 = J2 , and it follows that E1 = E2 . In passing, notice that the uniqueness of EA implies the uniqueness of the row pivot positions in any other row echelon form derived from A. If A ∼ U1 row and A ∼ U2 , where U1 and U2 are row echelon forms with different pivot positions, then Gauss–Jordan reduction applied to U1 and U2 would lead to two different reduced echelon forms, which is impossible. In §2.2 we observed the fact that the column relationships in a matrix A are exactly the same as the column relationships in EA . This observation is a special case of the more general result presented below.
136 Chapter 3 Matrix Algebra Column and Row Relationships row • If A ∼ B, then linear relationships existing among columns of A also hold among corresponding columns of B. That is, n n B∗k = αj B∗j if and only if A∗k = αj A∗j . (3.9.6) j=1 j=1 • In particular, the column relationships in A and EA must be iden- tical, so the nonbasic columns in A must be linear combinations of the basic columns in A as described in (2.2.3). col • If A ∼ B, then linear relationships existing among rows of A must also hold among corresponding rows of B. • Summary. Row equivalence preserves column relationships, and col- umn equivalence preserves row relationships. row Proof. If A ∼ B, then PA = B for some nonsingular P. Recall from (3.5.5) that the j th column in B is given by B∗j = (PA)∗j = PA∗j . Therefore, if A∗k = j αj A∗j , then multiplication by P on the left produces B∗k = j αj B∗j . Conversely, if B∗k = j αj B∗j , then multiplication on the left by P−1 produces A∗k = j αj A∗j . The statement concerning column equivalence follows by considering transposes. The reduced row echelon form EA is as far as we can go in reducing A by using only row operations. However, if we are allowed to use row operations in conjunction with column operations, then, as described below, the end result of a complete reduction is much simpler. Rank Normal Form If A is an m × n matrix such that rank (A) = r, then Ir 0 A ∼ Nr = . (3.9.7) 0 0 Nr is called the rank normal form for A, and it is the end product of a complete reduction of A by using both row and column operations.
3.9 Elementary Matrices and Equivalence 137 row Proof. It is always true that A ∼ EA so that there is a nonsingular matrix P such that PA = EA . If rank (A) = r, then the basic columns in EA are the r unit columns. Apply column interchanges to EA so as to move these r unit columns to the far left-hand side. If Q1 is the product of the elementary matrices corresponding to these column interchanges, then PAQ1 has the form Ir J PAQ1 = EA Q1 = . 0 0 Multiplying both sides of this equation on the right by the nonsingular matrix Ir −J Ir J Ir −J Ir 0 Q2 = produces PAQ1 Q2 = = . 0 I 0 0 0 I 0 0 Thus A ∼ Nr . because P and Q = Q1 Q2 are nonsingular.Example 3.9.3 Problem: Explain why rank A B 0 0 = rank (A) + rank (B). Solution: If rank (A) = r and rank (B) = s, then A ∼ Nr and B ∼ Ns . Consequently, A 0 Nr 0 A 0 Nr 0 ∼ =⇒ rank = rank = r + s. 0 B 0 Ns 0 B 0 Ns Given matrices A and B, how do we decide whether or not A ∼ B, row col A ∼ B, or A ∼ B? We could use a trial-and-error approach by attempting to reduce A to B by elementary operations, but this would be silly because there are easy tests, as described below. Testing for Equivalence For m × n matrices A and B the following statements are true. • A ∼ B if and only if rank (A) = rank (B). (3.9.8) row • A ∼ B if and only if EA = EB . (3.9.9) col • A ∼ B if and only if EAT = EBT . (3.9.10) Corollary. Multiplication by nonsingular matrices cannot change rank.
138 Chapter 3 Matrix Algebra Proof. To establish the validity of (3.9.8), observe that rank (A) = rank (B) implies A ∼ Nr and B ∼ Nr . Therefore, A ∼ Nr ∼ B. Conversely, if A ∼ B, where rank (A) = r and rank (B) = s, then A ∼ Nr and B ∼ Ns , and hence Nr ∼ A ∼ B ∼ Ns . Clearly, Nr ∼ Ns implies r = s. To prove (3.9.9), row row row suppose first that A ∼ B. Because B ∼ EB , it follows that A ∼ EB . Since a matrix has a uniquely determined reduced echelon form, it must be the case that EB = EA . Conversely, if EA = EB , then row row row A ∼ EA = EB ∼ B =⇒ A ∼ B. The proof of (3.9.10) follows from (3.9.9) by considering transposes because col T A ∼ B ⇐⇒ AQ = B ⇐⇒ (AQ) = BT row ⇐⇒ QT AT = BT ⇐⇒ AT ∼ BT .Example 3.9.4 Problem: Are the relationships that exist among the columns in A the same as the column relationships in B, and are the row relationships in A the same as the row relationships in B, where 1 1 1 −1 −1 −1 A = −4 −3 −1 and B = 2 2 2 ? 2 1 −1 2 1 −1 Solution: Straightforward computation reveals that 1 0 −2 EA = EB = 0 1 3, 0 0 0 row and hence A ∼ B. Therefore, the column relationships in A and B must be identical, and they must be the same as those in EA . Examining EA reveals that E∗3 = −2E∗1 + 3E∗2 , so it must be the case that A∗3 = −2A∗1 + 3A∗2 and B∗3 = −2B∗1 + 3B∗2 . The row relationships in A and B are different because EAT = EBT . On the surface, it may not seem plausible that a matrix and its transpose should have the same rank. After all, if A is 3 × 100, then A can have as many as 100 basic columns, but AT can have at most three. Nevertheless, we can now demonstrate that rank (A) = rank AT .
144 Chapter 3 Matrix Algebra where 1 0 0 ··· 0 21 1 0 ··· 0 L=I+ c 1 eT + c2 eT + ··· + cn−1 eT = 31 32 1 · · · 0 (3.10.6) 1 2 n−1 . . . . . . . . . . .. . . . n1 n2 n3 ··· 1 is the lower-triangular matrix with 1's on the diagonal, and where ij is precisely the multiplier used to annihilate the (i, j) -position during Gaussian elimination. Thus the factorization A = LU can be viewed as the matrix formulation of Gaussian elimination, with the understanding that no row interchanges are used. LU Factorization If A is an n × n matrix such that a zero pivot is never encountered when applying Gaussian elimination with Type III operations, then A can be factored as the product A = LU, where the following hold. • L is lower triangular and U is upper triangular. (3.10.7) • ii = 1 and uii = 0 for each i = 1, 2, . . . , n. (3.10.8) • Below the diagonal of L, the entry ij is the multiple of row j that is subtracted from row i in order to annihilate the (i, j) -position during Gaussian elimination. • U is the final result of Gaussian elimination applied to A. • The matrices L and U are uniquely determined by properties (3.10.7) and (3.10.8). The decomposition of A into A = LU is called the LU factorization of A, and the matrices L and U are called the LU factors of A. Proof. Except for the statement concerning the uniqueness of the LU fac- tors, each point has already been established. To prove uniqueness, observe that LU factors must be nonsingular because they have nonzero diagonals. If L1 U1 = A = L2 U2 are two LU factorizations for A, then L−1 L1 = U2 U−1 . 2 1 (3.10.9) Notice that L−1 L1 is lower triangular, while U2 U−1 is upper triangular be- 2 1 cause the inverse of a matrix that is upper (lower) triangular is again up- per (lower) triangular, and because the product of two upper (lower) trian- gular matrices is also upper (lower) triangular. Consequently, (3.10.9) implies L−1 L1 = D = U2 U−1 must be a diagonal matrix. However, [L2 ]ii = 1 = 2 1 [L−1 ]ii , so it must be the case that L−1 L1 = I = U2 U−1 , and thus L1 = L2 2 2 1 and U1 = U2 .
146 Chapter 3 Matrix Algebra set y 1 = b1 , y2 = b 2 − 21 y1 , y3 = b3 − 31 y1 − 32 y2 , etc. The forward substitution algorithm can be written more concisely as i−1 y 1 = b1 and y i = bi − ik yk for i = 2, 3, . . . , n. (3.10.10) k=1 After y is known, the upper-triangular system Ux = y is solved using the standard back substitution procedure by starting with xn = yn /unn , and setting n 1 xi = yi − uik xk for i = n − 1, n − 2, . . . , 1. (3.10.11) uii k=i+1 It can be verified that only n2 multiplications/divisions and n2 − n addi- tions/subtractions are required when (3.10.10) and (3.10.11) are used to solve the two triangular systems Ly = b and Ux = y, so it's relatively cheap to solve Ax = b once L and U are known—recall from §1.2 that these operation counts are about n3 /3 when we start from scratch. If only one system Ax = b is to be solved, then there is no significant difference between the technique of reducing the augmented matrix [A|b] to a row echelon form and the LU factorization method presented here. However, ˜ suppose it becomes necessary to later solve other systems Ax = b with the same coefficient matrix but with different right-hand sides, which is frequently the case in applied work. If the LU factors of A were computed and saved when the original system was solved, then they need not be recomputed, and ˜ the solutions to all subsequent systems Ax = b are therefore relatively cheap to obtain. That is, the operation counts for each subsequent system are on the order of n2 , whereas these counts would be on the order of n3 /3 if we would start from scratch each time. Summary • To solve a nonsingular system Ax = b using the LU factorization A = LU, first solve Ly = b for y with the forward substitution algorithm (3.10.10), and then solve Ux = y for x with the back substitution procedure (3.10.11). • The advantage of this approach is that once the LU factors for ˜ A have been computed, any other linear system Ax = b can be solved with only n multiplications/divisions and n − n ad- 2 2 ditions/subtractions.
148 Chapter 3 Matrix AlgebraExample 3.10.3 Computing A−1 . Although matrix inversion is not used for solving Ax = b, there are a few applications where explicit knowledge of A−1 is desirable. Problem: Explain how to use the LU factors of a nonsingular matrix An×n to compute A−1 efficiently. Solution: The strategy is to solve the matrix equation AX = I. Recall from (3.5.5) that AA−1 = I implies A[A−1 ]∗j = ej , so the j th column of A−1 is the solution of a system Axj = ej . Each of these n systems has the same coefficient matrix, so, once the LU factors for A are known, each system Axj = LUxj = ej can be solved by the standard two-step process. (1) Set yj = Uxj , and solve Lyj = ej for yj by forward substitution. (2) Solve Uxj = yj for xj = [A−1 ]∗j by back substitution. This method has at least two advantages: it's efficient, and any code written to solve Ax = b can also be used to compute A−1 . Note: A tempting alternate solution might be to use the fact A−1 = (LU)−1 = U−1 L−1 . But computing U−1 and L−1 explicitly and then multiplying the results is not as computationally efficient as the method just described. Not all nonsingular matrices possess an LU factorization. For example, there is clearly no nonzero value of u11 that will satisfy 0 1 1 0 u11 u12 = . 1 1 21 1 0 u22 The problem here is the zero pivot in the (1,1)-position. Our development of the LU factorization using elementary lower-triangular matrices shows that if no zero pivots emerge, then no row interchanges are necessary, and the LU factor- ization can indeed be carried to completion. The converse is also true (its proof is left as an exercise), so we can say that a nonsingular matrix A has an LU factorization if and only if a zero pivot does not emerge during row reduction to upper-triangular form with Type III operations. Although it is a bit more theoretical, there is another interesting way to characterize the existence of LU factors. This characterization is given in terms of the leading principal submatrices of A that are defined to be those submatrices taken from the upper-left-hand corner of A. That is, a11 a12 · · · a1k a11 a12 a21 a22 · · · a2k A1 = a11 , A2 = , . . . , Ak = . . . . .. . ,.... . a21 a22 . . . . ak1 ak2 · · · akk
3.10 The LU Factorization 149 Existence of LU Factors Each of the following statements is equivalent to saying that a nonsin- gular matrix An×n possesses an LU factorization. • A zero pivot does not emerge during row reduction to upper- triangular form with Type III operations. • Each leading principal submatrix Ak is nonsingular. (3.10.12) Proof. We will prove the statement concerning the leading principal submatri- ces and leave the proof concerning the nonzero pivots as an exercise. Assume first that A possesses an LU factorization and partition A as L11 0 U11 U12 L11 U11 ∗ A = LU = = , L21 L22 0 U22 ∗ ∗ where L11 and U11 are each k × k. Thus Ak = L11 U11 must be nonsingular because L11 and U11 are each nonsingular—they are triangular with nonzero diagonal entries. Conversely, suppose that each leading principal submatrix in A is nonsingular. Use induction to prove that each Ak possesses an LU fac- torization. For k = 1, this statement is clearly true because if A1 = (a11 ) is nonsingular, then A1 = (1)(a11 ) is its LU factorization. Now assume that Ak has an LU factorization and show that this together with the nonsingularity condition implies Ak+1 must also possess an LU factorization. If Ak = Lk Uk is the LU factorization for Ak , then A−1 = U−1 L−1 so that k k k Ak b Lk 0 Uk L−1 b k Ak+1 = = , (3.10.13) cT αk+1 cT U−1 k 1 0 αk+1 − cT A−1 b k where cT and b contain the first k components of Ak+1∗ and A∗k+1 , re- spectively. Observe that this is the LU factorization for Ak+1 because Lk 0 Uk L−1 b k Lk+1 = and Uk+1 = cT U−1 k 1 0 αk+1 − cT A−1 b k are lower- and upper-triangular matrices, respectively, and L has 1's on its diagonal while the diagonal entries of U are nonzero. The fact that αk+1 − cT A−1 b = 0 k follows because Ak+1 and Lk+1 are each nonsingular, so Uk+1 = L−1 Ak+1 k+1 must also be nonsingular. Therefore, the nonsingularity of the leading principal
150 Chapter 3 Matrix Algebra submatrices implies that each Ak possesses an LU factorization, and hence An = A must have an LU factorization. Up to this point we have avoided dealing with row interchanges because if a row interchange is needed to remove a zero pivot, then no LU factorization is possible. However, we know from the discussion in §1.5 that practical computa- tion necessitates row interchanges in the form of partial pivoting. So even if no zero pivots emerge, it is usually the case that we must still somehow account for row interchanges. To understand the effects of row interchanges in the framework of an LU decomposition, let Tk = I − ck eT be an elementary lower-triangular matrix as k described in (3.10.2), and let E = I − uuT with u = ek+i − ek+j be the Type I elementary interchange matrix associated with an interchange of rows k + i and k + j. Notice that eT E = eT because eT has 0's in positions k + i and k + j. k k k This together with the fact that E2 = I guarantees ETk E = E2 − Eck eT E = I − ck eT , k ˜ k where ˜ ck = Eck . In other words, the matrix ˜ Tk = ETk E = I − ck eT ˜ k (3.10.14) ˜ is also an elementary lower-triangular matrix, and Tk agrees with Tk in all positions except that the multipliers µk+i and µk+j have traded places. As be- fore, assume we are row reducing an n × n nonsingular matrix A, but suppose that an interchange of rows k + i and k + j is necessary immediately after the k th stage so that the sequence of left-hand multiplications ETk Tk−1 · · · T1 is applied to A. Since E2 = I, we may insert E2 to the right of each T to obtain ETk Tk−1 · · · T1 = ETk E2 Tk−1 E2 · · · E2 T1 E2 = (ETk E) (ETk−1 E) · · · (ET1 E) E ˜ ˜ ˜ = Tk Tk−1 · · · T1 E. In such a manner, the necessary interchange matrices E can be "factored" to ˜ the far-right-hand side, and the matrices T retain the desirable feature of be- ing elementary lower-triangular matrices. Furthermore, (3.10.14) implies that ˜ ˜ ˜ Tk Tk−1 · · · T1 differs from Tk Tk−1 · · · T1 only in the sense that the multipli- ers in rows k + i and k + j have traded places. Therefore, row interchanges in ˜ ˜ ˜ Gaussian elimination can be accounted for by writing Tn−1 · · · T2 T1 PA = U, where P is the product of all elementary interchange matrices used during the ˜ reduction and where the Tk 's are elementary lower-triangular matrices in which the multipliers have been permuted according to the row interchanges that were ˜ implemented. Since all of the Tk 's are elementary lower-triangular matrices, we may proceed along the same lines discussed in (3.10.4)—(3.10.6) to obtain PA = LU, where L = T−1 T−1 · · · T−1 . ˜ ˜ 1 2 ˜ n−1 (3.10.15)
3.10 The LU Factorization 151 When row interchanges are allowed, zero pivots can always be avoided when the original matrix A is nonsingular. Consequently, we may conclude that for every nonsingular matrix A, there exists a permutation matrix P (a product of elementary interchange matrices) such that PA has an LU factorization. Fur- thermore, because of the observation in (3.10.14) concerning how the multipliers ˜ in Tk and Tk trade places when a row interchange occurs, and because −1 T−1 = I − ck eT ˜ k ˜ k = I + ck eT , ˜ k it is not difficult to see that the same line of reasoning used to arrive at (3.10.6) can be applied to conclude that the multipliers in the matrix L in (3.10.15) are permuted according to the row interchanges that are executed. More specifically, if rows k and k+i are interchanged to create the k th pivot, then the multipliers ( k1 k2 ··· k,k−1 ) and ( k+i,1 k+i,2 ··· k+i,k−1 ) trade places in the formation of L. This means that we can proceed just as in the case when no interchanges are used and successively overwrite the array originally containing A with each mul- tiplier replacing the position it annihilates. Whenever a row interchange occurs, the corresponding multipliers will be correctly interchanged as well. The per- mutation matrix P is simply the cumulative record of the various interchanges used, and the information in P is easily accounted for by a simple technique that is illustrated in the following example.Example 3.10.4 Problem: Use partial pivoting on the matrix 1 2 −3 4 4 8 12 −8 A= 2 3 2 1 −3 −1 1 −4 and determine the LU decomposition PA = LU, where P is the associated permutation matrix. Solution: As explained earlier, the strategy is to successively overwrite the array A with components from L and U. For the sake of clarity, the multipliers ij are shown in boldface type. Adjoin a "permutation counter column" p that is initially set to the natural order 1,2,3,4. Permuting components of p as the various row interchanges are executed will accumulate the desired permutation. The matrix P is obtained by executing the final permutation residing in p to the rows of an appropriate size identity matrix: 1 2 −3 4 1 4 8 12 −8 2 4 8 12 −8 2 1 2 −3 4 1 [A|p] = −→ 2 3 2 1 3 2 3 2 1 3 −3 −1 1 −4 4 −3 −1 1 −4 4
154 Chapter 3 Matrix AlgebraExample 3.10.6 The LDU factorization. There's some asymmetry in an LU factorization be- cause the lower factor has 1's on its diagonal while the upper factor has a nonunit diagonal. This is easily remedied by factoring the diagonal entries out of the up- per factor as shown below: 1 u /u · · · u1n /u11 u11 u12 ··· u1n u11 0 ··· 0 12 11 0 u22 ··· u2n 0 u22 ··· 0 0 1 · · · u2n /u22 . . . = . . . . . . . . . . . .. . . . . . . . .. . . . . . .. . . . . . 0 0 · · · unn 0 0 · · · unn 0 0 ··· 1 Setting D = diag (u11 , u22 , . . . , unn ) (the diagonal matrix of pivots) and redefin- ing U to be the rightmost upper-triangular matrix shown above allows any LU factorization to be written as A = LDU, where L and U are lower- and upper- triangular matrices with 1's on both of their diagonals. This is called the LDU factorization of A. It is uniquely determined, and when A is symmetric, the LDU factorization is A = LDLT (Exercise 3.10.9).Example 3.10.7 22 The Cholesky Factorization. A symmetric matrix A possessing an LU fac- torization in which each pivot is positive is said to be positive definite. Problem: Prove that A is positive definite if and only if A can be uniquely factored as A = RT R, where R is an upper-triangular matrix with positive diagonal entries. This is known as the Cholesky factorization of A, and R is called the Cholesky factor of A. Solution: If A is positive definite, then, as pointed out in Example 3.10.6, it has an LDU factorization A = LDLT in which D = diag (p1 , p2 , . . . , pn ) is the diagonal matrix containing the pivots pi > 0. Setting R = D1/2 LT √ √ √ where D1/2 = diag p1 , p2 , . . . , pn yields the desired factorization because A = LD1/2 D1/2 LT = RT R, and R is upper triangular with positive diagonal 22 This is named in honor of the French military officer Major Andr´-Louis Cholesky (1875– e 1918). Although originally assigned to an artillery branch, Cholesky later became attached to the Geodesic Section of the Geographic Service in France where he became noticed for his extraordinary intelligence and his facility for mathematics. From 1905 to 1909 Cholesky was involved with the problem of adjusting the triangularization grid for France. This was a huge computational task, and there were arguments as to what computational techniques should be employed. It was during this period that Cholesky invented the ingenious procedure for solving a positive definite system of equations that is the basis for the matrix factorization that now bears his name. Unfortunately, Cholesky's mathematical talents were never allowed to flower. In 1914 war broke out, and Cholesky was again placed in an artillery group—but this time as the commander. On August 31, 1918, Major Cholesky was killed in battle. Cholesky never had time to publish his clever computational methods—they were carried forward by word- of-mouth. Issues surrounding the Cholesky factorization have been independently rediscovered several times by people who were unaware of Cholesky, and, in some circles, the Cholesky factorization is known as the square root method.
