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Discrete Mathematics
9780131593183
ISBN:
0131593188
Edition: 7 Pub Date: 2008 Publisher: Prentice Hall
Summary: This textbook provides an accessible introduction to discrete mathematics, using an algorithmic approach that focuses on problem-solving techniques. Each chapter has a special section dedicated to showing students how to attack and solve problems.
Johnsonbaugh, Richard is the author of Discrete Mathematics, published 2008 under ISBN 9780131593183 and 0131593188. Seven hundred twenty three Discrete Mathematic...s textbooks are available for sale on ValoreBooks.com, one hundred thirty five used from the cheapest price of $106.79, or buy new starting at $151
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're a Mathcad novice or veteran user, this first-of-its-kind book can help you quickly tap the awesome power of Mathcad, the world's most popular computer software for doing mathematics. Students and other new users are introduced to essential underlying concepts and key features of Mathcad in a user-friendly way--while the book's abundant sample problems from multiple disciplines, invaluable how-to tips, and accompanying CD-ROM exercises help experienced users discover and apply the power of previously unused Mathcad features to their daily work.
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Rates of Change and Limits Calculating Limits Using the Limit Laws Precise Definition of a Limit One-Sided Limits and Limits at Infinity Infinite Limits and Vertical Asymptotes Continuity Tangents and Derivatives
3. Differentiation
The Derivative as a Function Differentiation Rules The Derivative as a Rate of Change Derivatives of Trigonometric Functions The Chain Rule and Parametric Equations Implicit Differentiation Related Rates Linearization and Differentials
Estimating with Finite Sums Sigma Notation and Limits of Finite Sums The Definite Integral The Fundamental Theorem of Calculus Indefinite Integrals and the Substitution Rule Substitution and Area Between Curves
6. Applications of Definite Integrals
Volumes by Slicing and Rotation About an Axis Volumes by Cylindrical Shells Lengths of Plane Curves Moments and Centers of Mass Areas of Surfaces of Revolution and The Theorems of Pappus Work Fluid Pressures and Forces
Sequences Infinite Series The Integral Test Comparison Tests The Ratio and Root Tests Alternating Series, Absolute and Conditional Convergence Power Series Taylor and Maclaurin Series Convergence of Taylor Series; Error Estimates Applications of Power Series Fourier Series
Double Integrals Areas, Moments and Centers of Mass Double Integrals in Polar Form Triple Integrals in Rectangular Coordinates Masses and Moments in Three Dimensions Triple Integrals in Cylindrical and Spherical Coordinates Substitutions in Multiple Integrals
16. Integration in Vector Fields
Line Integrals Vector Fields, Work, Circulation, and Flux Path Independence, Potential Functions, and Conservative Fields Green's Theorem in the Plane Surface Area and Surface Integrals Parametrized Surfaces Stokes' Theorem The Divergence Theorem and a Unified Theory Appendices Mathematical Induction Proofs of Limit Theorems Commonly Occurring Limits Theory of the Real Numbers Complex Numbers The Distributive Law for Vector Cross Products Determinants and Cramer's Rule The Mixed Derivative Theorem and the Increment Theorem The Area of a Parallelogram's Projection on a Plane00
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First Course in Calculus (Undergraduate Texts in Mathematics) for an Amazon Gift Card of up to £3.02, which you can then spend on millions of items across the site. Trade-in values may vary (terms apply). Learn more
Book Description edition is similar in content to his book "Short Calculus", only about 3 times as big. Like "Short Calculus", it is intended as an introductory course - first year at University and possibly for good A' level students.
The material is presented in a very clear and easy to understand. It starts right at the begining. The first two chapters contains preliminary material essential to understand Calculus. The first 4 parts of the book has the same chapters as "Short Calculus", but containing extra material. If you have done both A'level and Further A' level mathematics, then you will have seen most of the topics in these first 4 parts of the book. The 5th part concerns functions of several variables. You will typically only see this at University.
Serge Lang was one of the main contributors to Nicholas Boubaki, and is both an eminent mathematician and teacher. So there is no surprise that the material contains rigour, even though the concepts are expressed so clearly and simply. Besides the clear explanations, there are some excellent proofs that are so much simpler than those I have seen in A'level texts. A good example is for addition formulae - cos(A+B)that only requires pythagoras theorem and a single identity that can itself be derived from pythagoras theorem for the proof.