3.10 The LU Factorization 155 entries. Conversely, if A = RRT , where R is lower triangular with a positive diagonal, then factoring the diagonal entries out of R as illustrated in Example 3.10.6 produces R = LD, where L is lower triangular with a unit diagonal and D is the diagonal matrix whose diagonal entries are the rii 's. Consequently, A = LD2 LT is the LDU factorization for A, and thus the pivots must be positive because they are the diagonal entries in D2 . We have now proven that A is positive definite if and only if it has a Cholesky factorization. To see why such a factorization is unique, suppose A = R1 RT = R2 RT , and factor out 1 2 the diagonal entries as illustrated in Example 3.10.6 to write R1 = L1 D1 and R2 = L2 D2 , where each Ri is lower triangular with a unit diagonal and Di contains the diagonal of Ri so that A = L1 D2 LT = L2 D2 LT . The uniqueness 1 1 2 2 of the LDU factors insures that L1 = L2 and D1 = D2 , so R1 = R2 . Note: More is said about the Cholesky factorization and positive definite matrices on pp. 313, 345, and 559.Exercises for section 3.10 1 4 5 3.10.1. Let A = 4 18 26 . 3 16 30 (a) Determine the LU factors of A. (b) Use the LU factors to solve Ax1 = b1 as well as Ax2 = b2 , where 6 6 b1 = 0 and b2 = 6 . −6 12 (c) Use the LU factors to determine A−1 . 3.10.2. Let A and b be the matrices 1 2 4 17 17 3 6 −12 3 3 A= and b = . 2 3 −3 2 3 0 2 −2 6 4 (a) Explain why A does not have an LU factorization. (b) Use partial pivoting and find the permutation matrix P as well as the LU factors such that PA = LU. (c) Use the information in P, L, and U to solve Ax = b. ξ 2 0 3.10.3. Determine all values of ξ for which A = 1 ξ 1 fails to have an 0 1 ξ LU factorization.
3.10 The LU Factorization 157 3.10.7. An×n is called a band matrix if aij = 0 whenever |i − j| > w for some positive integer w, called the bandwidth. In other words, the nonzero entries of A are constrained to be in a band of w diagonal lines above and below the main diagonal. For example, tridiagonal matrices have bandwidth one, and diagonal matrices have bandwidth zero. If A is a nonsingular matrix with bandwidth w, and if A has an LU factorization A = LU, then L inherits the lower band structure of A, and U inherits the upper band structure in the sense that L has "lower bandwidth" w, and U has "upper bandwidth" w. Illustrate why this is true by using a generic 5 × 5 matrix with a bandwidth of w = 2. 3.10.8. (a) Construct an example of a nonsingular symmetric matrix that fails to possess an LU (or LDU) factorization. (b) Construct an example of a nonsingular symmetric matrix that has an LU factorization but is not positive definite. 1 4 5 3.10.9. (a) Determine the LDU factors for A = 4 18 26 (this is the 3 16 30 same matrix used in Exercise 3.10.1). (b) Prove that if a matrix has an LDU factorization, then the LDU factors are uniquely determined. (c) If A is symmetric and possesses an LDU factorization, explain why it must be given by A = LDLT . 1 2 3 3.10.10. Explain why A = 2 8 12 is positive definite, and then find the 3 12 27 Cholesky factor R.
158 Chapter 3 Matrix Algebra As for everything else, so for a mathematical theory: beauty can be perceived but not explained. — Arthur Cayley (1821–1895)
CHAPTER 4 Vector Spaces4.1 SPACES AND SUBSPACES After matrix theory became established toward the end of the nineteenth century, it was realized that many mathematical entities that were considered to be quite different from matrices were in fact quite similar. For example, objects such as points in the plane 2 , points in 3-space 3 , polynomials, continuous functions, and differentiable functions (to name only a few) were recognized to satisfy the same additive properties and scalar multiplication properties given in §3.2 for matrices. Rather than studying each topic separately, it was reasoned that it is more efficient and productive to study many topics at one time by studying the common properties that they satisfy. This eventually led to the axiomatic definition of a vector space. A vector space involves four things—two sets V and F, and two algebraic operations called vector addition and scalar multiplication. • V is a nonempty set of objects called vectors. Although V can be quite general, we will usually consider V to be a set of n-tuples or a set of matrices. • F is a scalar field—for us F is either the field of real numbers or the field C of complex numbers. • Vector addition (denoted by x + y ) is an operation between elements of V. • Scalar multiplication (denoted by αx ) is an operation between elements of F and V. The formal definition of a vector space stipulates how these four things relate to each other. In essence, the requirements are that vector addition and scalar multiplication must obey exactly the same properties given in §3.2 for matrices.
160 Chapter 4 Vector Spaces Vector Space Definition The set V is called a vector space over F when the vector addition and scalar multiplication operations satisfy the following properties. (A1) x+y ∈ V for all x, y ∈ V. This is called the closure property for vector addition. (A2) (x + y) + z = x + (y + z) for every x, y, z ∈ V. (A3) x + y = y + x for every x, y ∈ V. (A4) There is an element 0 ∈ V such that x + 0 = x for every x ∈ V. (A5) For each x ∈ V, there is an element (−x) ∈ V such that x + (−x) = 0. (M1) αx ∈ V for all α ∈ F and x ∈ V. This is the closure property for scalar multiplication. (M2) (αβ)x = α(βx) for all α, β ∈ F and every x ∈ V. (M3) α(x + y) = αx + αy for every α ∈ F and all x, y ∈ V. (M4) (α + β)x = αx + βx for all α, β ∈ F and every x ∈ V. (M5) 1x = x for every x ∈ V. A theoretical algebraic treatment of the subject would concentrate on the logical consequences of these defining properties, but the objectives in this text 23 are different, so we will not dwell on the axiomatic development. Neverthe- 23 The idea of defining a vector space by using a set of abstract axioms was contained in a general theory published in 1844 by Hermann Grassmann (1808–1887), a theologian and philosopher from Stettin, Poland, who was a self-taught mathematician. But Grassmann's work was origi- nally ignored because he tried to construct a highly abstract self-contained theory, independent of the rest of mathematics, containing nonstandard terminology and notation, and he had a tendency to mix mathematics with obscure philosophy. Grassmann published a complete re- vision of his work in 1862 but with no more success. Only later was it realized that he had formulated the concepts we now refer to as linear dependence, bases, and dimension. The Italian mathematician Giuseppe Peano (1858–1932) was one of the few people who noticed Grassmann's work, and in 1888 Peano published a condensed interpretation of it. In a small chapter at the end, Peano gave an axiomatic definition of a vector space similar to the one above, but this drew little attention outside of a small group in Italy. The current definition is derived from the 1918 work of the German mathematician Hermann Weyl (1885–1955). Even though Weyl's definition is closer to Peano's than to Grassmann's, Weyl did not mention his Italian predecessor, but he did acknowledge Grassmann's "epoch making work." Weyl's success with the idea was due in part to the fact that he thought of vector spaces in terms of geometry, whereas Grassmann and Peano treated them as abstract algebraic structures. As we will see, it's the geometry that's important.
4.1 Spaces and Subspaces 161 less, it is important to recognize some of the more significant examples and to understand why they are indeed vector spaces.Example 4.1.1 Because (A1)–(A5) are generalized versions of the five additive properties of matrix addition, and (M1)–(M5) are generalizations of the five scalar multipli- cation properties given in §3.2, we can say that the following hold. • The set m×n of m × n real matrices is a vector space over . • The set C m×n of m × n complex matrices is a vector space over C.Example 4.1.2 The real coordinate spaces x1 x2 1×n = {( x1 x2 · · · xn ) , xi ∈ } and n×1 . , xi ∈ = . . xn are special cases of the preceding example, and these will be the object of most of our attention. In the context of vector spaces, it usually makes no difference whether a coordinate vector is depicted as a row or as a column. When the row or column distinction is irrelevant, or when it is clear from the context, we will use the common symbol n to designate a coordinate space. In those cases where it is important to distinguish between rows and columns, we will explicitly write 1×n or n×1 . Similar remarks hold for complex coordinate spaces. Although the coordinate spaces will be our primary concern, be aware that there are many other types of mathematical structures that are vector spaces— this was the reason for making an abstract definition at the outset. Listed below are a few examples.Example 4.1.3 With function addition and scalar multiplication defined by (f + g)(x) = f (x) + g(x) and (αf )(x) = αf (x), the following sets are vector spaces over : • The set of functions mapping the interval [0, 1] into . • The set of all real-valued continuous functions defined on [0, 1]. • The set of real-valued functions that are differentiable on [0, 1]. • The set of all polynomials with real coefficients.
162 Chapter 4 Vector SpacesExample 4.1.4 2 Consider the vector space , and let L = {(x, y) | y = αx} be a line through the origin. L is a subset of 2 , but L is a special kind of subset because L also satisfies the properties (A1)–(A5) and (M1)–(M5) that define a vector space. This shows that it is possible for one vector space to properly contain other vector spaces. Subspaces Let S be a nonempty subset of a vector space V over F (symbolically, S ⊆ V). If S is also a vector space over F using the same addition and scalar multiplication operations, then S is said to be a subspace of V. It's not necessary to check all 10 of the defining conditions in order to determine if a subset is also a subspace—only the closure conditions (A1) and (M1) need to be considered. That is, a nonempty subset S of a vector space V is a subspace of V if and only if (A1) x, y ∈ S =⇒ x + y ∈ S and (M1) x ∈ S =⇒ αx ∈ S for all α ∈ F. Proof. If S is a subset of V, then S automatically inherits all of the vector space properties of V except (A1), (A4), (A5), and (M1). However, (A1) together with (M1) implies (A4) and (A5). To prove this, observe that (M1) implies (−x) = (−1)x ∈ S for all x ∈ S so that (A5) holds. Since x and (−x) are now both in S, (A1) insures that x + (−x) ∈ S, and thus 0 ∈ S.Example 4.1.5 Given a vector space V, the set Z = {0} containing only the zero vector is a subspace of V because (A1) and (M1) are trivially satisfied. Naturally, this subspace is called the trivial subspace. Vector addition in 2 and 3 is easily visualized by using the parallelo- gram law, which states that for two vectors u and v, the sum u + v is the vector defined by the diagonal of the parallelogram as shown in Figure 4.1.1.
4.1 Spaces and Subspaces 163 u+v = (u1+v1, u2+v2) v = (v1,v2) u = (u1,u2) Figure 4.1.1 We have already observed that straight lines through the origin in 2 are subspaces, but what about straight lines not through the origin? No—they can- not be subspaces because subspaces must contain the zero vector (i.e., they must pass through the origin). What about curved lines through the origin—can some of them be subspaces of 2 ? Again the answer is "No!" As depicted in Figure 4.1.2, the parallelogram law indicates why the closure property (A1) cannot be satisfied for lines with a curvature because there are points u and v on the curve for which u + v (the diagonal of the corresponding parallelogram) is not on the curve. Consequently, the only proper subspaces of 2 are the trivial subspace and lines through the origin. u+v αu u+v v v u u Figure 4.1.2 P Figure 4.1.3 In 3 , the trivial subspace and lines through the origin are again subspaces, but there is also another one—planes through the origin. If P is a plane through the origin in 3 , then, as shown in Figure 4.1.3, the parallelogram law guarantees that the closure property for addition (A1) holds—the parallelogram defined by
164 Chapter 4 Vector Spaces any two vectors in P is also in P so that if u, v ∈ P, then u + v ∈ P. The closure property for scalar multiplication (M1) holds because multiplying any vector by a scalar merely stretches it, but its angular orientation does not change so that if u ∈ P, then αu ∈ P for all scalars α. Lines and surfaces in 3 that have curvature cannot be subspaces for essentially the same reason depicted in Figure 4.1.2. So the only proper subspaces of 3 are the trivial subspace, lines through the origin, and planes through the origin. The concept of a subspace now has an obvious interpretation in the visual spaces 2 and 3 —subspaces are the flat surfaces passing through the origin. Flatness Although we can't use our eyes to see "flatness" in higher dimensions, our minds can conceive it through the notion of a subspace. From now on, think of flat surfaces passing through the origin whenever you encounter the term "subspace." For a set of vectors S = {v1 , v2 , . . . , vr } from a vector space V, the set of all possible linear combinations of the vi 's is denoted by span (S) = {α1 v1 + α2 v2 + · · · + αr vr | αi ∈ F} . Notice that span (S) is a subspace of V because the two closure properties (A1) and (M1) are satisfied. That is, if x = i ξi vi and y = i ηi vi are two linear combinations from span (S) , then the sum x + y = i (ξi + ηi )vi is also a linear combination in span (S) , and for any scalar β, βx = i (βξi )vi is also a linear combination in span (S) . αu + βv βv αu v u Figure 4.1.4
4.1 Spaces and Subspaces 165 For example, if u = 0 is a vector in 3 , then span {u} is the straight line passing through the origin and u. If S = {u, v}, where u and v are two nonzero vectors in 3 not lying on the same line, then, as shown in Figure 4.1.4, span (S) is the plane passing through the origin and the points u and v. As we will soon see, all subspaces of n are of the type span (S), so it is worthwhile to introduce the following terminology. Spanning Sets • For a set of vectors S = {v1 , v2 , . . . , vr } , the subspace span (S) = {α1 v1 + α2 v2 + · · · + αr vr } generated by forming all linear combinations of vectors from S is called the space spanned by S. • If V is a vector space such that V = span (S) , we say S is a spanning set for V. In other words, S spans V whenever each vector in V is a linear combination of vectors from S.Example 4.1.6 (i) In Figure 4.1.4, S = {u, v} is a spanning set for the indicated plane. 1 2 (ii) S= , spans the line y = x in 2 . 1 2 1 0 0 (iii) The unit vectors e1 = 0 , e2 = 1 , e3 = 0 span 3 . 0 0 1 (iv) The unit vectors {e1 , e2 , . . . , en } in n form a spanning set for n . (v) The finite set 1, x, x2 , . . . , xn spans the space of all polynomials such that deg p(x) ≤ n, and the infinite set 1, x, x2 , . . . spans the space of all polynomials.Example 4.1.7 Problem: For a set of vectors S = {a1 , a2 , . . . , an } from a subspace V ⊆ m×1 , let A be the matrix containing the ai 's as its columns. Explain why S spans V if and only if for each b ∈ V there corresponds a column x such that Ax = b (i.e., if and only if Ax = b is a consistent system for every b ∈ V).
170 Chapter 4 Vector Spaces In particular, this result means that every matrix A ∈ m×n generates a subspace of m by means of the range of the linear function f (x) = Ax. 24 Likewise, the transpose of A ∈ m×n defines a subspace of n by means of the range of f (y) = AT y. These two "range spaces" are two of the four fundamental subspaces defined by a matrix. Range Spaces The range of a matrix A ∈ m×n is defined to be the subspace R (A) of m that is generated by the range of f (x) = Ax. That is, R (A) = {Ax | x ∈ n }⊆ m . Similarly, the range of AT is the subspace of n defined by R AT = {AT y | y ∈ m }⊆ n . Because R (A) is the set of all "images" of vectors x ∈ m under transformation by A, some people call R (A) the image space of A. The observation (4.2.2) that every matrix–vector product Ax (i.e., every image) is a linear combination of the columns of A provides a useful character- ization of the range spaces. Allowing the components of x = (ξ1 , ξ2 , . . . , ξn )T to vary freely and writing ξ1 ξ2 n Ax = A∗1 | A∗2 | · · · | A∗n . = . ξj A∗j . j=1 ξn shows that the set of all images Ax is the same as the set of all linear combi- nations of the columns of A. Therefore, R (A) is nothing more than the space spanned by the columns of A. That's why R (A) is often called the column space of A. Likewise, R AT is the space spanned by the columns of AT . But the columns of AT are just the rows of A (stacked upright), so R AT is simply 25 the space spanned by the rows of A. Consequently, R AT is also known as the row space of A. Below is a summary. 24 For ease of exposition, the discussion in this section is in terms of real matrices and real spaces, but all results have complex analogs obtained by replacing AT by A∗ . 25 Strictly speaking, the range of AT is a set of columns, while the row space of A is a set of rows. However, no logical difficulties are encountered by considering them to be the same.
4.2 Four Fundamental Subspaces 171 Column and Row Spaces For A ∈ m×n , the following statements are true. • R (A) = the space spanned by the columns of A (column space). • R AT = the space spanned by the rows of A (row space). • b ∈ R (A) ⇐⇒ b = Ax for some x. (4.2.3) • a∈R A T ⇐⇒ a = y A for some y . T T T (4.2.4)Example 4.2.1 1 2 3 Problem: Describe R (A) and R AT for A = 2 4 6 . Solution: R (A) = span {A∗1 , A∗2 , A∗3 } = {α1 A∗1 +α2 A∗2 +α3 A∗3 | αi ∈ }, but since A∗2 = 2A∗1 and A∗3 = 3A∗1 , it's clear that every linear combination of A∗1 , A∗2 , and A∗3 reduces to a multiple of A∗1 , so R (A) = span {A∗1 } . Geometrically, R (A) is the line in 2 through the origin and the point (1, 2). Similarly, R AT = span {A1∗ , A2∗ } = {α1 A1∗ + α2 A2∗ | α1 , α2 ∈ } . But A2∗ = 2A1∗ implies that every combination of A1∗ and A2∗ reduces to a multiple of A1∗ , so R AT = span {A1∗ } , and this is a line in 3 through the origin and the point (1, 2, 3). There are times when it is desirable to know whether or not two matrices have the same row space or the same range. The following theorem provides the solution to this problem. Equal Ranges For two matrices A and B of the same shape: row • R A T = R BT if and only if A ∼ B. (4.2.5) col • R (A) = R (B) if and only if A ∼ B. (4.2.6) row Proof. To prove (4.2.5), first assume A ∼ B so that there exists a nonsingular matrix P such that PA = B. To see that R AT = R BT , use (4.2.4) to write a ∈ R AT ⇐⇒ aT = yT A = yT P−1 PA for some yT ⇐⇒ aT = zT B for zT = yT P−1 ⇐⇒ a ∈ R BT .