Each chapter contains numerous exercises. These start of very easy and gradually get more difficult. In the appendix at the end of the book appears the answers to many of the questions in these exercises. It is the perfect book for self-study.
This has become one of my favourate introductory calculus texts. I highly recommend this book to all those readers interested in mathematics.
Serge Lang's text does an effective job of teaching you the skills you need to solve challenging calculus problems, while teaching you to think mathematically. The text is principally concerned with how to solve calculus problems. Key concepts are explained clearly. Methods of solution are effectively demonstrated through examples. The challenging exercises reinforce the concepts, while enabling you to develop the skills required to solve hard problems. Answers to the majority of exercises (not just the odd-numbered ones) are provided in a hundred page appendix, making this text suitable for self-study. In some sections, such as related rates and max-min problems, Lang provides many fully worked out solutions.
As effectively as Lang conveys the key concepts and teaches you how to solve problems, he does not neglect the subject's logical development. Topics are introduced only after their logical foundations have been laid. Results are derived. Theorems are proved when Lang feels that they will add to the reader's understanding. Through his exposition and his grouping of logically related exercises, Lang teaches the reader how a mathematician thinks about the subject.
The book is divided into five sections: review of basic material, differentiation and elementary functions, integration, Taylor's formula and series, and functions of several variables. The heart of the course is the middle three sections.
Most of the topics covered in the review of basic material should be familiar to most readers. However, it is still worth reading since there are challenging problems, properties of the absolute value function are derived from defining the absolute of a number as the square root of the square of the number, conic sections and dilations may be unfamiliar to some readers, and Lang views the material through the prism of a mathematician who knows what concepts are important for understanding higher mathematics.
Lang introduces the derivative as the slope of a curve in order to motivate the introduction of the idea of a limit. Next, Lang teaches you techniques of differentiation and shows you how to use them solve applications such as related rate problems. After a detailed discussion of the sine and cosine functions, Lang introduces the Mean Value Theorem and illustrates how it can be used for curve sketching and solving for maxima or minima. Lang covers properties of inverse functions before concluding the section by defining the natural logarithm of x as the area under the curve y = 1/x between 1 and x and defining the exponential function f(x) = e^x as its inverse.
The integral is introduced as the area under a curve, with the natural logarithm taken as the motivating example. Lang explains the relationship between integration and differentiation before introducing techniques of integration and their applications. Integration with respect to polar and parametric coordinates is introduced to expand the range of applications. The exercises introduce additional tricks that enable you to solve integrals that do not succumb to the basic techniques. A table of integrals is included on the inside of the book's front and back covers.
Lang's demonstrates the power of differential and integral calculus through his discussion of approximation of functions through their Taylor polynomials. This chapter should also give you an idea of how your calculator calculates square roots and the values of trigonometric, exponential, and logarithmic functions. The behavior of series, including convergence and divergence tests, concludes the material on single variable calculus.
The material on functions of several variables in the final section of the book is covered in somewhat greater detail in Lang's Calculus of Several Variables (Undergraduate Texts in Mathematics). Since the corresponding chapters in that text include additional sections on the cross product, repeated partial derivatives, and further techniques in partial differentiation and an expanded section on functions depending only on their distance from the origin, I chose to read these chapters in Lang's multi-variable calculus text. The material that is included here, on vectors, differentiation of vectors, and partial differentiation, should give the reader a solid foundation for a course in multi-variable calculus.
I have some caveats. There are numerous errors, including some in the answer key. Some terminology is nonstandard, notably the use of bending up (down) for concave up (down), or missing, limiting the text's usefulness as a reference. In the chapter on Taylor polynomials, when Lang requests an answer accurate to n decimal places, what he really means is that the error in the answer should be less than 1/10^n, which is not the same thing. At one point, Lang claims that the Extreme Value Theorem, which he leaves unnamed, is obvious. I turned to the more rigorous texts Calculus, Vol. 1: One-Variable Calculus with an Introduction to Linear Algebra (Second Edition) by Tom M. Apostol and Calculus by Michael Spivak, where I discovered proofs covering one and half pages of text of the Extreme Value Theorem and a preliminary result on which it depends that Lang does not state until an appendix much later in the book. Perhaps Lang meant the Extreme Value Theorem is intuitive. While I found much of the text to be clear, I sometimes found myself turning to Apostol's text for clarification when I read Lang's proofs.