172 Chapter 4 Vector Spaces Conversely, if R AT = R BT , then span {A1∗ , A2∗ , . . . , Am∗ } = span {B1∗ , B2∗ , . . . , Bm∗ } , so each row of B is a combination of the rows of A, and vice versa. On the basis of this fact, it can be argued that it is possible to reduce A to B by using row only row operations (the tedious details are omitted), and thus A ∼ B. The proof of (4.2.6) follows by replacing A and B with AT and BT .Example 4.2.2 Testing Spanning Sets. Two sets {a1 , a2 , . . . , ar } and {b1 , b2 , . . . , bs } in n span the same subspace if and only if the nonzero rows of EA agree with the nonzero rows of EB , where A and B are the matrices containing the ai 's and bi 's as rows. This is a corollary of (4.2.5) because zero rows are irrelevant in considering the row space of a matrix, and we already know from (3.9.9) that row A ∼ B if and only if EA = EB . Problem: Determine whether or not the following sets span the same subspace: 1 2 3 0 1 2 4 6 0 2 A = , , , B = , . 2 1 1 1 3 3 3 4 1 4 Solution: Place the vectors as rows in matrices A and B, and compute 1 2 2 3 1 2 0 1 A = 2 4 1 3 → 0 0 1 1 = EA 3 6 1 4 0 0 0 0 and 0 0 1 1 1 2 0 1 B= → = EB . 1 2 3 4 0 0 1 1 Hence span {A} = span {B} because the nonzero rows in EA and EB agree. We already know that the rows of A span R AT , and the columns of A span R (A), but it's often possible to span these spaces with fewer vectors than the full set of rows and columns. Spanning the Row Space and Range Let A be an m × n matrix, and let U be any row echelon form derived from A. Spanning sets for the row and column spaces are as follows: • The nonzero rows of U span R AT . (4.2.7) • The basic columns in A span R (A). (4.2.8)
174 Chapter 4 Vector Spaces Nullspace • For an m × n matrix A, the set N (A) = {xn×1 | Ax = 0} ⊆ n is called the nullspace of A. In other words, N (A) is simply the set of all solutions to the homogeneous system Ax = 0. • The set N AT = ym×1 | AT y = 0 ⊆ m is called the left- hand nullspace of A because N AT is the set of all solutions to the left-hand homogeneous system yT A = 0T .Example 4.2.4 1 2 3 Problem: Determine a spanning set for N (A), where A = 2 4 6 . Solution: N (A) is merely the general solution of Ax = 0, and this is deter- mined by reducing A to a row echelon form U. As discussed in §2.4, any such U will suffice, so we will use EA = 1 0 0 . Consequently, x1 = −2x2 − 3x3 , 0 2 3 where x2 and x3 are free, so the general solution of Ax = 0 is x1 −2x2 − 3x3 −2 −3 x2 = x2 = x2 1 + x3 0 . x3 x3 0 1 In other words, N (A) is the set of all possible linear combinations of the vecto
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This MOOC, offered by Coursra, was developed by Philip Klein of Brown University. From the Coursera page for this course: "Learn the concepts and methods of linear algebra, and how to use them to think about computational problems arising in computer science. Coursework includes building on the concepts to write small programs and run them on real data." The programming, in Python, appears to be a big part of this course.
Comments
The actual MOOC materials are not available until you register for the MOOC. The course outline suggests that the emphasis is on solving systems of equations, but does also emphasize vector spaces, dimension, basis, orthogonality, and inner-product spaces. The reduced mathematical content is assumed to have been done to make room in a typical course for the Python programming. Here's more from the Coursera website for this book: "In this class, you will learn the concepts and methods of linear algebra, and how to use them to think about problems arising in computer science. You will write small programs in the programming language Python to implement basic matrix and vector functionality and algorithms, and use these to process real-world data to achieve such tasks as: two-dimensional graphics transformations, face morphing, face detection, image transformations such as blurring and edge detection, image perspective removal, audio and image compression, searching within an image or an audio clip, classification of tumors as malignant or benign, integer factorization, error-correcting codes, secret-sharing, network layout, document classification, and computing Pagerank (Google's ranking method)." Without seeing the actual course materials, it's not really possible to make an informed rating.
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Mathematics is a tool for understanding our world and helping to solve its problems. Mathematical models are used in such diverse areas as determining the shape of aircraft wings for maximum lift, analyzing disease spread and control, and simulting network flow for efficient transportation systems. Math also requires imagination - necessitating abstract and formalized thought on one hand, creativity and intuition on the other. All people use math in their personal and professional lives every day. Thus, almost all university students will take formal courses to acquire the use of mathematical tools.
The math major introduces students to the areas of real and complex analysis, number theory, abstract algebra, topology, logic, and other fields. Mathematical study can be designed to support different career goals, such as industrial employment, pre-professional education, or preparation for graduate school.
Our faculty routinely conduct interdisciplinary research combining several of the areas list above or collaborating with other departments, schools, and institutes on campus. Most notably, the department is, jointly with the Department of Statistics, home of the Mathematical Biosciences Institute (MBI) - one of the 8 national math sciences institutes sponsored by the National Science Foundation (NSF).
OSU's Department of Mathematics is ranked 4th in the US in NSF funding for the Mathematical Sciences. Recognitions also include regular multiple invitations of our faculty members as speakers to the prestigious International Congress of Mathematics.
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Course 3: SOL
Strand 6: Patterns & Algebra
8.14 The student will a) describe and represent relations and functions, using tables, graphs, and rules; and b) relate and compare tables, graphs, and rules as different forms of representation for relationships.
8.15 The student will solve two-step equations and inequalities in one variable, using concrete materials, pictorial representations, and paper and pencil.
8.16 The student will graph a linear equation in two variables, in the coordinate plane, using a table of ordered pairs.
8.17 The student will create and solve problems, using proportions, formulas, and functions.
8.18 The student will use the following algebraic terms appropriately: domain, range, independent variable, and dependent variable.
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Thinking Mathematically - With 2 CDs - 4th edition
Summary: This general survey of mathematical topics helps a diverse audience, with different backgrounds and career plans, to understand mathematics. Blitzer provides the applications and technology readers need to gain an appreciation of mathematics in everyday life. Demonstrates how mathematics can be applied to readers' lives in interesting, enjoyable, and meaningful ways. Features abundant, step-by-step, annotated Examplesthat provide a problem-solving approach to reach the ...show moresolution; annotations are conversational in tone, explaining key steps and ideas as the problem is solved. Begins each section with a compelling vignette highlighting an everyday scenario, posing a question about it, and exploring how the chapter section subject can be applied to answer the question. A highly readable reference for anyone who needs to brush up their mathematics skills. ...show less
Book shows a small amount of wear to cover and binding. Some pages show signs of use. Sail the Seas of ValueForGoodwillGetJobs St Paul, MN
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Answered by
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In my 8th grade Advanced Algebra class, my group is moving extremely slowly due to constant chattering. I find it frustrating and would like to have the math skills to go into Advanced Geometry next...
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Can you think of one real-world example of when the concept of functions might be useful? Do you think you will ever use functions in your life to solve problems? If yes, explain how and why; if no,...
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Mathematics for Elementary Teachers: A Conceptual Approach
9780073519579
ISBN:
007351957X
Publisher: McGraw-Hill
Summary: Would you like to rent Mathematics for Elementary Teachers: A Conceptual Approach online from Valore Books now? If you would like to take advantage of discounted prices on pre-owned copies of this book published by McGraw-Hill, look at our selection now. Written by Albert B Bennett, Laurie J Burton and Leonard T Nelson, you can find the cheapest copies of this text book by using our site now. Buy Mathematics for Elem...entary Teachers: A Conceptual Approach online from us today and find out why so many people rent and buy books for college from us. Try our website now for the cheapest deals.
Bennett, Albert B. is the author of Mathematics for Elementary Teachers: A Conceptual Approach, published under ISBN 9780073519579 and 007351957X. Eight hundred twenty one Mathematics for Elementary Teachers: A Conceptual Approach textbooks are available for sale on ValoreBooks.com, two hundred sixty three used from the cheapest price of $61.97, or buy new starting at $112.11.[read more]
Ships From:Louisville, KYShipping:Standard, ExpeditedComments:9th edition. Does Not include CD's, DVD's, access codes or other supplements. PLEASE NOTE: Minor ... [more] Minor water damage is present along the side edge in some areas (minor). M [more]
my first college math book I've ever even opened. The examples are great! A lot of the odd numbered problems show the answers at the back of the book and that was a big help to know if we were doing the problem correct or not.
I was taking Math 155 & 156. It used the book all the time and it helps to get the manipulative kit.
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Mathematics for Every45
FREE
About the Book
Dispelling some of the subject's alarming aspects, this book provides, in a witty and engaging style, the fundamentals of mathematical operations. Topics include system of tens and other number systems, symbols and commands, first steps in algebra and algebraic notation, common fractions and equations, irrational numbers, much more. 1958 edition.
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Survey of Mathematics with Applications, A (9th Edition)
9780321759665
ISBN:
0321759664
Edition: 9 Pub Date: 2012 Publisher: Addison Wesley
Summary: This textbook serves as a broad introduction to students who are looking for an overview of mathematics. It is designed in such a way that students will actually find the text accessible and be able to easily understand and most importantly enjoy the subject matter. Students will learn what purpose math has in our lives and how it affects how we live and how we relate to it. It is not heavy on pure math; its purpose ...is as an overview of mathematics that will enlighten students without an intense background in math. If you want to obtain this and other cheap math textbooks we have many available to buy or rent in great condition online.
Allen R. Angel is the author of Survey of Mathematics with Applications, A (9th Edition), published 2012 under ISBN 9780321759665 and 0321759664. One thousand one hundred twenty one Survey of Mathematics with Applications, A (9th Edition) textbooks are available for sale on ValoreBooks.com, five hundred eighty used from the cheapest price of $72.75, or buy new starting at $163 ***Warning***Text Only. Still in Shrink Wrap Annotated Instructor's Copy, 9th edition but No Supplementary Materials otherwise same as student with help added tips,and answers.Shipping from California.[less me, the least helpful part was the first few pages of each section. It was almost as if trying too hard to explain something which would make easy things seem complicated. I deffinatly the practice questions were great practice for preparing for tests/quizzes and having some answers in the back of the book were helpful in checking myself.
The primary subject of this book was general college math and it was deffinatly effective. There is a lot of information in here that was very simple for me to understand and then there were other lessons that were much more complicated and requiread a lot more practice and hair pulling for me.
Honestly there is nothing I would change about this book. It worked great for the class I took. It explained the lesson in the beginning of the chapter then gave problems pertaining to the lesson. It was very helpful. Having the odd number answers in the back of the book helped a lot especially when you couldn't figure out how to do a problem. I would take the answer and answer the problem until I got that answer.
I learned how to do numerous things in this book, such as writing expressions, solving for X, rise over run and other problems. This book helped me a lot in my math skills. I haven't had math in a couple years, and this book helped refresh my math skills. Thanks to this book and the teacher I passed my HESI test for nursing with flying colors.
Adding because it help me learn the value of money and putting he decimals in the right place.It help me focus on adding nonstandard getting the right change back if you buy something in the near future.
I learn how to add, multiply, subtracting, and division in many ways the purpose of that is to help me build focus in this skill.Math is my favorite subject in this world I love it because without it we would not have paychecks ,banks or even stores who accept money.
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The dihedral group D8 (sometimes called D4), the group of symmetries of the square, is one of the simplest finite groups. In these three mathlets, we will explore different aspects of this symmetry group to get a better understanding of its structure.
This paper describes a Flash-based OSSLET that we have used at the United States Military Academy with first year calculus students as a vehicle for motivating vectors, matrices, and linear and affine transformations.
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Farmers Branch, TX ACT Math problems are a particular challenge for a number of students for whom the following steps must first be modeled: 1) drawing an appropriate diagram, if required, 2) establishing the correct algebraic representation(s) for the unknown(s) and then using this information to label the correct pa... This website has allowed me to have my own business
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Philosophy
of this Course. Read this!
The world is
changing rapidly. Everyday things like iPads, smartphones, apps,
Facebook, Twitter, and massive multi-player on-line games didn't even
exist 10 years ago. What you need to learn about mathematics is
changing rapidly too. Most of the math you learned in high school can
be done by machines. Most of what students used to learn in a typical
precalculus course or even a calculus course can be done by machines
(for free!) So, if you want to learn something with real value, you
need to think about mathematics differently. This course requires you to change your thinking
about math.
This course is different. It is intended to teach
you things that machines can not do for you. It is designed to teach
you what you need to know
about mathematics at the precalculus level that will make you capable
and valuable.
Software (like wolframalpha.com -- try it!)
can compute answers to symbolic algebra and calculus problems, if you know how to communicate with
it. So, learning to communicate mathematics to machines is one step.
But the real world does not pose problems in symbols, so another
important step is to learn how to take real-world problems and convert
them to mathematical notation (The term "indirect" in Section 1.1 refers to this.)
In the past math courses emphasized
1) computational skills (because, not many years
ago, computations were done by hand) and
2) facts.
But now if you want to solve an equation, there is
software that can
"do" that type of math problem. Now, everyone knows you can look up
facts on the web. If you can pose a math question well, you can
probably find an app or web page to answer it, if you can read
and write mathematics well enough. Nevertheless, there is still a lot
of math to learn.You need to learn how to communicate with your
software, interpret graphs, write and
read mathematics to understand how to formulate questions and
interpret answers. Also, you need to have the right question come
to mind, which is a non-trivial skill. We want you to have the
prior knowledge required to think of the right questions and
techniques.
In summary, if all you know are the facts and
computational skills that math classes previously emphasized (in your
school!), you have little value to anyone. The best person at
multiplying three-digit numbers can be replaced by a $5 calculator. The
best person at solving symbolic algebra problems can be replaced by
free software on a website! We want to add value to you, not
teach you valueless skills.
That is why this course is different--even from
what it used to be a few years ago. The world is changing and what
we
teach and what you need to learn are changing too.
1) This course encourages you to use machines to do problems that
are already given in symbols,
2) but, you will not get a lot of credit for being able to do
symbolic problems like you learned to do in high school.
3) The intention is to help you develop essential concepts,
abilities, and interpretation skills required to apply math to the real
world and do things machines won't do for you.
4) You will learn to have the right things "come to mind". (It is
one level of learning to know an answer when prompted, another to have
it come to mind without prompting.)
5) Exams will focus on concepts and skills you need to develop,
not on solving problems that machines can solve for free.
6) The point is to add value
to you (knowing how to solve for x in problems posed in symbols
is no longer a valuable skill -- sorry!)
7) Reading and writing math forces you to focus on essential,
valuable, mathematical concepts.
8) Expect exams to be quite different from math exams you have
taken before (even different from previous exams given in this class!)
If you have read your text closely, written symbolic math daily on
homework, and assimilated the concepts covered in the text, you will do
well. However, if all you can do is compute numerical answers to
symbolic problems, you will not do well (because the value you added to
yourself would be close to zero.)
Take a look at this brilliant artwork created from words
on this page. How much do you suppose it cost to have this complex work
created?
It cost nothing. It was free on a
web site! I just copied and dropped
in the text
of this page and it arranged the words with size corresponding to word
frequency.
This (previously) $1000 artwork was free! The point is, machines can do
a lot of what we used to train people to
do (and pay them well for--but not anymore!). This course is aware of
that and will seem
different to you because it teaches useful and valuable skills you were
not taught in high school.
There are big changes from a few years ago. You need a tool
that can do what a graphing calculator can do, but it need not be the
Texas Instruments graphing calculator we will use in class (a TI-83 or
84). If you find a great app, tell your instructor and your
friends! I like "Free Graphing Calculator" (by William Jockusch)
and we have had the app "Graphing Calculator HD" recommended, but there
are many that would work fine. There will be some changes to exams too.
There will be fewer questions asking for calculations that machines can
do. Calculators can help you learn concepts, so we want you to use
calculators a lot, but your understanding of the concepts you develop
can be tested without using calculators.
When we do graphing-calculator activities in class,
you may use your laptop or iPad or Smartphone or calculator to
participate. However, using technology in class for non-math activities
is prohibited and a severe breach of etiquette.
Section
summaries: Your
job is to 1.1. Learn what it means for a problem to
be indirect and to become comfortable with working indirectly, that is,
writing about operations you don't actually do.
1.2. Learn how order is expressed in written mathematics and on your
personal calculator so well that you can rapidly evaluate complicated
expressions correctly. This requires learning how to insert parentheses
that are not in the usual written mathematics. Practice until you can
get five problems from B2-B16 correct in a row.
1.3. Learn how functional notation is used to express sequences
of operations. Learn to distinguish the function from the notation used
to express it, and how to apply that function to expressions other than
"x". (This is also discussed in Section 2.2.)
1.4. Learn how to read definitions and theorems (They express
mathematical methods). Learn how to write mathematical methods in
symbolic notation. (This is a course-long project, begun in Section 1.1
and treated as the focus of Section 1.4. Reading and writing are
essential to word problems, so learning to read and write symbolically
is critical.)
1.5. Learn to read graphs (that is, extract information they
contain). Learn how to graph with your graphing calculator and obtain a
"representative" graph. Learn to select and modify windows, and use the
calculator's capabilities to determine key points on graphs. (This is
continued in Section 2.1.)
1.6. Recall the usual algebraic ways to solve equations. Learn to
rapidly classify equations according to which way to solve them, chosen
from the "four ways to solve an equation." (This is a section on making
good algebraic decisions.)
2.1. Learn how to choose windows that make graphs
"representative"
or have a particular look. Learn how changes in the window will change
the appearance of graphs. (This section continues Section 1.5)
2.2. Learn how notation expresses functional composition. Learn
how given graphs are affected by composition with addition,
subtraction, multiplication, division, and attaching a negative sign.
2.3. Learn that solutions to "f(x) = c"
may not be unique if the
function is not "one-to-one." Learn how to recognize when a function is
not one-to-one and how to deal with the complications that occur when
your calculator has an inverse function but the original function is
not one-to-one.
[to be continued]
Precalculus
requires
you to change!
It is not just more of the
same--it is not just more math methods.
You must change the way
you think about math symbols to focus on operations and order.
You must change how you
learn math to also learning by reading the text outside of class. (This
is not high school where your teacher has enough time to cover it all.)
You must change your study habits
to devote hours to learning and practicing outside class (even if it is
not "fun"). Learning is uncomfortable. In sports
you don't get stronger until you lift heavy weights and you get sore
with pain. In math you don't get better by doing the easy, painless,
work. It takes concentration and, yes, discomfort. Accept that fact and
you will do much better.
Thoughts about Learning.
Read
each
section.
Do
not
skip the harder parts. In fact, when the going gets rough you need to
slow down and read it several times until it makes sense. If it remains
unclear, ask!
Reading
takes a lot of
effort! But, you will be learning an extremely valuable skill.
Don't just
skim.
Don't expect
that only high points are important
(Don't read only
the bold parts).
Don't skip
the rest of the paragraph because you
want to move
along to the next high point.
Read it all!
Advice, designed for this course,
about how to learn math. (Read this!
It has some helpful, and perhaps surprising, ideas. Here is a copy in Word.) Advice from
previous students about how to do well in
this course. Believe it! Links to articles on learning:
Learning while multitasking. Recently
the news has had quite a bit about research on multitasking. I
summarized some of it here, and provide links.
"The
huge finding is, the more media people use the worse they are at using
any media. We were totally shocked." "What's
new is that even if you can learn while distracted, it changes how you
learn"--making the learning "less efficient and useful."
A summary of new research on multitasking
says it has a negative effect on learning. You will be better at what
you do if you do one thing at a time. (For example, don't switch
attention to texting [at all!] or Facebook while you are studying.)