Despite my reservations, I think this text is well worth reading. Reading the text and working through the exercises gives you a good understanding of the key concepts and techniques in calculus, enables you to develop strong problem solving skills, prepares you well for more advanced mathematics courses, and gives you a sense of how mathematicians think about the subject.
13 of 13 people found the following review helpful
5.0 out of 5 starsshines without all the bling and flash24 Jun 2007
By tech book guy - Published on Amazon.com
Format:Hardcover
This book by the late Prof. Lang covers calculus in a clear and concise manner. I own more than a few calculus books and this book is one of my favorites. The book looks like a math book in that it is not a 1200 page glossy coloring book with multi-colored inserts on every page. I think that the style of this book is a hugh improvement over most of the books on the market. I think a student who buys this book along with a good calculus study guide would be very well set.
27 of 31 people found the following review helpful
4.0 out of 5 starsCalculus for beginning college students28 Aug 2002
By A Customer - Published on Amazon.com
Format:Hardcover|Verified Purchase
I needed to bring my high school calculus up to speed for first year physics studies and found this to be the only book which covered the necessary ground. The material is presented in a thorough manner with the great majority of topics shown with proofs. The book is very well organized and there are abundant worked examples. Some problems are offered which deal with matters not covered in the text, but usually there is a worked example given among the answers. Lang deals with the material in a clear fashion so that the subject matter is usually not difficult to follow.On the negative side I can say that there is no human touch between the covers. His sole attempt at humor is an item following a list of problems in which he notes "relax". In the foreword he exhibits his firm belief that many freshmen arrive unprepared for college calculus, which may be true. But nowhere in the book is there a note of encouragement, so it cannot be described as reader friendly. Finally the index is pathetic--just three pages for a book of 624 pages, so that finding things can be frustrating.
12 of 12 people found the following review helpful
5.0 out of 5 starsPromotes real understanding of calculus26 Mar 2008
By Coleman Nee - Published on Amazon.com
Format:Hardcover
I had to take a refresher calculus course as a prerequisite to get into graduate school, but the assigned text (Edwards and Penney) was horrible. Like every other mass market calculus, it was filled with colorful diagrams and digressions on how to use calculators, but little in the way of explanation. Fortunately I found Lang's calculus in the university book store and it cured all of my problems. Unlike the bloated E&P, Lang's book is clear and concise. E&P covers more material to be sure, but for the essentials nothing beats Lang. After reading this book calculus became easy for me again. Which is as it should be, since calculus is a surprisingly simple subject if expalined well.
5 of 5 people found the following review helpful
3.0 out of 5 starsGood book, not great14 Feb 2010
By W. Ghost - Published on Amazon.com
Format:Hardcover
The book is OK, but I wouldn't say it's great. There are lots of exercises that ask you to do simple symbolic manipulation so you'll remember rules -- but there are too few exercises that require the reader to actually think harder and be creative. The explanations are often shallow and not as stimulating as they could be, in my opinion.
Some examples of sections that I think are not well written are the one about implicit differentiation (the discussion is too short and not clear, and there are less exercises in this section than in others); the one about rate of change (some examples are boring, like "find the rate of change of the area of a circle given the rate of change of its diameter"; he does not make it clear that he's always derives with relation to time and that, for example, the radius and height of a cylinder should be understood as functions of time, so there's a feeling of sloppiness).
It's a good book,anyway. Now, it becomes a really great book when compared to the colorful, flashy books available today.
| 677.169 | 1 |
Tagged Questions
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| 677.169 | 1 |
Modern Geometries - 5th edition
Summary: This comprehensive, best-selling text focuses on the study of many different geometries -- rather than a single geometry -- and is thoroughly modern in its approach. Each chapter is essentially a short course on one aspect of modern geometry, including finite geometries, the geometry of transformations, convexity, advanced Euclidian geometry, inversion, projective geometry, geometric aspects of topology, and non-Euclidean geometries. The Fifth Edition reflects the re...show morecommendations of the COMAP proceedings on "Geometry's Future," the NCTM standards, and the Professional Standards for Teaching Mathematics. ...show less
Introduction to Geometry. Development of Modern Geometries. Introduction to Finite Geometries. Four-Line and Four-Point Geometries. Finite Geometries of Fano and Young. Finite Geometries of Pappus and Desargues. Other Finite Geometries.
2. GEOMETRIC TRANSFORMATIONS.
Introduction to Transformations. Groups of Transformations. Euclidean Motions of the Plane. Sets of Equations for Motions of the Plane. Applications of Transformations in Computer Graphics. Properties of the Group of Euclidean Motions. Motions and Graphics of Three-Space. Similarity Transformations. Introduction to the Geometry of Fractals and Fractal Dimension. Examples and Applications of Fractals.
The Philosophy of Constructions. Constructible Numbers. Constructions in Advanced Euclidean Geometry. Constructions and Impossibility Proofs. Constructions by Paper Folding and by Use of Computer Software. Constructions with Only One Instrument.
6. THE TRANSFORMATION OF INVERSION.
Basic Concepts. Additional Properties and Invariants under Inversion. The Analytic Geometry of Inversion. Some Applications of Inversion.
Foundations of Euclidean and Non-Euclidean Geometries. Introduction to Hyperbolic Geometry. Ideal Points and Omega Triangles. Quadrilaterals and Triangles. Pairs of Lines and Area of Triangular Regions. Curves. Elliptic Geometry. Consistency; Other Modern Geometries.
1997 Hardcover69.2177.66 +$3.99 s/h
Good
TextbookBarn Woodland Hills, CA
05343518
| 677.169 | 1 |
Overview
Designed to supplement an ordinary differential equations textbook, this book focuses on the features of Mathematica that are useful for analyzing differential equations to deepen the reader's understanding.
More About
This Book
Overview
Designed to supplement an ordinary differential equations textbook, this book focuses on the features of Mathematica that are useful for analyzing differential equations to deepen the reader's understanding.
Editorial Reviews
Booknews
Uses the mathematical software system to introduce numerical methods, geometric interpretation, symbolic computation, and qualitative analysis. Assumes no prior experience with Mathematica and includes instructions for using it on Macintosh, Windows, NeXT, and the X Window System. Contains a glossary and sample notebook solutions
| 677.169 | 1 |
Course Syllabus
Course Description
Elementary draw conclusions from data.
The course introduces the student to applications in engineering, business, economics, medicine, education, the sciences, and other related fields. The use of technology (computers or graphing calculators) will be required in certain applications.
Texts, Materials, and Plug-ins
Texts
These course materials are designed for use with Collaborative Statistics by Barbara Illowsky and Susan Dean. This text may be used online or can be downloaded in PDF format at no cost through the Connexions website, or you may choose to purchase a low-cost printed copy using the "Order Printed Copy" link provided on the collection home page linked above.
Instructors wishing to customize this textbook can do so by creating a Connexions account. Connexions accounts are free and allow users to rip, mix, and burn content by updating modules and creating custom collections of educational content. Please see the Connexions website to learn more about Connexions and how you can use it to customize your students' learning experience at absolutely no cost.
Materials
Required Calculator: The TI-83 calculator is required. There are many examples that use the TI-83 calculator and contain the calculator instructions. YOU WILL BE TAUGHT HOW TO USE THE CALCULATOR IN THE COURSE LESSONS.
Throughout the course, you will be given instructions for the TI-83 Calculator. Labs and projects make use of the TI graphing calculator and may be done individually or in groups of up to four.
Please download the TI-83 calculator guidebook from this TI-83 Site. Follow the links for "TI-83 Plus Silver Edition" OR "TI-83 Plus" and use the "Guidebooks" link.
Homework and Suggested Grading
The purpose of homework is to help you learn the material in the course. You learn the most and do your best if you do the homework problems. You are expected to do the chapter PRACTICE in the workbook before attempting the homework. The answers to the Practice are in the back of the workbook. Then do the assigned odd numbered homework problems in the text and check those answers in the back of the text.
Table 1
Homework
Total Points
Lowest Points (out of 700) for:
Percentage
Exams (3 @ 100 points each)
300
A: 630
90-100%
Quizzes (12 @ 10 points each,
3 lowest dropped)
90
B: 546
78-89%
Labs (2 @ 30 points each)
60
C: 462
66-77%
Projects (2 @ 75 points each)
150
D: 385
55-65%
Final Exam
100
F: Below 385
0-54%
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Geometry Guide illustrates every geometric principle, formula, and problem type tested on the GMAT to help you understand and master the intricacies of shapes, planes, lines, angles, and objects. Each chapter builds comprehensive content understanding by providing rules, strategies, and in-depth examples of how the GMAT tests a given topic and how you can respond accurately and quickly. The Guide contains a total of 83 "In-Action" problems of increasing difficulty with detailed answer explanations. The content of the book is aligned to the latest Official Guides from GMAC (12th edition). Purchase of this book includes one year of access to Manhattan GMAT's online practice exams and Geometry question bank.