And, you will get as much or more done. Don't kid yourself that
multitasking is somehow efficient. It is not.
Required
text: Precalculus,6th edition,
by
Warren Esty.
(The 5th or 4th editions will serve just fine, but correct their typos). Required graphing-calculator capability:
Calculators play a large
role, and you must have access to graphing-calculator functionality,
but this semester you do not need to buy a stand-alone
calculator if you have an iPad or Smartphone or laptop with an
equivalent calculator app or software.(The iPad app "Graphing
Calculator HD" will serve. I have had the app "Algeo calculator" highly
recommended and it looks good and is free. Probably many other app
would work fine too. If you get an app, you must take the time to learn
to use it!)
You must
satisfy our
special
prerequisite to stay in Math
M-151. Have you satisfied it?
(Many incoming
students who imagine they have actually have not. Check it!)
"I
took
precalculus (or
calculus) in High
School, so I have satisfied the prerequisite, right?" No! You
must test into the course.
What you took in high school
is does not
count. What you know
counts. Here are the rules
about prerequisites.
Prerequisite
(you must satisfy the prerequisite!), Work, Calculators, Exams and Grading, Course GoalsIn this course, calculators
are a
learning tool, not just a calculating tool. Calculators help in
two main ways. By making lower-level work less time-consuming, we can Calculators.
In this course you are supposed to
develop essential algebraic concepts. Graphing calculators or graphing
apps can help and are required. We will use a stand-alone TI calculator
in class, however, you may use any technology which is more or less
equivalent, including smart apps or any internet-based graphing
program. If you already own an iPad, why pay $100 for a calculator when
you effectively already have one?
In this
course, calculators
are a
learning tool, not just a calculating tool. Calculators help in
two main ways. By making lower-level work less time-consuming, we can
1) Concentrate attention on
essential points, and
2) Increase the rate at which
students gather experience with the subject.
Other important
information that you will want to know. For example, copies of previous
exams are available on reserve in the Library. They are also on-line here.
We have free
tutoring! The Math Learning Center (1-110 Wilson) has free
tutoring 8:30am -9:00 pm M-Th and 8:30-2:00
Fridays. Click here
for more about its hours and when you can find a Precalculus instructor
there.
are common-hour exams given at 6:00 pm. The dates are on our calendar.
Mark your personal calendar with these dates and times.
Be there! If you have an unavoidable
academic conflict, or a disability, see here.
If you are taking other common-hour exam
courses, you
may have a conflict. Look up their exam times
now and
see. If you have an academic conflict, you may be able
to resolve it by signing up (with Dr. Esty in 2-238 Wilson Hall) for
our alternative exam time (probably 4:45 the same day for common-hour
exams). However, you must sign up well in advance. Signing up
the last day is not an option.
We use calculators a great deal. Instructors will use the TI-83 or
TI-84, but you may use other models or iPad or SmartPhone apps. Learn
to use technology. If you use a TI calculator, one program you will
need many times is given next.
Activities. Chapter 1
Program
your calculator with
the Quadratic Formula. Here is a simple
four-line program for the TI-83 or 84. Here it is:
Prompt A,
B, C (-B+√(B2-4*A*C))/(2*A) -> P
(-B-√(B2-4*A*C))/(2*A) -> M
Disp P, M ______________Here
is how
to program it: Hit PRGM Follow each line here withENTER. Comments you do not type are in green. Arrow right to NEW
ENTER Enter the name, letter by letter, say, QUAD (the blinking "A" means Alphabetic mode which
refers to the letters in green on your keyboard) ENTER Prompt A, B, C To
find the Prompt command, while writing the program, hit PRGM
(again) which brings up a menu. Arrow
right to I/O (for Input/Output) and down to Prompt. There
is a comma key above the 7 key. For "A", type ALPHA A, (then ALPHA B, ALPHA C, then ENTER)
(-B+√(B2-4*A*C))/(2*A) -> P [again "B" is ALPHA B] The
"->" command is for STOre (it appears as an arrow), on a key
near the bottom left. It
stores numbers in memory We use "P" for "plus" and "M" for "minus".
(-B-√(B2-4*A*C))/(2*A) -> M
ENTER
Disp P, M The
Disp command is for Display, which is also under I/O (hit PRGM, arrow over to I/O, and down to DispENTER). At this line, you can QUIT (2nd QUIT)
If
something goes wrong, don't worry. Just
QUIT (= 2nd
QUIT in yellow) and resume from where
you were by hitting PRGM and,
this time, EDIT (instead
of NEW).
[Now "quit" and try it out on an
example where you know the answer. For example, to run it, hit
PRGM
arrow to QUAD and hit
ENTER and ENTER again.
Try to
solve x2 - 8x + 15 = 0. Did you get 5, 3?
If not, check your keystrokes.] If you want to do a second
example, you need not begin over, just hit ENTER and it will ask you
for the next value of "A".]
Prior to each exam there will be review sessions [dates and times to be announced] You must
come
prepared to show you have seriously tried the problems you ask
about. Do not kid yourself that asking questions on Monday is good
preparation for a Tuesday exam. Most of your preparation should have
been done long before Monday!
We have free tutoring all day long in the "Math
Learning Center". (However, it is closed the hour before the exam.)
Exam 1, like previous exams, comes with instructions. It says, "Show
clear supporting work on problems with several steps. Algebraic problems that display little
or no supporting work will get little or no credit. You do not
need to show work on one-step calculator problems. To solve
numerical problems guess-and-check is legal unless you are requested to
solve them 'algebraically.'" You
should look at old versions of Exam 1. View them on-line here.
Be careful when you study the old exams. Not all old
first exams cover the same sections. This semester the first exam
covers Sections 1.1 through 2.2. (Spring 2012 only covered through
2.1.) Each exam says on the top which sections it covers. You are supposed
to know a lot of algebra already. There are several levels of algebra,
and most of the algebra you learned in school is at a lower level--a
level that will not be
emphasized on the exam. The exam tests you on higher-level skills. It
tests you on material newly
learned in this course.
Be sure you can do the "B" problems. If there is
something you don't know, or don't know how to do, be sure to study
that. Don't be content with the algebra you knew before you signed up
for this course.
You are responsible for
reading and writing mathematics. On the exam we will
state theorems or definitions that you have not seen before and
ask you to read them and use them. This is a skill that cannot be
picked up in an hour or two. You learn to read by reading. We strongly
recommend you learn to read math by reading your text.
On the exam we may ask you to state methods
symbolically (as in Section 1.4).
Be sure to bring your graphing calculator with the
Quadratic Formula programmed into it.
There are many problems you will find hard if you
have not put a lot of time in playing with your calculator and reading
your text. You do not develop reading skills (required on the exam) by
watching your instructor. You do not develop calculator skills without
practicing a lot.
Other Resources: The Education Portal Academy
has some slick videos
on Precalculus topics. If you want to supplement the lectures, take
a look.
The Kahn Academy has Precalculus videos
too, but most of what we are do is in their algebra and trig sections.
Unfortunately, they have very many videos and many seem to develop the
topics slowly and often in a disorganized fashion, so I doubt anyone
would want to wade through them all.
In college your instructor
does not have enough class time to cover all the material. You are responsible for all the material in
the text anyway. You are expected to learn the rest outside
of class by reading the text. The homework and exam questions are all
closely related to things discussed in the text. Read it! Then, if
something is not clear, put in the time and effort to
figure it out. Make sure you have a Quadratic Formula program in your
calculator. ************************************************************************************************ This is
the end of the required Precalculus material at this time. Check back
for updates, especially when exams are about to happen and when we
get to Chapter 6 on trig.
You can
quit here. The
rest gives some
interesting links, relevant to education, but not required for
Math 151.
Brain Rules: 12 Principles for
Surviving and Thriving at Work, Home, and School, by John
Medina.
"The brain is an amazing thing. Most of us have no idea what's
really going on inside our heads.
"How do we
learn? What
exactly do sleep and stress do to our brains? Why is multi-tasking a
myth? Why is it so easy to forget—and so important to repeat new
knowledge?
"Brain Rules
is about what
we know for sure, and what we might do about it." Here
is the fascinating site. Learn about how to learn. Pay attention to
the "12 rules".
The famous essayist and
"MacArthur Genius Grant" recipient, David Foster Wallace said, in an
interview,
"At a certain point we either gonna have to put away
childish things
and discipline ourself about how much time do I spend being passively
entertained? And, how much time do I spend doing stuff that
actually isn't that much fun minute by minute, but that builds certain
muscles in me as a grown up and a human being? And, if we don't do that
... the cultures going to grind to a halt. Because we're gonna get so
interested in entertainment that we're not gonna want to do that work
that generates the income that buys the products that pays for the
advertising that disseminates the entertainment. .... It won't be
anybody else doing it to us, we will have done it to ourselves."
Practice beats talent
when talent doesn't practice
-- unknown author
Index:
This site has information about Calculator programs (like the
Quadratic Formula) we use frequently.
Textbook: textbook and calculating
technology requirements (a stand-alone calculator is no longer
required--a iPad app or internet-capable phone will suffice. You must
have some technology that allows you to do what a graphing calculator
does. We use a TI-84 in class.) Click here if (and only
if) your text is the 4th or 5th edition and not the current 6th
edition.
| 677.169 | 1 |
Homework
Syllabus
The goals of this course are for you to learn a significant amount of
important mathematics, to prepare yourself for further study of mathematics and
its applications, to further improve your problem solving, critical thinking and
reasoning skills, and most of all to have fun!
We will follow Mathematics 3, written by the Mathematics Department at
Philips Exeter Academy. Be sure to have the July 2010 edition. It is also
available online. You will also need a graphing calculator.
We'll do roughly 8 problems per class period, both from the text and
handouts. The exact problems assigned for each class period will be posted in
advance on the web.
Preparation for each class consists of working through the assigned problems
and writing up solutions in preparation to present them to the class. Class time
will be devoted to discussing the solutions, discovering what is interesting
about each problem, and how it connects to other problems and mathematics in
general. I will generally not collect homework but may ask that a few problems
be turned in.
There will be 4 tests and 4 quizzes each semester, as well as a final exam.
Grading
Homework is worth 15% of each quarter grade, quizzes are worth 10%, and tests
75%. The semester grade is then based on the quarter grades (80%) and the final
(20%).
Although homework seems to be weighted low it is by far the most important
part of the course. The best way to prepare for the tests is to work hard on the
homework throughout the semester.
Participation
Participation is expected of everyone and is not explicitly graded. It will
however affect grades in borderline cases. Everyone is expected to take turns
presenting solutions, and everyone should participate in discussions.
Laptop Policy
Laptops are normally not allowed in class and must be kept closed. You
do not need to bring them to class normally. There may be a few classes in
which we will use them, and I will let you know in advance if you'll need to
bring your laptop. When laptops are allowed in class, they may only be
used for the task specified.
Collaboration
I encourage you to work with fellow students on homework. However you should
attempt each problem on your own first, and all write ups must be your own work.
For any problems you hand in, you must specify any help you received, whether
from another person or persons, a book, a web site, or any other source. No
collaboration is allowed on tests, of course.
| 677.169 | 1 |
Basics of Real Numbers Study Guide
Student Contributed
Basics of Real Numbers Study Guide
This study guide reviews the different types of rational numbers and some of their properties: rational number, integer, natural number, whole number, non-integer, fraction, and irrational number. It also looks at symbols used in algebra and sets.
| 677.169 | 1 |
Summary: Provides completely worked-out solutions to all odd-numbered exercises within the text, giving you a way to check your answers and ensure that you took the correct steps to arrive at an answer
| 677.169 | 1 |
Solving Systems with No or Infinitely Many Solutions Using Graphing Stop and Jot Table
Solving Systems with No or Infinitely Many Solutions Using Graphing Stop and Jot Table
Strengthen ability to analyze word meaning and symbolic or mathematical notation from context and introduce students to unfamiliar vocabulary or notation. Further solidify understanding by defining new vocabulary words and notation as well as generating personal examples of words, concepts, and notation usage.
| 677.169 | 1 |
Book Description: The Homework Practice Workbook contains two worksheets for every lesson in the Student Edition. This workbook helps students: Practice the skills of the lesson, Use their skills to solve word problems.
Geometry, Homework Practice Workbook
Book Description: The Homework Practice Workbook contains two worksheets for every lesson in the Student Edition. This workbook helps students: Practice the skills of the lesson, Use their skills to solve word problems
| 677.169 | 1 |
Thank you for visiting us. We are currently updating our shopping cart and regret to advise that it will be unavailable until September 1, 2014. We apologise for any inconvenience and look forward to serving you again.
Description
Are you preparing for calculus? This easy-to-follow, hands-on workbook helps you master basic pre-calculus concepts and practice the types of problems you'll encounter in your cour sework. You get valuable exercises, problem-solving shortcuts, plenty of workspace, and step-by-step solutions to every problem. You'll also memorize the most frequently used equations, see how to avoid common mistakes, understand tricky trig proofs, and much more.
100s of Problems!
Detailed, fully worked-out solutions to problems
The inside scoop on quadratic equations, graphing functions, polynomials, and more
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Tilghman ACTPrealgebra includes a long list of introductory topics in both algebra and geometry. Topics covered include, but or not limited to the following:
Algebra:
- Mathematical operations of signed numbers
- Order of Operations (PEMDAS)
- Classification of Numbers
- Simplifying algebraic expressions
-Dear Prospective Tutee,
Get ready to learn, have fun, and gain confidence in your ability to do math?and raise your grades too! I
| 677.169 | 1 |
Overview
Word problems are the most difficult part of any math course –- and the most important to both the SATs and other standardized tests. This book teaches proven methods for analyzing and solving any type of math word problem.
More About
This Book
Overview
Word problems are the most difficult part of any math course –- and the most important to both the SATs and other standardized tests. This book teaches proven methods for analyzing and solving any type of math word problem.
Related Subjects
Meet the Author
Allan G. Bluman taught mathematics and statistics in high school, college, and graduate school for 39 years. He is the recipient of "An Apple for the Teacher Award" for bringing excellence to the learning environment and the "Most Successful Revision of a Textbook" award from McGraw-Hill. Mr. Bluman's biographical record appears in Who's Who in American Education, 5th Edition. He is the author of three mathematics textbooks and several highly successful books in the DeMYSTiFieD series, including Pre-Algrebra DeMYSTiFieD and Business Math DeMYSTiFie
| 677.169 | 1 |
Volume 8, Number 36
8 September 2003 Vol. 8, No. 36
THE MATH FORUM INTERNET NEWS
T2T FAQ | Polyhedra - Jorge Rezende
Preparing for University Calculus | NOVA Teachers
THE MATH FORUM'S TEACHER2TEACHER FAQ
As many of you greet the new school year, browsing the
Teacher2Teacher FAQ can provide you with new ideas:
- First Day of School
- First Year Suggestions
Pages to share with parents:
- Homework Help Ideas
- Fraction Help
- Multiplication Help
- Searching for Information
Pages to plan ahead with:
- 100th Day of School
- Assessment Preparation
- Literature and Mathematics
- Metric Week
- Pi Day
Have a question about teaching? Submit it here:
-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-
POLYHEDRA - JORGE REZENDE
Professor Rezende of the Mathematics Department of the
Faculty of Science of the University of Lisbon has written a
variety of puzzles with polyhedra and numbers.
Download PDFs, print them out, and then follow Rezende's
instructions to build the polyhedra. For each puzzle, place
the number plates over the polyhedron faces in such a way
that the two numbers near each edge of the polyhedron are
equal.
-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-
PREPARING FOR UNIVERSITY CALCULUS
The Mathematics and Statistics Committee of the Atlantic
Provinces Council on the Sciences has posted a booklet
entitled "Preparing for University Calculus." Neither a
textbook nor a formal diagnostic test, the booklet offers
sample questions and hints of the sort often found on
calculus placement tests.
Their web page, "Preparing for University Calculus," answers
frequently asked questions about university level calculus
courses, such as "What math will I need in university?" and
"What is calculus?"
-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-\-/-|-
NOVA TEACHERS - PBS
Teachers' featured stories of a lesson plan using one of the
NOVA films. Lesson titles include:
- Students Inclined to Study Galileo
- Exploring Research Bias
- Teaching Science and Literacy
- Placing Value on Maya Math
- Latin Classes and Roman Baths
- What's Up with the Weather
- Teaching with Trebuchets
- Bridges Form Foundation for Geometry Unit
- Melting the Iceman
- Teaching with Sextants
- Operation ObeliskMath Tools
Teacher2Teacher
Discussion Groups
Join the Math Forum
Send comments to the Math Forum Internet Newsletter editors
Donations
Ask Dr. Math Books
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Mathematical Thinking: Problem-Solving and Proofs
Aims to prepare students in the logical thinking skills, necessary to understand the fundamental ideas and proofs in mathematics. This text discusses ...Show synopsisAims to prepare students in the logical thinking skills, necessary to understand the fundamental ideas and proofs in mathematics. This text discusses mathematical language and proof techniques, applies them to questions in elementary number theory, and then develops additional techniques of proof through discrete and continuous mathematics Mathematical Thinking: Problem-Solving and Proofs. This...Good. Mathematical Thinking: Problem-Solving and Proof
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Summary: This clear, accessible treatment of beginning algebra features an enhanced problem-solving strand highlighted by A Mathematics Blueprint for Problem Solving that helps students determine where to begin the problem-solving process, as well as how to plan subsequent problem-solving steps. Also includes Step-by-Step Procedure, realistic Applications, and Cooperative Learning Activities in Putting Your Skills to Work.
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Napa CalCore concepts in Pre-Calculus include algebra in the coordinate plane, system of equations, roots of polynomials, functions & their inverses such as polynomials & rational functions, exponential & log functions, and an introduction to trigonometry, with some variations among different Pre-Calculu, photo manipulation, and photo editing. Another passion of mine is learning new languages and discovering new cultures.
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This is an interesting example of what can be done at the high-school level with almost no prerequisites. It can be studied after an introduction to algebra and geometry. It is primarily a proofs book, but also includes much discussion of how you can discover things. By "geometric inequalities" the author means inequalities that have a geometric interpretation, and most of the discussion is also based on synthetic geometry.
The book begins with a chapter on the arithmetic mean–geometric mean inequality. This is the one part of the book that is primarily algebraic rather than geometric, but this does give a good opportunity for a careful look at the number line, the concept of inequalities, and the algebraic rules for manipulating and proving inequalities.
The center of the book is the isoperimetric problem: what plane figure of a given perimeter has the largest area? The problem is tackled "in the spirit of Steiner" (p. 31), that is, from a synthetic geometry viewpoint rather than an algebraic or analytic one, and works up gradually from solving the problem for simpler polygons to the general case. Steiner assumed the existence of a figure with maximal area and was later criticized for this. The book devotes several pages to understanding why existence really is an issue that has to be dealt with, and considers several other problems where there is indeed not a maximal figure.
The isoperimetric chapter is followed by a chapter on further uses of reflection and symmetry in optimization. This is a more miscellaneous chapter than the preceding ones, and is tied together by the "reflection principle" (sometimes called the "mirror trick"), of reflecting part of a figure through a line to get a more tractable problem.
The book includes many problems, most asking for a proof, and about a quarter of the book is devoted to complete solutions to most of these problems, with hints for the rest.
Two somewhat similar books are Beckenbach & Bellman's An Introduction to Inequalities (the preceding volume in the New Mathematical Library series), that studies many of the same problems but with algebraic rather than geometric methods; and Niven's Maxima and Minima Without Calculus, that extends the ideas here to solve even more problems.