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Featuring humor, easy-to-understand explanations, and silly illustrations, Life of Fred is guaranteed to make your math studies come alive! Each text is written as a novel, including a hilarious story line based on the life of Fred Gauss. As Fred encounters the need for math during his daily exploits, he learns the methods necessary to solve his predicaments – plus loads of other interesting facts! Filled with plenty of solved examples, each book is self-teaching and reusable – perfect for families full of learners.
Introduce your students to Fred today and see how his fun, lighthearted approach to learning is revolutionizing mathematics!
Life of Fred: Beginning Algebra Expanded Edition covers the following concepts:
Finite/Infinite Numbers
Natural numbers
Whole numbers
Integers
Adding signed numbers
Ratios
Multiplying signed numbers
Proportions
Inequalities in the integers
Continued Ratios
Adding like terms
Rectangles
Trapezoids
Sectors
Symmetric Law of Equality
Order of operations
Solving Equations
Rational numbers
Set builder notation
Distance-rate-time problems
Distributive law
Reflexive Law of Equality
Proof of the distributive law
Coin problems
Age problems
Transposing
Solving systems of equations by elimination
Work problems in two unknowns
Graphs
Plotting points
Averages
Graphing linear equations
Graphing any equation
Solving systems of equations by graphing
Solving systems of equations by substitution
Inconsistent and Dependent systems of equations
Factorial function
Areas and volumes
Commutative laws
Negative exponents
Multiplying polynomials
Solving quadratic equations by factoring
Common factors factoring
Easy trinomial factoring
Difference of squares factoring
Grouping factoring
Harder trinomial factoring
Solving fractional equations
Simplifying rational expressions
Adding and subtracting rational expressions
Multiplying and dividing rational expressions
Solving pure quadratic equations
Square roots
Pythagorean theorem
Real numbers
Irrational numbers
Fractional exponents
Solving radical equations
Rationalizing the denominator
Quadratic equations in everyday life
Solving quadratic equations by completing the square
Quadratic formula
Long division of polynomials
Functions
Slope
Finding slope of a line from its equation
Slope-intercept form of a line
Range of a function
Fast way to graph y = mx + b
Fahrenheit-Celsius conversions
Graphing inequalities
Why you can't divide by zero
Absolute value
Solving inequalities in one unknown.
Product:
Life Of Fred: Beginning Algebra Expanded Edition - Grades 8-10
Vendor:
Z Twist Books
Minimum Grade:
8th Grade
Maximum Grade:
10th Grade
Weight:
2.73 pounds
Length:
10.25 inches
Width:
7 inches
Height:
1.5 inches
Subject:
Math
Curriculum Name:
Life of Fred
Learning Style:
Auditory, Visual
Teaching Method:
Charlotte Mason, Unschooling
There are currently no reviews for Life Of Fred: Beginning Algebra Expanded Edition - Grades 8-10.
| 677.169 | 1 |
1
00:00:00,000 --> 00:00:02,560
PROFESSOR: Welcome
to class again.
2
00:00:02,560 --> 00:00:05,560
This time it's not Professor
Forney it's me, so my name is
3
00:00:05,560 --> 00:00:06,810
Ralf Koetter.
4
00:00:06,810 --> 00:00:11,650
5
00:00:11,650 --> 00:00:14,150
You guys have some substantial
chalk here at MIT.
6
00:00:14,150 --> 00:00:20,130
7
00:00:20,130 --> 00:00:24,390
And I'm visiting here from the
University of Illinois, so
8
00:00:24,390 --> 00:00:26,860
Professor Forney thought I could
teach this class here.
9
00:00:26,860 --> 00:00:31,530
10
00:00:31,530 --> 00:00:31,722
All right, let's see.
11
00:00:31,722 --> 00:00:35,070
So I understand that last time,
last Wednesday, you went
12
00:00:35,070 --> 00:00:40,060
through all the finite field
stuff, meaning, so you know
13
00:00:40,060 --> 00:00:46,080
what that would mean,
the finite field.
14
00:00:46,080 --> 00:00:48,620
There's p elements, p
to the m elements.
15
00:00:48,620 --> 00:00:53,190
Whatever q you have here, is a
power of a prime in order to
16
00:00:53,190 --> 00:00:54,440
be a field.
17
00:00:54,440 --> 00:00:56,710
18
00:00:56,710 --> 00:01:02,740
So this one, as a notation,
is a ring of polynomials.
19
00:01:02,740 --> 00:01:06,010
You've seen that too.
20
00:01:06,010 --> 00:01:09,170
So I assume you know everything
about finite fields
21
00:01:09,170 --> 00:01:12,660
that you will need to know here,
at least, except for one
22
00:01:12,660 --> 00:01:14,460
more theorem which Professor
Forney told
23
00:01:14,460 --> 00:01:17,870
me he did not cover.
24
00:01:17,870 --> 00:01:21,192
And this is the fundamental
theorem of algebra.
25
00:01:21,192 --> 00:01:23,630
I have to write a little bit
smaller with this thing here,
26
00:01:23,630 --> 00:01:26,626
otherwise I'll run out.
27
00:01:26,626 --> 00:01:27,876
AUDIENCE:
[UNINTELLIGIBLE PHRASE]
28
00:01:27,876 --> 00:01:29,950
29
00:01:29,950 --> 00:01:33,440
PROFESSOR: Oh, I know, that's
probably better.
30
00:01:33,440 --> 00:01:34,690
Better.
31
00:01:34,690 --> 00:01:40,540
32
00:01:40,540 --> 00:01:53,175
With the algebra, at least
that's what it's often called,
33
00:01:53,175 --> 00:01:57,420
and really, about 60 percent of
all the proofs in algebra
34
00:01:57,420 --> 00:01:59,520
eventually boil down
to this here.
35
00:01:59,520 --> 00:02:30,510
And what it says is, polynomial
of degree m, f beta
36
00:02:30,510 --> 00:02:46,490
equals zero, at most,
m of beta.
37
00:02:46,490 --> 00:02:48,310
At least, that's one way
to formulate it.
38
00:02:48,310 --> 00:02:49,560
Let me see.
39
00:02:49,560 --> 00:02:52,420
40
00:02:52,420 --> 00:02:54,400
So that's fine.
41
00:02:54,400 --> 00:02:57,490
So what it says is a polynomial
of degree m has at
42
00:02:57,490 --> 00:03:00,560
most m roots.
43
00:03:00,560 --> 00:03:02,910
Once you all have seen that,
probably one way or another,
44
00:03:02,910 --> 00:03:06,250
but because of its importance,
I want to
45
00:03:06,250 --> 00:03:07,500
emphasize it once more.
46
00:03:07,500 --> 00:03:11,140
47
00:03:11,140 --> 00:03:14,670
Do we need a proof of this?
48
00:03:14,670 --> 00:03:17,940
In true MIT spirit we do.
49
00:03:17,940 --> 00:03:23,460
And the proof would go
something like this.
50
00:03:23,460 --> 00:03:27,020
You look at problem number one
in your homework assignment,
51
00:03:27,020 --> 00:03:30,630
and from problem number one, I
could prove that here, too,
52
00:03:30,630 --> 00:03:32,280
but since it's in the
homework, I won't.
53
00:03:32,280 --> 00:03:35,790
54
00:03:35,790 --> 00:03:38,170
You can write the following
given any beta.
55
00:03:38,170 --> 00:03:42,810
56
00:03:42,810 --> 00:03:57,440
Write f of x as f of x is
equal to plus alpha.
57
00:03:57,440 --> 00:04:01,064
58
00:04:01,064 --> 00:04:05,340
Alphas are the field so that's
by some sort of long division
59
00:04:05,340 --> 00:04:07,754
you get to that.
60
00:04:07,754 --> 00:04:09,480
That's what I'm not
going to prove.
61
00:04:09,480 --> 00:04:17,710
Then f of beta is equal zero
is the same as saying that
62
00:04:17,710 --> 00:04:19,160
alpha is equal to zero.