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realize the importance of making sure young students have a total grasp of math concepts, as this is the foundation for future success in the higher level math courses, which will follow them throughout life. As a Mathematics/Computer science major in undergraduate school, I graduated with a 3
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MERLOT Search - category=2513&materialType=Presentation&createdSince=2012-10-17&sort.property=dateCreated
A search of MERLOT materialsCopyright 1997-2014 MERLOT. All rights reserved.Wed, 20 Aug 2014 06:37:49 PDTWed, 20 Aug 2014 06:37:49 PDTMERLOT Search - category=2513&materialType=Presentation&createdSince=2012-10-17&sort.property=dateCreated
4434Lesson 5 - Polynomials
This lesson received an honorable mention in the 2014 SoftChalk Lesson Challenge.'We have seen quadratic functions which are either concave up or concave down, but we may need more flexibility than that. In this section we are going to discuss higher orderpolynomial functions and models. Linear functions are actually a first order polynomial and quadratics are a second order polynomial.'Section 3 - Introduction to Linear Functions
This lesson won an honorable mention in the 2014 SoftChalk Lesson Challenge.This lesson is an introduction to linear functions. Topics included are: Topic 1: Definitions of Relations and FunctionsTopic 2: Domain and Range of FunctionsTopic 3: Vertical Line TestTopic 4: Function NotationMAT 155 Statistical Analysis - What's the Message?
This lesson received an honorable mention in the 2014MANOVA
MANOVA is the sixth Linear Regression
The fourth volume in a series of Advanced Educational Statistics, Multiple Linear Regression builds Analyses & ANOVA AnalysisLinear Regression to Quantitative Statistics: Terms & Definitions, Vol. 2
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How the Program solves: Two masses a pulley and an inclined plane
~ Bring up the menu and scroll to what's asked for in the problem
~ Scroll to the specific combination of variables given in the problem
~ Input the variables given in the problem, by pressing alpha before
~ Program will review and ask for any corrections
~ Shows the specific formula required
~ Program produces the answer with all step by step calculations
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Get 90+ Programs Inside your TI-89: BUY NOW
Winter 2014 Comments
Tom-
I showed my ex, who is a calculus professor, and he was waaaaaaay impressed. And he is an arrogant ass, who never helped me ever...I could tell he wanted to hate on it, but he couldnt. -- Kristin P
You are an angel sent from above TOM!!! Thank you so much for being patient with me. I got the programs to work and I am very confident I am going to pass this class once and for all. The Double and Triple Integrals programs are a life saver! Thank You Thank You Thank You!
-Cotto
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The Pythagorean Relationship - MAT-956 does A-squared plus B-squared equals C-squared really mean? After teaching with hands-on activities, video demonstrations, animations, and comics, your students will be able to answer that question and apply the relationship in problem solving situations. This course is built around core propositions from the National Board for Professional Teaching Standards as well as national content standards
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Cliffs Quick Review for Geometry - 01 edition
Summary: When it comes to pinpointing the stuff you really need to know, nobody does it better than CliffsNotes. This fast, effective tutorial helps you master core geometry concepts -- from perimeter, area, and similarity to parallel lines, geometric solids, and coordinate geometry -- and get the best possible grade.
At CliffsNotes, we're dedicated to helping you do your best, no matter how challenging the subject. Our authors are veteran teachers and talented wri...show moreters who know how to cut to the chase -- and zero in on the essential information you need to succeed. ...show less
Ed Kohn, MS is an outstanding educator and author with over 33 years experience teaching mathematics. Currently, he is the testing coordinator and math department chairman at Sherman Oaks Center for Enriched Studies
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Find a BurlingtonAlso, there is a study on radicals, roots, complex numbers, combinational math, permutation, probability, and matrices. In addition, it consists of rules on logarithms, circle, eclipse, and hyperbola. It contains topics like solution of system equation graphs, asymptotes, and basics on trigonometric identities.
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GeoGebra is a free and multi-platform dynamic mathematics software for education in secondary schools that joins geometry,...
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Geo finds derivatives and integrals of functions and offers commands like Root or Extremum. These two views are characteristic of GeoGebra: an expression in the algebra window corresponds to an object in the geometry window and vice versa.
This site contains an extensive collection of java applets involving probability miscellany and puzzles. Written by Alexander...
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This site contains an extensive collection of java applets involving probability miscellany and puzzles. Written by Alexander Bogomolny, these applets are designed to engage the reader in interactive investigations. Background material is provided and the site serves as an excellent educational resource.
This site contains an extensive collection of java applets involving goemetry miscellany and puzzles. Written by Alexander...
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This site contains an extensive collection of java applets involving goemetry miscellany and puzzles. Written by Alexander Bogomolny, these applets are designed to engage the reader in interactive investigations. Background material is provided and the site serves as an excellent educational resource.
In this applet, the user applies Euler's Method to modeling population growth using the Malthus exponential model and the...
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In this applet, the user applies Euler's Method to modeling population growth using the Malthus exponential model and the Verhulst constrained growth model. After finding the Euler solution, the user can "check" the solution with the Adaptive Euler Approximation or with a slope field. Also, the user can enter an exact solution obtained from separating variables (or whatever) and again check the Euler solution graphically.
The material presented in the following pages are for middle school students, high school students, college students, and all...
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The material presented in the following pages are for middle school students, high school students, college students, and all who are interested in mathematics. You will find interactive programs that you can manipulate and a lot of animation that helps you to grasp the meaning of mathematical ideas.
LiveGraphics3D is a non-commercial Java 1.1 applet that enables users to put almost any three-dimensional graphics computed...
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LiveGraphics3D is a non-commercial Java 1.1 applet that enables users to put almost any three-dimensional graphics computed by Mathematica directly onto an HTML page. It can then be viewed and interactively rotated without any additional software. LiveGraphics3D also shows animations and supports parametrized graphics.
This applet is a web based lab that explores the properties of linear functions. It is one in a series of other precalculus...
see more
This applet is a web based lab that explores the properties of linear functions. It is one in a series of other precalculus labs by the same author. The directions for using Graph Explorer are contained in the Cartesian Coordinates applet.
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Saxon Math 76 Student Book (6th Grade) 4th Edition
Customized for homeschooling, Investigations and an Illustrated
Glossary, 120 Lessons: Each daily lesson includes warm-up activities,
teaching of the new concept, and practice of new and previous material,
two-color format, 744 pages.
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will show how logarithms can be introduced without
the apparatus of calculus, i.e., by only using the four basic arithmetic operations.
This is a useful exercise because it provides an alternative, and perhaps
simpler, way to broach the subject.
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Taking calculus? Then you need the Wolfram Calculus Course Assistant. This definitive app for calculus–from the world leader in math software–will help you work through your homework problems, ace your tests, and learn calculus concepts. Forget canned examples! The Wolfram Calculus Course Assistant solves your specific Calculus problems on the fly including step-by-step guidance for
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Summary: Featuring updated content, vivid applications, and integrated coverage of graphing utilities, the ninth edition of this hands-on trigonometry text guides readers step by step, from the right triangle to the unit-circle definitions of the trigonometric functions. Examples with matched problems illustrate almost every concept and encourage readers to be actively involved in the learning process. Key pedagogical elements, such as annotated examples, think boxes, cautio...show moren warnings, and reviews, help readers comprehend and retain the material. ...show less
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Precalculus : Funcations and Graphs - 4th edition
Summary: Dugopolski'sPrecalculus: Functions and Graphs, Fourth Edition, gives you the essential strategies you need to make the transition to calculus. Throughout this book, you will find carefully placed learning aids and review tools to help them learn the math without getting distracted. The new edition includes over 900 additional exercises that are specifically designed to increase student understanding and retention of the concepts. Along the way, you'll see how the algebra conne...show morects to your future calculus course, with tools like Foreshadowing Calculus and Concepts of Calculus. Dugopolski's emphasis on problem solving and critical thinking helps you be successful in this course, as well as in future calculus89431129.85 +$3.99 s/h
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High School Pre-Algebra Tutor
Specifically designed to meet the needs of high school students, REA's High School Pre-Algebra Tutor presents hundreds of solved problems with step ...Show synopsisSpecifically designed to meet the needs of high school students, REA's High School Pre-Algebra Tutor presents hundreds of solved problems with step-by-step and detailed solutions. Almost any imaginable problem that might be assigned for homework or given on an exam is covered. Builds problem-solving skills and a strong foundation for studying more advanced math topics. Valuable for students who are already studying algebra but are having difficulty with the basic concepts. The following topics are covered: integers, fractions, decimals, ratio sand proportions, percents, roots and exponents, algebraic equations, inequalities, word problems, plotting graphs, and geometry. Fully indexed for locating specific problems rapidly
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best-selling book provides an accessible introduction to discrete mathematics through an algorithmic approach that focuses on problem- solving techniques. This edition has the techniques of proofs woven into the text as a running theme and each chapter has the problem-solving corner. The text provides complete coverage of: Logic and Proofs; Algorithms; Counting Methods and the Pigeonhole Principle; Recurrence Relations; Graph Theory; Trees; Network Models; Boolean Algebra and Combinatorial Circuits; Automata, Grammars, and Languages; Computational Geometry. For individuals interested in mastering introductory discrete mathematics.
Editorial Reviews
Booknews
New edition of a time-tested text first published in 1984 in response to a need for a course that extended students' mathematical maturity and ability to deal with abstraction and included useful topics such as combinatorics, algorithms, and graphs. Intended for a one-or two- term introductory course, the text does not require knowledge of calculus, and there are no computer science prerequisites. Annotation c. by Book News, Inc., Portland, Or.
Related Subjects
Meet the Author
Richard Johnsonbaugh is Professor Emeritus of Computer Science, Telecommunications and Information Systems, DePaul University, Chicago. Prior to his 20-year service at DePaul University, he was a member and sometime chair of the mathematics departments at Morehouse College and Chicago State University. He has a B.A. degree in mathematics from Yale University, M.A. and Ph.D. degrees in mathematics from the University of Oregon, and an M.S. degree in computer science from the University of Illinois, Chicago. His most recent research interests are in pattern recognition, programming languages, algorithms, and discrete mathematics. He is the author or co-author of numerous books and articles in these areas. Several of his books have been translated into various languages. He is a member of the Mathematical Association of America.
Read an Excerpt introductory a fresh werePreface introductory afresh were worse textbook I have encountered
I agree with the others that this book needs to be burned. I have taken Discrete Mathematics one using this textbook and I also tried looking at the authors earlier edition, but it was a worthless attempt. I have to resort to using other resources such as the internet, other text, and individuals who have a firm undersatnding of the subject. The same individuals who are Computer Science or Math majors all agree that this book is trash. The author is probably a very intelligent individual but he does not explain much of anything in great detail. That is something that a novice needs to grasp a full understanding of the concepts. Without concepts the foundation is weak and clarity is non-existent. To whom it may concern choose another text book.
Was this review helpful? YesNoThank you for your feedback.Report this reviewThank you, this review has been flagged.
Anonymous
Posted July 28, 2001
I don't know what the rest of you are talking about.
I am a Comp Sci student from WCU in Pa., this is a required reading for a 100 level math course (MAT151 to be exact). I found this book to be extremely informative and helpful, especially chapter 5 on Recurrence Relations. Perhaps the people who didn't do so well were either not ready for the course material or not devoted to their studying. Either way, good or bad, go to the B.N. store and read some of the book before you buy to see if you like it or not.
Was this review helpful? YesNoThank you for your feedback.Report this reviewThank you, this review has been flagged.
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proven motivator for readers of diverse mathematical backgrounds, this book explores mathematics within the context of real life using understandable, realistic applications consistent with the abilities of any reader. Graphing techniques are emphasized, including a thorough discussion of polynomial, rational, exponential, and logarithmic functions and conics. Includes Case Studies; New design that utilizes multiple colors to enhance accessibility; Multiple source applications; Numerous graduated examples and exercises; Discussion, writing, and research problems; Important formulas, theorems, definitions, and objectives; and more. For anyone interested in precalculus.
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old Jacobs's Geometry created a revolution in the approach to teaching this subject, one that gave rise to many ideas now seen in the NCTM Standards. Since its publication nearly one million students have used this legendary text. Suitable for either classroom use or self-paced study, it uses innovative discussions, cartoons, anecdotes, examples, and exercises that unfailingly capture and hold student interest.
This focuses on guided discovery to help students develop geometric intuition
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Could it be that they refer to Texas Instruments' TI-83, -85, etc. (and
Casio makes some, too) graphing calculators? They are equipped with
matrix algebra functions, and allow the user to specify several matrices,
including column matrices such as vectors. There may even be models which
can handle non-numerical elements in the matrices, such as algebraic
expressions.
A guess, but my feeling is that it's right.
____________________________________________________
Jason St. John 617.353.2634 stjohn@bu.edu
Boston University Physics Lecture Demonstrations
On 2002-10-26.16:36 owner-tap-l@listproc.appstate.edu sent:
The preface to the 6th ed. of Fundamentals of Physics by Halliday,
Resnick, etc.
mentions "vector-capable calculators". What does this mean?
Can someone give me some examples of these?
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App Detail » Pre-Calculus
App Description
Calculus may not seem very important to you but the lessons and skills you learn will be with for your whole lifetime!
Calculus is the mathematical study of continuous change. It helps you practice and develop your logic/reasoning skills. It throws challenging problems your way which make you think. Although you may never use calculus ever again after school or college, you will definitely hold on to the lessons that calculus teaches you.
Things like time management, how to be organized, how to accomplish things on time, how to perform under pressure, how to be responsible are just some of the things Calculus helps you become proficient in. Traits that will help you succeed.
Calculus plays a big role in most universities today as students in the fields of economics, science, business, engineering, computer science, and so on are all required to take Calculus as prerequisites.
Our Pre-Calculus guide is a preliminary version of Calculus containing over 300 rules, definitions, and examples that provides you with a broad and general introduction of this subject. A valuable reference guide to have on your phone.
Even if you dont use Calculus, this app sure is a cool way to show-off some high IQ!
Like all our 'phoneflips', this fast and lightweight application navigates quick, has NO Adverts, NO In-App purchasing, never needs an internet connection and will not take up much space on your iPhone!
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Description:Fair. Book is in acceptable condition; cover shows signs of wear...Fair. Book is in acceptable condition; cover shows signs of wear. Pages are unmarked by pen or highlighter. PAGES 1, 2, 3, ARE MARKED***
Description:New. New unread book. Might have light shelf wear to cover....New. New unread book. Might have light shelf wear to cover. Featuring comprehensive instruction and practice with word problems that were developed with the latest standards-based teaching methods, Spectrum Enrichment Math provides examples of how the math skills students learn in school apply to everyday life with challenging, multi-step problems. Perfect as a supplement to classroom work or as a home school resource, as well as for proficiency test preparation, these workbooks are the essential source for parents and teachers to help bring math skills out of the textbook and into the student's
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More About
This Textbook
Overview
Introductory Algebra is typically a 1-semester course that provides a solid foundation in algebraic skills and reasoning for students who have little or no previous experience with the topic. The goal is to effectively prepare students to transition into Intermediate Algebra.
Related Subjects
Meet the Author
Elayn Martin-Gay, University of New Orleans
An award-winning instructor and best-selling author, Elayn Martin-Gay has taught mathematics at the University of New Orleans for more than 25 years. Her numerous teaching awards include the local University Alumni Association's Award for Excellence in Teaching, and Outstanding Developmental Educator at University of New Orleans, presented by the Louisiana Association of Developmental Educators.
Prior to writing textbooks, Elayn developed an acclaimed series of lecture videos to support developmental mathematics students in their quest for success. These highly successful videos originally served as the foundation material for her texts. Today, the videos are specific to each book in the Martin-Gay series. Elayn also originated the Chapter Test Prep Video CDs to help students during their most "teachable moment" —-as they prepare for a test.
Elayn's experience has made her aware of how busy instructors are and what a difference quality supports make. For this reason, she created the Instructor-to-Instructor video series. These videos provide instructors with suggestions for presenting specific math topic and concepts in basic mathematics, prealgebra, beginning algebra, and intermediate algebra. Seasoned instructors can use them as a source for alternate approaches in the classroom. New or adjunct faculty may find the CDs useful for review. They are a great resource for suggestions regarding areas they may wish to emphasize, or common trouble areas students experience, that instructors my wish to highlight.
With her textbooks series, the Chapter Test Prep Video Cd, and CD Lecture series, Elayn has sought to put success within the reach of every student and instructor.
Introduction
About This Book
Introductory Algebra, was written to provide a solid foundation in algebra for students who might have had no previous experience in algebra. Specific care was taken to ensure that students have the most up-to-date relevant text
The many factors that contributed to the success of the first edition have been retained. In preparing the Second Edition, I considered comments and suggestions of colleagues, students, and many users of the prior edition throughout the country.
Introductory Algebra, Second Edition is part of a series of texts that can include Basic College Mathematics, Second Edition; Prealgebra, Third Edition; Intermediate Algebra, Second Edition; and a combined text, Algebra A Combined Approach, Second Edition. Throughout the series pedagogical features are designed to develop student proficiency in algebra and problem solving, and to prepare students for future courses.
Key Pedagogical Features and Changes in the Second Edition
Readability and Connections. I
Problem-Solving Process. This is formally introduced in Chapter 2 with a four-step process that is integrated throughout the text. The four steps are Understand, Translate, Solve, and Interpret. The repeated use of these steps in a variety of examples shows their wide applicability. Reinforcing the steps can increase students' comfort level and confidence in tackling problems.
Applications and Connections. Every effort was made to include as many interesting and relevant real-life applications as possible throughout the text in both worked-out examples and exercise sets. In the Second Edition, the applications have been thoroughly revised and updated, and the number of applications
Practice Problems. Throughout the text, each worked-out example has a parallel Practice Problem placed next to the example in the margin. These invite students to be actively involved in the learning process before beginning the end-of-section exercise set. Practice Problems immediately reinforce a skill after it is developed. Answers appear at the bottom of the page for quick reference.
Concept Checks. These margin exercises are appropriately placed throughout the text. They allow students to gauge their grasp of an idea as it is being explained in the text. Concept Checks stress conceptual understanding at the point of use and help suppress misconceived notions before they start. Answers appear at the bottom of the page.
Increased Integration of Geometry Concepts. In addition to the traditional topics in introductory and intermediate algebra courses, this text contains a strong emphasis on problem solving and geometric concepts, which are integrated throughout. The geometry concepts presented are those most important to a student's understanding of algebra, and I have included many applications and exercises devoted to this topic. These are marked with the geometry icon. Also, geometric figures, and a review of angles, lines, and special triangles are covered in the appendices.
Helpful Hints. Helpful Hints contain practical advice on applying mathematical concepts. These are found throughout the text and strategically placed where students are most likely to need immediate reinforcement. Helpful Hunts are highlighted for quick reference.
Visual Reinforcement of Concepts. The Second Edition contains a wealth of graphics, models, photographs, and illustrations to visually clarify and reinforce concepts. These include new and updated bar graphs, line graphs, calculator screens, application illustrations, and geometric figures.
Calculator and Graphing Calculator Explorations. These optional explorations offer point-of-use instruction, through examples and exercises, on the proper use of scientific and graphing calculators as tools in the mathematical problem-solving process. Placed appropriately throughout the text, Calculator and Graphing Calculator Explorations also reinforce concepts learned in the corresponding section and motivate discovery-based learning.
Additional exercises building on the skill developed in the Explorations may be found in exercise sets throughout the text. Exercises requiring a calculator are marked with the calculator icon. Exercises requiring a graphing calculator are marked with the calculator icon. An Introduction to Using a Graphing Utility is included in the appendix.
Study Skills Reminders. New Study Skills Reminder boxes are integrated throughout the text. They are strategically placed to constantly remind and encourage students as they hone their study skills. A new Section 1.1, Tips on Success in Mathematics, provides an overview of the Study Skills needed to succeed in math. These are reinforced by the Study Skills Reminder boxes throughout the text.
Focus On. Appropriately placed throughout each chapter, these are divided into Focus on Mathematical Connections, Focus on Business and Career, Focus on the Real World, and Focus on History. They are written to help students develop effective habits for engaging in investigations of other branches of mathematics, understanding the importance of mathematics in various careers and in the world of business, and seeing the relevance of mathematics in both the present and past through critical thinking exercises and group activities.
Chapter Highlights. Found at the end of each chapter, these contain key definitions, concepts, and examples to help students understand and retain what they have learned and help them organize their notes and study for tests.
Chapter Activity. These features occur once per chapter at the end of the chapter, often serving as a chapter wrap-up. For individual or group completion, the Chapter Activity, usually hands-on or data-based, complements and extends to concepts of the chapter, allowing students to make decisions and interpretations and to think and write about algebra.
Integrated Reviews. These "mid-chapter reviews" are appropriately placed once per chapter. Integrated Reviews allow students to review and assimilate the many different skills learned separately over several sections before moving on to related material in the chapter.
Pretests. Each chapter begins with a pretest that is designed to help students identify areas where they need to pay special attention in the upcoming chapter.
Chapter Review and Test. The end of each chapter contains a review of topics introduced in the chapter. The Chapter Review offers exercises that are keyed to sections of the chapter. The Chapter Test is a practice test and is not keyed to sections of the chapter.
Cumulative Review. These features are found at the end of each chapter (except Chapters R and 1). Each problem contained in the Cumulative Review is an earlier worked example in the text that is referenced in the back of the book along with the answer. Students who need to see a complete worked-out solution, with explanation, can do so by turning to the appropriate example in the text.
Student Resource Icons. At the beginning of each section, videotape and CD, tutorial software, Prentice Hall Tutor Center, and solutions manual icons are displayed. These icons help reinforce that these learning aids are available should students wish to use them to help them review concepts and skills at their own pace. These items have direct correlation to the text and emphasize the text's methods of solution.
Functional Use of Color and New Design. Elements of this text are highlighted with color or design to make it easier for students to read and study. Special care has been taken to use color within solutions to examples or in the art to help clarify, distinguish, or connect concepts.
Exercise Sets. Each text section ends with an Exercise Set. Each exercise in the set, except those found in parts labeled Review and Preview or Combining Concepts, is keyed to one of the objectives of the section. Wherever possible, a specific example is also referenced. In addition to the approximately 4400 exercises in end-of-section exercise sets, exercises may also be found in the Pretests, Integrated Reviews, Chapter Reviews, Chapter Tests, and Cumulative Reviews.
Exercises and examples marked with a video icon have been worked out step-by-step by the author in the videos that accompany this text.
Throughout the exercises in the text there is an emphasis on data and graphical interpretation via tables, charts, and graphs. The ability to interpret data and read and create a variety of types of graphs is developed gradually so students become comfortable with it. Similarly, geometric concepts—such as perimeter and area—are integrated throughout the text. Exercises and examples marked with a geometry icon have been identified for convenience.
Each exercise set contains one or more of the following features.
Mental Math. Found at the beginning of an exercise set, these mental warmups reinforce concepts found in the accompanying section and increase students' confidence before they tackle an exercise set. By relying on their own mental skills, students increase not only their confidence in themselves but also their number sense and estimation ability.
Review and Preview. These exercises occur in each exercise set (except for those in Chapters R and 1) after the exercises keyed to the objectives of the section. Review and Preview problems are keyed to previous sections and review concepts learned earlier in the text that are needed in the next section or in the next chapter. These exercises show the links between earlier topics and later material.
Combining Concepts. These exercises are found at the end of each exercise set after the Review and Preview exercises. Combining Concepts exercises require students to combine several concepts from that section or to take the concepts of the section a step further by combining them with concepts learned in previous sections. For instance, sometimes students are required to combine the concepts of the section with the problem-solving process they learned in Chapter 2 to try their hand at solving an application problem.
Writing Exercises. These exercises occur in almost every exercise set and are marked with an icon. They require students to assimilate information and provide a written response to explain concepts or justify their thinking. Guidelines recommended by the American Mathematical Association of Two Year Colleges (AMATYC) and other professional groups recommend incorporating writing in mathematics courses to reinforce concepts.
Vocabulary Checks. Vocabulary Checks, new to this edition, provide an opportunity for students to become more familiar with the use of mathematical terms as they strengthen their verbal skills. These appear at the end of the chapter before the Chapter Highlights.
Data and Graphical Interpretation. There is an emphasis on data interpretation in exercises via tables and graphs. The ability to interpret data and read and create a variety of types of graphs is developed gradually so students become comfortable with it.
Internet Excursions. These exercises occur once per chapter. Internet Excursions require students to use the Internet as a data-collection tool to complete the exercises, allowing students first-hand experience with manipulating and working with real data.
Key Content Features in the Second Edition
Overview. This new edition retains many of the factors that have contributed to its success. Even so, every section of the text was carefully reexamined. Throughout the new edition you will find numerous new applications, examples, and many real-life applications and exercises. For example, look at the exercise sets of Sections 1.8, 2.5, 2.6, 6.1, 6.4, or 8.2. Some sections have internal re-organization to better clarify and enhance the presentation.
Chapter 1 now begins with Tips for Success in Mathematics (Section 1.1). New applications, and real data enhance the chapter.
New Study Skills Reminder boxes have been inserted throughout the text. These boxes reinforce the tips from Section 1.1. They are placed strategically to encourage students to hone their study skills.
Increased Integration of Geometry Concepts. The geometry concepts presented are those most important to a student's understanding of algebra, and I have included many applications and exercises devoted to this topic. These are marked with a geometry icon. Also, geometric figures and a review of angles, lines, and special triangles are covered in the appendices.
New Examples. Detailed step-by-step examples were added, deleted, replaced, or updated as needed. Many of these reflect real life.
Exercise Sets Revised and Updated. The exercise sets have been carefully examined and extensively revised. The real-world and real-data applications have been thoroughly updated and many new applications are included. In addition, an increased number of challenging problems have been included in the new edition. Writing exercises are now included in most exercise sets and new Vocabulary Checks have been added to the end of the chapter to help students become proficient in the language of mathematics.
Enhanced Supplements Package. The Second Edition is supported by a wealth of supplements designed for added effectiveness and efficiency. New items include the MathPro 5 on-line tutorial with diagnostic and unique video clip feature, a new computerized testing system (TestGen-EQ with QuizMaster), Prentice Hall Tutor Center, digitized videos on CD, and Instructor to Instructor, Teaching Mathematics CD Series. Please see the list of supplements for descriptions.
Options for On-line and Distance Learning
For maximum convenience, Prentice Hall offers on-line interactivity and delivery options for a variety of distance learning needs. Instructors may access or adopt these in conjunction with this text.
The Companion Web site includes basic distance learning access to provide links to the text's Internet Excursions and a selection of on-line self quizzes. Email is available.
WebCT. WebCT includes distance learning access to content found in the Martin-Gay companion Web site plus more. WebCT provides tools to create, manage, and use on-line course materials. Save time and take advantage of items such as on-line help, communication tools, and access to instructor and student manuals. Your college may already have WebCT's software installed on their server or you may choose to download it.
Annotated Instructor's Edition (0-13-067684-5)
• Answers to all exercises printed on the same text page.
• Teaching Tips throughout the text placed at key points in the margin.
Instructor's Resource Manual with Tests (0-13-067692-6)
• Notes to the Instructor that include an introduction to Interactive Learning, Interpreting Graphs and Data, Alternative Assessment, Using Technology, and Helping Students Succeed.
• Two free-response Pretests per chapter.
• Eight Chapter Tests per chapter (3 multiple-choice, 5 free-response).
• Two Cumulative Review Tests (one multiple-choice, one free-response) every two chapters (after chapters 2,4,6,9).
• Eight Final Exams (3 multiple-choice, 5 free-response).
• Twenty additional exercises per section for added test exercises if needed.
• Group Activities (an average of two per chapter; providing short group activities in a convenient, ready-to-use format).
• Answers to all items.
Media Supplements
TestGen-EQ with QuizMaster CD-ROM (Windows/Macintosh) (0-13-067697-7)
• Algorithmically driven, text-specific testing program.
• Networkable for administering tests and capturing grades on-line.
• Edit and add your own questions to create a nearly unlimited number of tests and worksheets.
• Use the new "Function Plotter" to create graphs.
• Tests can be easily exported to HTML so they can be posted to the Web for student practice.
• Includes an email function for network users, enabling instructors to send a message to a specific student or an entire group.
• Network-based reports and summaries for a class or student and for cumulative or selected scores are available.
MathPro Explorer 4.0
• Network Version IBM/Mac 0-13-067696-9.
• Enables instructors to create either customized or algorithmically generated practice quizzes from any section of a chapter.
• Includes email function for network users, enabling instructors to send a message to a specific student or to an entire group.
• Network-based reports and summaries for a class or student and for cumulative or selected scores.
MathPro 4.0 Student Version (0-13-067688-8)
• Available on CD-ROM for stand alone use or can be networked in the school laboratory.
• Text specific tutorial exercises and instructions at the objective level.
• Algorithmically generated Practice Problems.
• "Watch" screen videoclips by K. Elayn Martin-Gay.
Videotape Series (0-13-067686-1)
• Written and presented by Elayn Martin-Gay.
• Keyed to each section of the text.
• Step-by-step solutions to exercises from each section of the text. Exercises that are worked in the videos are marked with a video icon.
New Digitized Lecture Videos on CD-ROM (0-13-067691-8)
• The entire set of Introductory Algebra, Second Edition lecture videotapes in digital form.
• Convenient access anytime to video tutorial support from a computer at home or on campus.
• Available shrink-wrapped with the text or stand-
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SMP 11-16 is a mathematics course for secondary schools which emphasises the relationship between mathematics and the world around us. The course materials fall into two parts. Part 1, covering the first two years, consists mainly of topic booklets arranged in strands, which enable pupils to work…
Originally published in 1992, this is the revised third edition of a text which provides a complete course for VCE mathematical methods units 1 and 2. Takes into account syllabus changes arising out of the course re-accreditation from 1997 to 2000. Each chapter contains a list of objectives and…
"We must use the ideas and experiences in this marvelous book as pilots, beacons, and exemplars for our own best practices in our schools and communities. . . . We all owe a lot to the authors, the teachers, the kids, and the founding fathers and mothers of this school for having the courage to…
Metals are a huge group of materials. This book tells you everything you need to know about them. There are loads of photos and facts. This will help you to get to grips with the topic and find answers quickly. Includes exiting photos, bite-sized chunks of information, and tips for further
| 677.169 | 1 |
Precise Calculator has arbitrary precision and can calculate with complex numbers, fractions, vectors and matrices. Has more than 150 mathematical functions and statistical functions and is programmable (if, goto, print, return, for).
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Precalculus
9780077221294
ISBN:
007722129X
Edition: 3 Pub Date: 2008 Publisher: McGraw-Hill Companies, The
Summary: The Barnett Graphs & Modelsseries in college algebra and precalculus maximizes student comprehension by emphasizing computational skills, real-world data analysis and modeling, and problem solving rather than mathematical theory. Many examples feature side-by-side algebraic and graphical solutions, and each is followed by a matched problem for the student to work. This active involvement in the learning process helps... students develop a more thorough understanding of concepts and processes. A hallmark of the Barnett series, the function concept serves as a unifying theme. A major objective of this book is to develop a library of elementary functions, including their important properties and uses. Employing this library as a basic working tool, students will be able to proceed through this course with greater confidence and understanding as they first learn to recognize the graph of a function and then learn to analyze the graph and use it to solve the problem. Applications included throughout the text give the student substantial experience in solving and modeling real world problems in an effort to convince even the most skeptical student that mathematics is really useful.
Barnett, Raymond A. is the author of Precalculus, published 2008 under ISBN 9780077221294 and 007722129X. One hundred ninety seven Precalculus textbooks are available for sale on ValoreBooks.com, one hundred used from the cheapest price of $34.94, or buy new starting at $214.99
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A Beginner's Guide to Graph Theory
This introduction to graph theory presents only those topics and material that are accessible to the true beginner in graph theory -- making the ...Show synopsisThis introduction to graph theory presents only those topics and material that are accessible to the true beginner in graph theory -- making the subject accessible to those in mathematics, computer science, engineering, and management science. The initial chapters cover the main ideas of graph theory and are followed by more specialized, application-oriented discussions. Featuring numerous examples, diagrams, illustrations, and exercises, this gentle introduction to graph theory has a distinctly applied flavor that will be welcomed by those new to the field. This item is printed on demand. Graph theory continues to...New. This item is printed on demand. Graph theory continues to be one of the fastest growing areas of modern mathematics because of its wide applicability in such diverse disciplines as computer science, engineering, chemistry, management science, social sc
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Michael Sullivan's time-tested approach focuses students on the fundamental skills they need for the course: preparing for class, practicing with homework, and reviewing the concepts. The Enhanced with Graphing Utilities Serieshas evolved to meet today's course needs by integrating the usage of graphing calculator, active-learning, and technology in new ways to help students be successful in their course, as well as in their future endeavors.
Book Description:no binding. Book Condition: NEW. This is an ebook. This is a complete solutions manual to the textbook. Solution manual ONLY, not textbook. Including very detailed worked out solutions to all the problems. The solutions manual is in high quality PDF format and will be sent to your email box within few hours. Worldwide shipping is free. Customer satisfaction guaranteed. All sales are final. book. Bookseller Inventory # etbu10276
Book Description:no binding. Book Condition: NEW. This is an ebook. This is a Test Bank to the textbook. Not textbook. The Test Bank is in high quality PDF format and will be sent to your email box within few hours. Worldwide shipping is free. Customer satisfaction guaranteed. All sales are final. book. Bookseller Inventory # etbu10277
Book Description:U.S.A.: Pearson, 2012. Hardcover. Book Condition: New. 5th or later Edition. (5b) PLEASE READ BEFORE ORDERING: This book is an ANNOTATED INSTRUCTOR'S EDITION. New and unused 6th edition which ships lightning quick (your order ships within 24-hours on any given business day). Choose EXPEDITED shipping (USPS Priority Mail), as delivery takes a mere 2-5 business days. Media Mail can take up to 10 business days. This book's paging is the same as the student edition. May contain black tape on front, back and spine of book; may contain some or all answers within the body, front and back pages of the text; may contain annotations within page margins; may not contain Infotrac, MyMathLab, Connect and/or other online accessed codes; may not include other study or lab manuals required by your instructor as course materials. Please do not order this Annotated Instructor's Edition if you're having any doubts about this item. A return and refund is a time and cost consuming option for us all. If you have any questions PLEASE inquire before ordering. Bookseller Inventory # ABE-13654075928
Book Description:Pearson. Hardcover. Book Condition: New. 0321795466 AtAGlance Books -- Orders ship next business day, with tracking numbers, from our warehouse in upstate NY. This book is in brand new condition. Bookseller Inventory # 9780321795465N
Book Description:Pearson. Hardcover. Book Condition: New. 032179546695466ZN
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101 Mathematical Projects
Contains descriptions of over one hundred projects which may be used in the classroom to introduce students to various math topics, focusing on ...Show synopsisContains descriptions of over one hundred projects which may be used in the classroom to introduce students to various math topics, focusing on lessons that demonstrate the relevance of mathematics to the real 172 p. Contains: Illustrations, black & white
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9780131871410183.33Elementary Linear Algebra : A Matrix Approach
Summary
Ideal as a reference or quick review of the fundamentals of linear algebra, this book offers amatrix-oriented approach--with more emphasis on Euclidean n-space, problem solving, and applications, and less emphasis on abstract vector spaces. It features a variety of applications, boxed statements of important results, and a large number of numbered and unnumbered examples.Matrices, Vectors, and Systems of Linear Equations. Matrices and Linear Transformations. Determinants. Subspaces and Their Properties. Eigenvalues, Eigenvectors, and Diagonalization. Orthogonality. Vector Spaces. Complex Numbers.A professional reference for computer scientists, statisticians, and some engineers.
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032153557X
9780321535573
Worksheets for Classroom or Lab Practice for Applied Basic Mathematics, Applied Basic Mathematics:These lab- and classroom- friendly workbooks offer extra practice exercises for every section of the text, with ample space for students to show their work. The worksheets list the learning objectives and key vocabulary terms for every text section and provide extra vocabulary practice.
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textbook offered by BookBoon'In this book, which is basically self-contained, the following topics are treated...
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This is a free textbook offered by BookBoon'This is a free textbook offered by BookBoon.'In this book, which is basically self-contained, the following topics are...
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This is a free textbook offered by BookBoon.'״Created by a high school math teacher with over 25 years of experience in the classroom. Develop your algebraic equation...
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״Created by a high school math teacher with over 25 years of experience in the classroom. Develop your algebraic equation solving skills through playing a Bingo game. This game will take you step by step through the process of learning how to solve the most basic equation up through multiple-step equations by way of 13 different levels and it will allow you to choose from a variety of Bingo games from straight-line Bingo through multiple patterns of Bingo all the way up to Black-out Bingo if you choose. The game will keep track of your time and points to be able to compete with others as well as yourself.״״This game is great for Pre-Algebra and Algebra I students who are just learning how to solve equations as well as for Algebra II and even Precalculus students needing to review their equation solving skills. It can also be used an Algebra tutorial for anyone preparing for a standardized test such as the S.A.T. or A.C.T.״This app costs $.99
This is a free online course offered by the Saylor Foundation.'This introductory mathematics course is for you if you have a...
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This is a free online course offered by the Saylor Foundation.'This introductory mathematics course is for you if you have a solid foundation in arithmetic (that is, you know how to perform operations with real numbers, including negative numbers, fractions, and decimals). Numbers and basic arithmetic are used often in everyday life in both simple situations, like estimating how much change you will get when making a purchase in a store, as well as in more complicated ones, like figuring out how much time it would take to pay off a loan under interest.The subject of algebra focuses on generalizing these procedures. For example, algebra will enable you to describe how to calculate change without specifying how much money is to be spent on a purchase–it will teach you the basic formulas and steps you need to take no matter what the specific details of the situation are. Likewise, accountants use algebraic formulas to calculate the monthly loan payments for a loan of any size under any interest rate. In this course, you will learn how to work with formulas that are already known from science or business to calculate a given quantity, and you will also learn how to set up your own formulas to describe various situations by translating verbal descriptions to mathematical language. In the later units of this course, you will discover another tool used in mathematics to describe numbers and analyze relationships: graphing. You will learn that any pair of numbers can be represented by a point on a coordinate plane and that a relationship between two quantities can be represented by a line or a curve.Units 6, 7, and 8 may seem more abstract than the earlier ones, as you will deal with expressions that contain mostly variables and not too many numbers. While the procedures you will master in these units might seem to have little practical application, you have to keep in mind that they result in formulas that describe very real situations in business, accounting, and science. Knowing how to perform various operations with algebraic expressions will eventually enable you to solve quadratic and even more complex equations. You will explore a variety of real-world scenarios that can be described by these kinds of equations. For example, if a ball is thrown up in the air, solving a quadratic equation will help you find out when it will hit the ground. As another example, if you know the area of a rectangular garden, then you can use a quadratic equation to find the length of each side.'
Ths is a free version of the Boundless Algebra book that can be downloaded from Amazon for a Kindle.'The Boundless Algebra...
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Ths is a free version of the Boundless Algebra book that can be downloaded from Amazon for a Kindle of solution
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Geometry 2 continues the study of geometry after Math Mammoth Geometry 1, and is suitable for grades 6-7. It concentrates on two broad and important topics: area and volume of all common shapes.