63
00:04:19,160 --> 00:04:29,570
So if either is a root of the
polynomial, zero, it follows
64
00:04:29,570 --> 00:04:39,464
that f of x has this thing here
as a factor, this h of x,
65
00:04:39,464 --> 00:04:52,010
x minus beta, where because of
the degree properties of
66
00:04:52,010 --> 00:05:01,740
polynomials, h of
x is m minus 1.
67
00:05:01,740 --> 00:05:05,740
And so the rest follows
by induction.
68
00:05:05,740 --> 00:05:08,580
So basically, then we can prove
that this polynomial
69
00:05:08,580 --> 00:05:10,870
has, at most, m minus
1 roots, and so on.
70
00:05:10,870 --> 00:05:13,450
And you can descend this route,
and so the rest follows
71
00:05:13,450 --> 00:05:14,700
by induction.
72
00:05:14,700 --> 00:05:18,480
73
00:05:18,480 --> 00:05:33,180
In particular we can say if f of
x has m distinct roots beta
74
00:05:33,180 --> 00:05:51,040
one, beta m, then it factors
completely into the linear
75
00:05:51,040 --> 00:05:53,310
factors like this.
76
00:05:53,310 --> 00:05:59,490
So I just wanted to quickly
state the fundamental theorem
77
00:05:59,490 --> 00:06:03,380
of algebra, since we need it
in a proof later on, and I
78
00:06:03,380 --> 00:06:04,630
think you didn't
go through it.
79
00:06:04,630 --> 00:06:09,660
80
00:06:09,660 --> 00:06:09,760
OK.
81
00:06:09,760 --> 00:06:13,080
So last time, you learned
everything about fields,
82
00:06:13,080 --> 00:06:17,010
finite fields, extension fields,
so chapter eight is
83
00:06:17,010 --> 00:06:18,750
pretty much what we
have to cover now.
84
00:06:18,750 --> 00:06:22,310
85
00:06:22,310 --> 00:06:25,120
What is the whole idea
of chapter eight?
86
00:06:25,120 --> 00:06:41,190
It's linear codes, codes, MDS
codes, and redundant codes.
87
00:06:41,190 --> 00:06:44,520
Oh, by the way, do you have any
questions about this here?
88
00:06:44,520 --> 00:06:45,020
That in any way?
89
00:06:45,020 --> 00:06:46,270
It's pretty straight, right?
90
00:06:46,270 --> 00:06:48,350
91
00:06:48,350 --> 00:06:52,280
OK, so I understand in chapter
six or so, you had already
92
00:06:52,280 --> 00:06:53,876
linear codes over the
binary fields.
93
00:06:53,876 --> 00:06:56,540
94
00:06:56,540 --> 00:07:05,610
So let's just define codes over
a larger field, formally,
95
00:07:05,610 --> 00:07:12,380
a linear code C of length n.
96
00:07:12,380 --> 00:07:31,180
97
00:07:31,180 --> 00:07:39,620
No subspace of Fn.
98
00:07:39,620 --> 00:07:41,050
So whatever the field is.
99
00:07:41,050 --> 00:07:43,890
So F could be any extension
field, could be the binary
100
00:07:43,890 --> 00:07:49,080
field, so it really generalizes
a definition of
101
00:07:49,080 --> 00:07:51,670
code, of what a linear
code is.
102
00:07:51,670 --> 00:07:54,330
OK, so it's a subspace.
103
00:07:54,330 --> 00:07:57,690
What can be derived from that?
104
00:07:57,690 --> 00:08:00,580
Since it's a subspace,
it's a group.
105
00:08:00,580 --> 00:08:05,440
And then we can derive minimum
distance properties.
106
00:08:05,440 --> 00:08:09,620
So let's first define it again,
since it's slightly
107
00:08:09,620 --> 00:08:11,865
different than the definition
for binary codes.
108
00:08:11,865 --> 00:08:21,310
109
00:08:21,310 --> 00:08:32,480
Between Fn, say Fqn.
110
00:08:32,480 --> 00:08:35,530
So I denote the vectors
with an underscore.
111
00:08:35,530 --> 00:08:38,840
I think in the notes, it's
boldface notation, so
112
00:08:38,840 --> 00:08:41,835
translate that online
as I go here.
113
00:08:41,835 --> 00:08:53,920
The distance between two words
x and y, given as dx, the
114
00:08:53,920 --> 00:09:00,786
number of positions that
x_i is unequal to y_i.
115
00:09:00,786 --> 00:09:02,540
AUDIENCE: What's
the subscript?