The PDF version of this book is enabled for annotation. This means that if you prefer, your student can fill it in on the computer, using the typewriter and drawing tools in Adobe Reader version 9 or greater.
In the first section of the book, which is also the longest, students learn to calculate the area of all common shapes:
triangles, parallelograms, other polygons, and circles. They also learn about Pi and the circumference of a circle.
Next, we study nets and surface area of common solids. Naturally the student needs to know how to calculate the
area of two-dimensional shapes by this point (specifically, the area of rectangles, triangles, and circle).
There is one more section about area, in which we learn how to convert between various units of area, both metric
and customary.
Lastly, the book teaches about volume of common solids. I assume the students already know how to find the
volume of a right rectangular prism (a box). First we expand this topic by calculating volumes of rectangular prisms
with fractional edge lengths. Then we go on to calculate volumes of other solids: prisms, cylinders, pyramids, and
cones.
Besides simple calculation exercises, the lessons contain many real-life applications, word problems, and
mathematical problems concerning area and volume. I have tried to create a variety of problems to encourage
students' problem-solving skills.
These topics (area and volume) involve lots of calculations, and the calculator is allowed in the problems that are
marked with a little calculator image.
The earlier version of this book (before March 2014 or before Edition 3) covered a different list of geometry topics.
After completing the order at Kagi, you will see the download links on the receipt page. You will also receive an email with download links for each book you bought, or for a zip file for the packages. You click on the links and download the books to your computer's hard drive.
In case of any problems with the download, you can always email me, and I can email the books to you directly. My contact info will be in the email you receive.
CurrClick.com carries all Blue, Golden, and Light Blue series books as downloads. They accept credit cards and Paypal. You will be able to download the products immediately upon the purchase, and also return to your account at CurrClick to redownload.
Lulu offers printed copies for the Blue, Golden, and Light Blue series books.
Rainbow Resource carries printed copies for the Light Blue series books, plus Light Blue CDs. You can also find links to Rainbow Resource on the individual product pages on this site.
By purchasing any of the books, permission IS granted for the teacher (or parent) to reproduce this material to be used with his/her students in a teaching situation; not for commercial resale. However, you are not permitted to share the material with another teacher.
In other words, you are permitted to make copies for the students/children you are teaching, but not for other teachers' usage.
Math Mammoth books are PDF files. You will need Adobe Reader to view them, including if you use a Mac or Linux. You can try other PDF viewers, but they seem to either omit or mess up some of the images personally
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Tenafly PrealgebraThe course begins with a study of algebraic expressions and integers, allowing students to combine the abstract thinking, involved with variables and the more concrete thinking involved in operations with integers. Students then progress to solving equations and inequalities as they apply such c...
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MATH 080Elementary Algebra I•
5 Cr.
Department
Division
Description:
First in a two-quarter sequence of basic algebra using a lecture/workshop format. Topics include lines and graphs, systems of equations, linear equations, and applications. Format includes self/group study and individual assistance. Intended for students with little or no algebra. Students must complete both MATH 080 and 085 to have the equivalent of MATH 097. Recommended: Basic arithmetic skills.
Outcomes:
After completing this class, students should be able to:
Perform the operations of addition, subtraction, multiplication and division with real numbers.
Evaluate numerical and algebraic expressions using the Order of Operations.
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Cliffside Park Calculus like most people, it probably seemed really hard. But years later, is it hard now? After elementary school, these are routine math problems most people can easily accomplish
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groups of managers, clients, customers, and the general public. Able to apply concepts of basic algebra and geometry such as fractions, percentages, ratios, and proportions to practical situations. Able to calculate figures and amounts...
| 677.169 | 1 |
Elementary Statistics-Text - 8th edition
Summary: ELEMENTARY STATISTICS: A STEP BY STEP APPROACH is for introductory statistics courses with a basic algebra prerequisite. The book is non-theoretical, explaining concepts intuitively and teaching problem solving through worked examples and step-by-step instructions. In recent editions, Al Bluman has placed more emphasis on conceptual understanding and understanding results, along with increased focus on Excel, MINITAB, and the TI-83 Plus and TI-84 Plus graphing calculators; computing ...show moretechnologies commonly used in such39.93 +$3.99 s/h
Good
Recycle-A-Textbook Lexington, KY
007338610339.97 +$3.99 s/h
LikeNew
Nivea Books Lynnwood, WA
Hardcover Fine 0073386103 Like New copy, without any marks or highlights. Has shelf wear on covers. Has Formula Card. NO CD. This is Student US Edition. A+ Customer Service!
$41.31 +$3.99 s/h
Good
Penntext Downingtown, PA
Sorry, CD MISSING. This is an INSTRUCTOR COPY. May have minimal notes/highlighting, minimal wear/tear. Please contact us if you have any Questions.
$48.07 +$3.99 s/h
Good
TextbookBarn Woodland Hills, CA
007338610349.49 +$3.99 s/h
Acceptable
ACME Roadrunner BEEP ZIP BANG Lawrenceville, GA
Hardcover Reader copy No highlighting or markings, but slight separation of front cover from spine-fixed with tape and hot glue. Totally readable
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Peer Review
Ratings
Overall Rating:
This site visually presents lectures on pre-algebra and algebra usually between 10 and 20 minutes in length. In each movie, a teacher is interacting with a student as the lesson is presented and discussed. The notes for each class are also made available to print so that a student watching the video can easily follow along.
Every lecture provides a set of homework exercises for students to practice and reinforce their understanding of the material. Each learning session also culminates in an online quiz that helps the student assess his/her learning of the material.
Learning Goals:
To teach and assess student learning of pre-algebra.
Target Student Population:
Students in middle school and possibly high school who are in the process of learning pre-algebra or algebra concepts. The site also recommends being used by college students taking college pre-algebra or other students in business or science who just want to brush up on their math.
Prerequisite Knowledge or Skills:
Prerequisites would be similar to those for regular pre-algebra or algebra classes. That would include some type of basic math skills.
Type of Material:
Online course, Lecture/Presentation with Notes & Worksheets, Quizzes
Recommended Uses:
These lessons can be used as online modules or used simply as supplemental material to reinforce concepts a student needs help with. Good tool for assessment of student learning.
Evaluation and Observation
Content Quality
Rating:
Strengths:
The lessons are nicely sequenced in a logical order. Each lesson describes the concepts in detail in an interactive lecture format. There are also notes included (in both English and Spanish) that are well organized and can be printed for the student to use while listening to the lesson.
Content appears to be in an ongoing process of making improvements. One of these improvements is using frames to have the lecture and the notes and tools all visible on the screen simultaneously.
The information in the video aligns with the assessment. Content is accurate and complete for pre-algebra concepts.
The learner, or instructor, can select a learning path by choosing a specific unit. The units build on previous concepts as would be common in a math course. The format of the lecture is a cross between a reality show and a sit-com. It is engaging with humor, drama.
Good presentation design minimizes visual search. Text is legible. Images are labeled and free of clutter. Color and use of symbols are aesthetically pleasing. No obvious split-attention or redundancy effects that distract from learning.
Concerns:
While the pre-algebra course is complete, the algebra class lessons are still being developed. The scope and sequence has been defined and the list is on display, but not all of the sections have the lessons with them yet.
It is not very interactive; a student just watches the video. Interaction is just press start or pause button.
Feedback on quiz gives correct answer, but not how to get it. The feedback does not inform learners of their level of competence relative to learning goals
Students may get bored watching and not doing.
Potential Effectiveness as a Teaching Tool
Rating:
Strengths:
Units have quizzes that can be taken to assess the student's learning. There are usually 3 different versions of the quiz, so if students do poorly on a quiz, they can review the material and come back and take a different version of the quiz for that unit.
Exam reviews are also helpful in preparing the student to take exams. Learning objectives are well laid out and the material progresses at a decent pace for the average student.
Concerns:
Students would not know what they did wrong on a quiz, just given the correct answer. Would like to see feedback specific for the error the student had.
Ease of Use for Both Students and Faculty
Rating:
Strengths:
The course is nicely laid out in sequence so students can start at the beginning and complete the whole course, or pick and choose units or sections they want to use as refreshers to review particular concepts they need extra help with.
This module is easy to use and navigate through. The instructions are clear and the videos are appealing and engaging.
The bilingual feature is also very nice, making it available to Spanish speaking students, who may have difficulty in the regular class if the instructor is speaking English.
Concerns:
None
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MULTIPLY your chances of understanding DISCRETE MATHEMATICS If you're interested in learning the fundamentals of discrete mathematics but can't seem to get your brain to function, then here's your solution. Add this easy-to-follow guide to the equation and calculate how quickly you learn the essential concepts.
Written by award-winning math professor... more...
Clear, rigorous definitions of mathematical terms are crucial to good scientific and technical writing-and to understanding the writings of others. Scientists, engineers, mathematicians, economists, technical writers, computer programmers, along with teachers, professors, and students, all have the need for comprehensible, working definitions of mathematical... more...
You know mathematics. You know how to write mathematics. But do you know how to produce clean, clear, well-formatted manuscripts for publication? Do you speak the language of publishers, typesetters, graphics designers, and copy editors? Your page design-the style and format of theorems and equations, running heads and section headings, page breaks,... more...
This text explores the many transformations that the mathematical proof has undergone from its inception to its versatile, present-day use, considering the advent of high-speed computing machines. Though there are many truths to be discovered in this book, by the end it is clear that there is no formalized approach or standard method of discovery to... more...
This book treats all of the most commonly used theories of the integral. After motivating the idea of integral, we devote a full chapter to the Riemann integral and the next to the Lebesgue integral. Another chapter compares and contrasts the two theories. The concluding chapter offers brief introductions to the Henstock integral, the Daniell integral,... more...
This text provides a masterful and systematic treatment of all the basic analytic and geometric aspects of Bergman's classic theory of the kernel and its invariance properties. These include calculation, invariance properties, boundary asymptotics, and asymptotic expansion of the Bergman kernel and metric. Moreover, it presents a unique compendium... more...
Foundations of Analysis covers the basics of real analysis for a one- or two-semester course. In a straightforward and concise way, it helps students understand the key ideas and apply the theorems. The book?s accessible approach will appeal to a wide range of students and instructors.
Each section begins with a boxed introduction that familiarizes... more...
Requiring only basic background in real analysis and a little linear algebra, this book presents an analytic way of looking at convexity theory. It also provides some background in classical geometric theory. The book illustrates how modern mathematics is developed and how geometric ideas can be studied analytically. It guides readers in thinking... more...
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This student-focused text addresses individual learning styles through the use of a complete study system that starts with a learning styles inventory and presents targeted learning strategies designed to guide students toward success in this and future college-level courses.
Comes With13 CD's. Each CD has about an hour of video. Digital Video Tutor is included with the textbook: Elementary and Intermediate Algebra: Graphs & Models 2nd edition Published by Pearson / Addison Wesley. A study guide is also included.
Students who approach math with trepidation will find that Elementary and Intermediate Algebra, Second Edition, builds competence and confidence. The study system, introduced at the outset and used consistently throughout the text, transforms the student experience by applying time-tested strategies to the study of mathematics. Learning strategies dovetail nicely into the overall system and build on individual learning styles by addressing students' unique strengths. The authors talk to students in their own language and walk them through the concepts, showing students both how to do the math and the reasoning behind it. Tying it all together, the use of the Algebra Pyramid as an overarching theme relates specific chapter topics to the 'big picture' of algebra.
The Carson algebra series addresses two fundamental issues – individual learning styles and comprehension of key mathematical concepts – to meet the needs of today's readers. Carson's Study System, presented in the To the Student section at the front of the text, adapts to the way each reader learns to ensure their success in this and future courses. The consistent emphasis on the big picture of algebra, with pedagogy and support that helps readers put each new concept into proper context, encourages conceptual understanding.
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...
More About
This Book
and complex systems.
In addition to offering a unified presentation of continuous and discrete time systems, this treatment integrates computing comfortably into the text. Appendixes feature important background material, including a gentle introduction to differential equations and explanations of how to write MATLAB, Mathematica, and C programs to compute dynamical systems. Prerequisites for advanced undergraduates and graduate students include two semesters of calculus and one semester of linear algebra.
Editorial Reviews
Booknews
A textbook on dynamical systems, for sophomore-junior level students who want to continue exploring mathematics beyond linear algebra, but who may not be ready for highly abstract material. It presents both the "classical" theory of linear systems and the "modern" theory of nonlinear and chaotic systems; works with both continuous and discrete time systems; integrates computing into the text; and includes a wide variety of topics, such as bifurcation, symbolic dynamics, fractals, and complex systems. A collection of programs written in MATLAB is available as a supplement. Four stunning color plates (2pp
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Stay mobile, continue learning - Transfer class assignments from handheld to computer. Complete work outside of school using student software. On the desktop at home or a laptop on the bus, at the library, coffee shop or wherever. Explore higher-level math concepts - Explore symbolic algebra and symbolic calculus, in addition to standard numeric calculations. View exact values - in the form of variables such as x and y, radicals and pi - when doing step-by-step arithmetic, algebraic and calculus calculations.Visualize in full color - Color-code equations, objects, points and lines on the full-color, backlit display. Make faster, stronger connections between equations, graphs and geometric representations on screen. Real-world images - Use digital images or your own photos. Overlay and color-code math and science concepts. Discover real-world connections. Recharge with ease - The installed TI-Nspire Rechargeable Battery is expected to last up to two weeks of normal use on a single charge. No alkaline batteries needed. Calculate in style - The sleek TI-Nspire CX handheld is the thinnest and lightest TI graphing calculator model to date. It's also the brightest with a high-resolution, full-color display that makes it easy to see every exponent, variable and line. 3D Graphing - Graph and rotate (manually and automatically) 3D functions. Change the wire or surface color of your 3D graph
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College Algebra : Graphs and Models - 3rd edition
ISBN13:978-0077221287 ISBN10: 0077221281 This edition has also been released as: ISBN13: 978-0073051956 ISBN10: 0073051950
Summary: TheBarnett Graphs & Modelsseries in college algebra and precalculus maximizes student comprehension by emphasizing computational skills, real-world data analysis and modeling, and problem solving rather than mathematical theory. Many examples feature side-by-side algebraic and graphical solutions, and each is followed by a matched problem for the student to work. This active involvement in the learning process helps students develop a more thorough understanding of concepts and proce...show moresses.A hallmark of the Barnett series, the function concept serves as a unifying theme. A major objective of this book is to develop a library of elementary functions, including their important properties and uses. Employing this library as a basic working tool, students will be able to proceed through this course with greater confidence and understanding as they first learn to recognize the graph of a function and then learn to analyze the graph and use it to solve the problem. Applications included throughout the text give the student substantial experience in solving and modeling real world problems in an effort to convince even the most skeptical student that mathematics is really useful221281 Item in good condition. Textbooks may not include supplemental items i.e. CDs, access codes etc... All day low prices, buy from us sell to us we do it all!!
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0471534595 Used texts do not include any supplemental material such as access codes, info-trac, CDs, etc...All text is legible and may contain; markings, highlighting, bookstore ...stickers, worn corners, folded pages etc. Satisfaction Guaranteed. Orders ship same or next business day w/ trackingRead moreShow Less
Good Ships from UK in 48 hours or less (usually same day). Your purchase helps support the African Children's Educational Trust (A-CET). Ex-library, so some wear and internal ...barcode tAdvanced calculus has had a fundamental and seminal role in the development of the basic theory underlying statistical methodology. The last thirty years have seen a remarkable growth in the field of statistics, further underscoring the interconnection between the disciplines and the need for students and professionals to acquire a facility in the fundamentals of advanced calculus. Filling the information gulf often faced by graduate students in statistics with an experience of only introductory calculus, this applications-oriented text is a clear, well-paced, and highly rigorous introduction to the themes and topics central to advanced calculus. The book's careful theoretical explanation makes it also suitable for students in mathematics, especially those who may be interested in a minor in statistics. In addition, the book can serve as a reference for a wide spectrum of advanced calculus topics for practicing statisticians. If used as a text, the entire book would be suitable for a two-semester or three-quarter course. Each chapter in the book features a complete section describing specific applications in statistics of the material given. Numerous exercises, all classified by discipline (mathematics and statistics), are provided, in addition to an annotated bibliography for the benefit of the reader. The book's sequence of topics builds gradually from the fundamentals of advanced calculus, progressing finally to more specialized subject areas. The material in Chapters 1-6 cover the core of advanced calculus including continuity, differentiation, integration, and the theory of infinite sequences and series. Multidimensional calculus is discussed in Chapter 7. Chapters 8 and 9, more specialized than 1-7, cover certain aspects of optimization in statistics and approximation of continuous functions. This is the only advanced calculus book that emphasizes applications in statistics. Students and professionals will find Advanced Calculus with Applications in Stati
From the Publisher
Editorial Reviews
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An application-oriented and highly rigorous introduction to the central themes of advanced calculus for statistics students, with enough theoretical explanation to be suitable for mathematics students. Topics include an introduction to set theory, linear algebra, limits and continuity, differentiation, infinite sequences and series, integration, multidimensional calculus, optimization in statistics, and approximation of functions. Exercises and an annotated bibliography are included. Annotation c. Book News, Inc., Portland, OR (booknews.com)
From the Publisher
"This is an exceptional book, which I would recommend for anyone beginning a career in statistical research." (Journal of the American Statistical Association, September 2004)
Read an Excerpt
Advanced Calculus with Applications in Statistics
John Wiley & Sons
Chapter One
The origin of the modern theory of sets can be traced back to the Russian-born German mathematician Georg Cantor (1845-1918). This chapter introduces the basic elements of this theory.
1.1. THE CONCEPT OF A SET
A set is any collection of well-defined and distinguishable objects. These objects are called the elements, or members, of the set and are denoted by lowercase letters. Thus a set can be perceived as a collection of elements united into a single entity. Georg Cantor stressed this in the following words: "A set is a multitude conceived of by us as a one."
If x is an element of a set A, then this fact is denoted by writing x [element of] A. If, however, x is not an element of A, then we write x [??] A. Curly brackets are usually used to describe the contents of a set. For example, if a set A consists of the elements [x.sub.1], [x.sub.2],..., [x.sub.n], then it can be represented as A = {[x.sub.1], [x.sub.2],..., [x.sub.n]}. In the event membership in a set is determined by the the satisfaction of a certain property or a relationship, then the description of the same can be given within the curly brackets. For example, if A consists of all real numbers x such that [chi square] > 1, then it can be expressed as A = {x|[chi square] > 1}, where the bar | is used simply to mean "such that." The definition of sets in this manner is based on the axiom of abstraction, which states that given any property, there exists a set whose elements are just those entities having that property.
Definition 1.1.1. The set that contains no elements is called the empty set and is denoted by [empty set].
Definition 1.1.2. A set A is a subset of another set B, written symbolically as A [subset] B, if every element of A is an element of B. If B contains at least one element that is not in A, then A is said to be a proper subset of B.
Definition 1.1.3. A set A and a set B are equal if A [subset] B and B [subset] A. Thus, every element of A is an element of B and vice versa.
Definition 1.1.4. The set that contains all sets under consideration in a certain study is called the universal set and is denoted by [Omega].
1.2. SET OPERATIONS
There are two basic operations for sets that produce new sets from existing ones. They are the operations of union and intersection.
Definition 1.2.1. The union of two sets A and B, denoted by A [union] B, is the set of elements that belong to either A or B, that is,
A [union] B = {x|x [element of] A or x [element of] B}.
This definition can be extended to more than two sets. For example, if [A.sub.1], [A.sub.2],..., [A.sub.n] are n given sets, then their union, denoted by [[union].sup.n.sub.i=1][A.sub.i], is a set such that x is an element of it if and only if x belongs to at least one of the [A.sub.i] (i = 1, 2,..., n).
Definition 1.2.2. The intersection of two sets A and B, denoted by A [intersection] B, is the set of elements that belong to both A and B. Thus
A [intersection] B = {x|x [element of] A and x [element of] B}.
This definition can also be extended to more than two sets. As before, if [A.sub.1], [A.sub.2],..., [A.sub.n] are n given sets, then their intersection, denoted by [[intersection].sup.n.sub.i]=1 [A.sub.i], is the set consisting of all elements that belong to all the [A.sub.i] (i = 1, 2,..., n).
Definition 1.2.3. Two sets A and B are disjoint if their intersection is the empty set, that is, A [intersection] B = [empty set].
Definition 1.2.4. The complement of a set A, denoted by [bar.A], is the set consisting of all elements in the universal set that do not belong to A. In other words, x [element of] [bar.A] if and only if x [??] A.
The complement of A with respect to a set B is the set B - A which consists of the elements of B that do not belong to A. This complement is called the relative complement of A with respect to B.
From Definitions 1.1.1-1.1.4 and 1.2.1-1.2.4, the following results can be concluded:
Result 1.2.1. The empty set [empty set] is a subset of every set. To show this, suppose that A is any set. If it is false that [empty set] [subset] A, then there must be an element in [empty set] which is not in A. But this is not possible, since [empty set] is empty. It is therefore true that [empty set] [subset] A.
Definition 1.2.5. Let A and B be two sets. Their Cartesian product, denoted by A x B, is the set of all ordered pairs (a, b) such that a [element of] A and b [element of] B, that is,
A x B = {(a, b)|a [element of] A and b [element of] B}.
The word "ordered" means that if a and c are elements in A and b and d are elements in B, then (a, b) = (c, d) if and only if a = c and b = d.
The preceding definition can be extended to more than two sets. For example, if [A.sub.1], [A.sub.2],..., [A.sub.n] are n given sets, then their Cartesian product is denoted by [x.sup.n.sub.i=1][A.sub.i] and defined by
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]
Here, ([a.sub.1], [a.sub.2],..., [a.sub.n]), called an ordered n-tuple, represents a generalization of the ordered pair. In particular, if the [A.sub.i] are equal to A for i = 1, 2,..., n, then one writes [A.sup.n] for [x.sup.n.sub.i=1]A.
The following results can be easily verified:
Result 1.2.13. A x B = [empty set] if and only if A = [empty set] or B = [empty set].
Definition 1.3.1. A relations [rho] from A to B is a subset of A x B, that is, [rho] consists of ordered pairs (a, b) such that a [element of] A and b [element of] B. In particular, if A = B, then [rho] is said to be a relation in A.
Whenever [rho] is a relation and (x, y) [element of] [rho], then x and y are said to be [rho]-related. This is denoted by writing x p y.
Definition 1.3.2. A relation [rho] in a set A is an equivalence relation if the following properties are satisfied:
1. [rho] is reflexive, that is, a [rho] a for any a in A.
2. [rho] is symmetric, that is, if a [rho] b, then b [rho] a for any a, b in A.
3. [rho] is transitive, that is, if a [rho] b and b [rho] c, then a [rho] c for any a, b, c in A.
If [rho] is an equivalence relation in a set A, then for a given [a.sub.0] in A, the set
ITLITL([a.sub.0]) = {a [element of] A|[a.sub.0] [rho] a},
which consists of all elements of A that are [rho]-related to [a.sub.0], is called an equivalence class of [a.sub.0].
Result 1.3.1. a [element of] ITLITL(a) for any a in A. Thus each element of A is an element of an equivalence class.
Result 1.3.2. If ITLITL([a.sub.1]) and ITLITL([a.sub.2]) are two equivalence classes, then either ITLITL([a.sub.1]) = ITLITL([a.sub.2]), or ITLITL([a.sub.1) and ITLITL([a.sub.2]) are disjoint subsets.
It follows from Results 1.3.1 and 1.3.2 that if A is a nonempty set, the collection of distinct [rho]-equivalence classes of A forms a partition of A.
As an example of an equivalence relation, consider that a [rho] b if and only if a and b are integers such that a - b is divisible by a nonzero integer n. This is the relation of congruence modulo n in the set of integers and is written symbolically as a [equivalent to] b (mod n). Clearly, a [equivalent to] a (mod n), since a - a = 0 is divisible by n. Also, if a [equivalent to] b (mod n), then b [equivalent to] a (mod n), since if a - b is divisible by n, then so is b - a. Furthermore, if a [equivalent to] _ b (mod n) and b [equivalent to] c (mod n), then a [equivalent to] c (mod n). This is true because if a - b and b - c are both divisible by n, then so is (a - b) + (b - c) = a - c. Now, if [a.sub.0] is a given integer, then a [rho]-equivalence class of [a.sub.0] consists of all integers that can be written as a = [a.sub.0] + kn, where k is an integer. This in this example ITLITL([a.sub.0]) is the set {[a.sub.0] + kn|k [element of] J}, where J denotes the set of all integers.
Definition 1.3.3. Let [rho] be a relation from A to B. Suppose that [rho] has the property that for all x in A, if x [rho] y and x [rho] z, where y and z are elements in B, then y = z. Such a relation is called a function.
Thus a function is a relation [rho] such that any two elements in B that are [rho]-related to the same x in A must be identical. In other words, to each element x in A, there corresponds only one element y in B. We call y the value of the function at x and denote it by writing y = f(x). The set A is called the domain of the function f, and the set of all values of f(x) for x in A is called the range of f, or the image of A under f, and is denoted by f(A). In this case, we say that f is a function, or a mapping, from A into B. We express this fact by writing f: A [right arrow] B. Note that f(A) is a subset of B. In particular, if B = f(A), then f is said to be a function from A onto B. In this case, every element b in B has a corresponding element a in A such that b = f(a).
Definition 1.3.4. A function f defined on a set A is said to be a one-to-one function if whenever f([x.sub.1]) = f([x.sub.2]) for [x.sub.1], [x.sub.2] in A, one has [x.sub.1] = [x.sub.2]. Equivalently, f is a one-to-one function if whenever [x.sub.1] [not equal to] [x.sub.2], one has f([x.sub.1])[not equal to]f([x.sub.2]).
Continues...
Excerpted from Advanced Calculus with Applications in Statistics by André I. Khuri
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The Curriculum Foundations Workshop in Computer Science
By Allen Tucker
Curriculum development has been an obsession for the computer science education community during the last four decades, due to the enormous and rapid rate of evolution that has occurred in the discipline itself. During this period, the Association for Computing Machinery (ACM) has developed various curriculum standards for undergraduate programs, first in 1968, and then again in 1978 and 1991. A new standard was published in December 2001 (see for more information).
Central to each of these curriculum efforts is the important discussion of the role and flavor of mathematics within the computer science curriculum. Recently, the undergraduate mathematics curriculum committees (CUPM and CRAFTY) reached out to computer science educators and raised the central question, "what mathematics is needed for an undergraduate major in computer science, and how should it be taught?"
In response to this question, an eight-member panel of computer science educators gathered for the first of a series of Curriculum Foundations Workshops, which took place at Bowdoin College on October 28-31, 1999. The group included Owen Astrachan (Duke University), Doug Baldwin (SUNY Geneseo), Kim Bruce (Williams College), Peter Henderson (Butler University), Charles F. Kelemen (Swarthmore College), Dale Skrien (Colby College), Allen Tucker (Bowdoin College), and Charles Van Loan (Cornell University).
In the first two years of a computer science major, students should be comfortable with abstract thinking and with mathematical notation and its meaning. They should be able to both generalize from examples and create examples from generalizations. To estimate the complexity of algorithms, they should understand functions that represent different rates of growth (e.g., logarithmic, polynomial, exponential). To reason effectively about the complexity and correctness of programs, they should gain facility with formal proofs, especially induction proofs. The same kind of clear and careful thinking and expression needed for a coherent mathematical argument is needed for the effective design of a computer program.
For mathematical problem solving skills, students should be able to represent "real-world" problem situations using discrete structures such as arrays, linked lists, trees, finite graphs, other multi-linked structures, and matrices. They should be able to develop and analyze algorithms that operate on these structures. They should understand what a mathematical model is and be able to relate mathematical models to real problem domains. General problem solving strategies such as divide-and-conquer and backtracking strategies are also essential.
For specific topics, students should master the following in their first two years: logical reasoning (propositions, DeMorgan's laws, including negation with quantifiers), functions, relations (equivalence relations and partitions), sets, notation for functions and for set operations, mathematical induction (structural, strong, weak), combinatorics, finite probability, asymptotic notation (e.g., O(n2) and O(2n)), recurrence/difference equations, graphs, trees, and number systems.
Later in their undergraduate careers, students should gain experience in the following topics in order to master additional intermediate and advanced computer science coursework:
induction and diagonalization proofs, the use of counterexamples and proof by contradiction, for algorithms and theory of computation courses.
speech understanding and synthesis algorithms use transforms.
compression algorithms use wavelets
encryption algorithms use group and ring theory
The panel's general conclusion is that undergraduate computer science majors need to acquire mathematical maturity and skills, especially in discrete mathematics, early in their college education. The following topics are likely to be used in the first three courses of a computer science major: logical reasoning, functions, relations, sets, mathematical induction, combinatorics, finite probability, asymptotic notation, recurrence/difference equations, graphs, trees, and number systems. Ultimately, calculus, linear algebra, and statistics topics are also used, but none of these is needed earlier than discrete mathematics. Thus, a discrete mathematics course should ideally be offered in the first semester of college, and its prerequisites and conceptual level should be the same as the Calculus I course.
Allen Tucker is Bass Professor of Natural Science at Bowdoin College. He was the local organizer for the Curriculum Foundations Workshop in Computer Science, held at Bowdoin College in October 1999.
This is one of a series of articles on Curriculum Foundations, a project of CRAFTY, the MAA Committee on Calculus Reform and the First Two Years. Earlier articles have described the project as a whole (FOCUS, November 2000) and the workshop on the mathematics courses needed by physics students (FOCUS, March 2001 and also online). Future articles will focus on other client disciplines. CRAFTY is a subcommittee of CUPM, the Committee on the Undergraduate Program in Mathematics, which is undertaking a review of the whole undergraduate curriculum.
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Advanced undergraduates and graduate students studying quantum mechanics will find this text a valuable guide to mathematical methods. Emphasizing the unity of a variety of different techniques, it is enduringly relevant to many physical systems outside the domain of quantum theory.Concise in its presentation, this text covers eigenvalue problems in classical physics, orthogonal functions and expansions, the Sturm-Liouville theory and linear operators on functions, and linear vector spaces. Appendixes offer useful information on Bessel functions and Legendre functions and spherical harmonics. This introductory text's teachings offer a solid foundation to students beginning a serious study of quantum mechanics.
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Foundations of Analysis
Edition:
2
Publisher:
Dover Publications
Number of Pages:
427
Price:
24.95
ISBN:
9780486462967
Despite the imposing title, this book is "designed as a first encounter with rigorous, formal mathematics for students with one year of calculus". I think the authors succeed quite well in this goal in first half of the book, which treats the usual topics of introductory analysis in the real line in a leisurely way that includes many details. They are less successful in the second half of the book, which treats calculus in two dimensions, including line integrals and Green's theorem, and introductory complex analysis, up to Cauchy's integral formula. Although the authors strive to keep the leisurely pace in that second part, they succeed only partially in their treatment of calculus in the plane (their motivation for generalizations of the Fundamental Theorem of Calculus is good) and mostly not in their introduction to complex analysis: they do a lot of preparation and very few applications, except for a proof of the Fundamental Theorem of Algebra and the computation of one improper real integral. One cannot really do justice to complex analysis in fifty pages.
On the positive side, the prose is pleasant throughout the book. The first chapter, which discusses the real number system, is particularly nice, if probably a bit challenging for the intended audience. It starts with a discussion of irrational numbers, with the mandatory proof of the irrationality of the square root of 2 (including a proof of the Fundamental Theorem of Arithmetic), and then surprisingly moves directly to a construction of the reals via Dedekind cuts. The authors then consider the real numbers again, but this time axiomatically as a complete ordered field, with the least upper bound axiom as the expression of completeness. Dedekind cuts are no longer mentioned. The chapter ends with a proof of the Heine-Borel theorem, which, while not particularly difficult, will probably seem mysterious and unmotivated at that point. (It won't be used until the end of the second chapter when discussing uniform continuity and the boundedness of continuous function in compact intervals.)
The book is useful as a starting point for beginners in analysis and can be used for independent study and reference, but cannot really (and does not intend to) compete with Spivak's Calculus or Abbott's Understanding Analysis. On the other hand, the Dover price does add to its attractiveness.
Luiz Henrique de Figueiredo is a researcher at IMPA in Rio de Janeiro, Brazil. His main interests are numerical methods in computer graphics, but he remains an algebraist at heart. He is also one of the designers of the Lua language.
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{"currencyCode":"GBP","itemData":[{"},{"priceBreaksMAP":null,"buyingPrice":15.73,"ASIN":"0521665671","isPreorder":0}],"shippingId":"0521558751::rkgWoMeeKY%2FnRJRa5hFBlGg3NpMmKg2vlmzzkT9VUbzlioET7V8BPtqGo0GbFF20JSPjDnQcJT6MH1yWAGtWEDjJv3nyqVqC,0521585686::LGdBUH87mwO2RW0c%2B3AyeoUKh64gRVewKr2WzqS5%2FHoXMUM%2F9Tw2hLPva87rB%2BT0cvqDFWvnY5BUXlB5V9tlWjEo3G5jp17q,0521665671::ItaR4PtuTCf%2FvEv2%2FOanXx8F%2FH128IOp8z7lc6N8J4hO2D30gepkWbDZxA9YtTHNrikP96xcqlCpzZvFZRQg5pcN95NGLY"...a valuable addition to the resources of any teacher preparing high school students for mathematics contests. ...an ideal addition to high school libraries." CRUX with Mayhem
Book Description
This book contains almost 600 unusual and challenging multiple-choice problems designed for. The first part consists of past papers (1988–93) for the annual UK Schools Mathematical Challenge.The second part contains forty-two short papers (10 questions each) in the same style.
The book consists of question papers of the UKMT junior maths challenge for the years 1988 to 1993. In addition to this there are 42 question papers of 10 questions each. The book has 570 questions in total which is a good value for money. The second book in this series More Mathematical Challenges has slightly more challenging problems and is designed for junior maths olympiad preparation.
4.0 out of 5 starsBritish mathematics competition problems for middle school students.15 April 2007
By N. F. Taussig - Published on Amazon.com
Format:Paperback
This text contains papers from the U. K. School Mathematics Challenge for the years 1989 - 1994, each of which contains 25 multiple choice problems meant to be done without a calculator by middle school students, and 42 additional 10 problems papers in the same format provided for additional practice. The problems, which are succinctly stated and sometimes humorous, require good arithmetic skills, number sense, a rudimentary knowledge of algebra and geometry, and the willingness to think through a multi-step problem. The problems range from routine calculations to problems that require considerable ingenuity to solve.
The U. K. School Mathematical Challenge papers consist of 25 problems. The 15 problems in part A of the examination are relatively routine; the 10 problems in part B are not. Given the difficulty of the problems in part B and the time limit of one hour to complete the examination, students are encouraged to check that their answers to the problems in part A are correct before they delve into the more difficult problems in part B.
The second part of the book consists of miniature examinations. Each paper has 10 problems that increase in difficulty. The papers themselves also increase in difficulty. Some problems are more challenging than those that appeared in the competitions.
Answers to all problems are provided; solutions are not. This is the chief limitation of the book, because it is not always possible to discern a solution to a problem even when you know the correct answer. Even with this limitation, working through this text is a great way for students of this age to prepare for mathematics competitions. Their teachers could use this text as a source of enrichment, whether in class or in mathematics clubs. The companion volume, More Mathematical Challenges, to this text consists of the open-ended problems given to students who scored well enough on the U. K. School Mathematics Challenge to qualify for the U. K. Junior Mathematical Olympiad.
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Math Made Nice & Easy series simplifies the learning and use of math and lets you see that math is actually interesting and fun. This series is for people who have found math scary, but nevertheless need some understanding of math without having to deal with the complexities found in most math textbooks. Topics in Book 7 include Trigonometric Identities and Equations, Straight Lines, Conic Sections.
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Product Description
This book of tests accompanies Singapore Math's sold-separately Primary Mathematics Standards Edition 5A Textbook. These tests follow the concepts taught in the textbook, with each chapter including a Test A and Test B; each unit also includes a Cumulative Test.
Test A consists of questions that assess students' grasp of mathematical concepts while developing problem-solving skills. Test B is optional (and may be used as a re-test, if needed) and consists of multiple-choice questions aimed at testing students' comprehension of key concepts.
In Cumulative Tests A and B, questions from earlier units are incorporated into each test, focusing on review through integrated concepts and strands.
262 pages, perforated and three-hole-punched, softcover. Grades 5-6. A line-listed answer key is provided at the end of the book
| 677.169 | 1 |
MATLAB: An Introduction with Applications
9780471439974
ISBN:
0471439975
Edition: 1 Pub Date: 2003 Publisher: Wiley & Sons, Incorporated, John
Summary: This practical guide offers a beginner2s introduction to understanding and using MATLAB®. Starting with basic features, the book covers everything needed to use the program effectively, from simple arithmetic operations with scalars to creating and using arrays to three-dimensional plots and solving differential equations. Detailed images of computer screens, tutorials, worked examples, and homework questions in math..., science, and engineering make mastering the program efficient and thorough. Users gain experience running MATLAN® with examples incorporated throughout the book. Topic explanations within framed boxes help users learn the program and its commands in an easy-to-use format. Sample programs, applications, and homework problems allows instructors to show how MATLAB® is used in science and engineering. Subject matter includes script files, 2-D and 3-D plotting, function files, programming (flow control), polynomials, curve fitting, interpolation, and applications in numerical analysis.
Gilat, Amos is the author of MATLAB: An Introduction with Applications, published 2003 under ISBN 9780471439974 and 0471439975. Twenty eight MATLAB: An Introduction with Applications textbooks are available for sale on ValoreBooks.com, twenty one used from the cheapest price of $0.01, or buy new starting at $14
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College Geometry Using The Geometer's Sketchpad -With CD - 06 edition
ISBN13:978-0470412176 ISBN10: 0470412178 This edition has also been released as: ISBN13: 978-1931914543 ISBN10: 1931914540
Summary: College Geometry Using The Geometer's Sketchpad(R) is unique in its truly discovery based approach to learning geometry with complete integration of Sketchpad. Through explorations of topics ranging from triangles and circles to transformational, taxicab, and hyperbolic geometries, students hone their understanding of geometry and their ability to write rigorous mathematical proofs. It is an ideal text for students planning to teach at the middle or secondary level. ...show moreEach chapter follows the same format to best promote active learning: --Activities open each chapter allowing students the opportunity to explore topics and make conjectures using Sketchpad. --Discussions cover important theorems and concepts that solidify the conjectures made in the activities. --Exercises reinforce the discussions and advance student expertise with Sketchpad. Each new copy of the book is packaged with a valuable CD-ROM with Sketchpad documents that relate directly to the material in the text. These multi-page documents provide a starting point for every activity and provide dynamic interactive versions of all the figures in the text
| 677.169 | 1 |
all new edition of Trigonometry, derived from the author's popular Algebra & Trigonometry, Third Edition, helps students visualize mathematics for better comprehension. By creating algebraic visual side-by-sides to solve various problems in the examples, the authors show students the relationship of the algebraic solution with the visual, often graphical, solution. In addition, a variety of new tools help students better use the book for maximum effectiveness to not only pass the course, but truly understand the material.
| 677.169 | 1 |
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