116
00:09:02,540 --> 00:09:04,230
PROFESSOR: There, a q.
117
00:09:04,230 --> 00:09:06,650
Oh, this is another thing
I should warn you about.
118
00:09:06,650 --> 00:09:10,890
My handwriting is bound to
deteriorate during class.
119
00:09:10,890 --> 00:09:13,830
So I usually start out
reasonably okay, towards the
120
00:09:13,830 --> 00:09:15,570
end of the class it's --
121
00:09:15,570 --> 00:09:18,370
I tell my students to throw
little pieces of chalk at me
122
00:09:18,370 --> 00:09:23,710
when it gets too bad and I'm
not facing them, so please
123
00:09:23,710 --> 00:09:25,400
just say something if
it gets too bad.
124
00:09:25,400 --> 00:09:28,720
125
00:09:28,720 --> 00:09:30,595
So distance is defined
as that, quickly.
126
00:09:30,595 --> 00:09:33,770
127
00:09:33,770 --> 00:09:36,290
So it doesn't really matter
what the values are here.
128
00:09:36,290 --> 00:09:39,700
The x_i and the y_i could
assume different values.
129
00:09:39,700 --> 00:09:44,050
It's a somewhat coarse measure
for the real, the difference
130
00:09:44,050 --> 00:09:49,520
between code words, or
difference between words.
131
00:09:49,520 --> 00:09:52,560
Why do you think I
say it's coarse?
132
00:09:52,560 --> 00:09:53,965
In digital communications
in particular?
133
00:09:53,965 --> 00:09:58,760
134
00:09:58,760 --> 00:09:59,670
Good question, right?
135
00:09:59,670 --> 00:10:04,600
In the end, we want to map that
into a modulation scheme.
136
00:10:04,600 --> 00:10:06,870
In the end, we want to map our
codes that we are deriving
137
00:10:06,870 --> 00:10:08,210
here into modulation schemes.
138
00:10:08,210 --> 00:10:09,630
In the end, we want
to embed them into
139
00:10:09,630 --> 00:10:11,610
some Euclidean space.
140
00:10:11,610 --> 00:10:15,060
Now, different elements of our
alphabet we will map to
141
00:10:15,060 --> 00:10:17,900
different elements in
Euclidean space.
142
00:10:17,900 --> 00:10:22,650
So basically, approximating
their distance relation in
143
00:10:22,650 --> 00:10:25,410
Euclidean space, which we are
really interested in with the
144
00:10:25,410 --> 00:10:30,630
Hamming distance here is pretty
coarse, but we can do,
145
00:10:30,630 --> 00:10:31,290
so we do that.
146
00:10:31,290 --> 00:10:32,540
It's an approximation,
at least.
147
00:10:32,540 --> 00:10:35,660
148
00:10:35,660 --> 00:10:36,145
That clear?
149
00:10:36,145 --> 00:10:37,770
All set?
150
00:10:37,770 --> 00:10:38,285
All right.
151
00:10:38,285 --> 00:10:38,855
AUDIENCE:
[UNINTELLIGIBLE PHRASE] the
152
00:10:38,855 --> 00:10:39,926
Hamming distance
[UNINTELLIGIBLE] the same as
153
00:10:39,926 --> 00:10:42,450
the Euclidean distance?
154
00:10:42,450 --> 00:10:44,035
PROFESSOR: Well, it depends
on the modulation scheme.
155
00:10:44,035 --> 00:10:45,970
It very much depends on
the modulation scheme.
156
00:10:45,970 --> 00:10:54,890
If you have a 8-PSK scheme,
where you would label, put in
157
00:10:54,890 --> 00:10:58,730
the words, here, with three
bit symbols, or with the
158
00:10:58,730 --> 00:11:05,900
symbol from F8, then it's
definitely different.
159
00:11:05,900 --> 00:11:08,690
It's definitely different.
160
00:11:08,690 --> 00:11:10,460
So if you do anti-polar
signaling, then
161
00:11:10,460 --> 00:11:13,970
it's directly reflected.
162
00:11:13,970 --> 00:11:17,510
OK, I'm starting to
digress already.
163
00:11:17,510 --> 00:11:22,930
164
00:11:22,930 --> 00:11:32,440
So just for completeness,
minimum distance, minimum
165
00:11:32,440 --> 00:12:01,540
Hamming, of a code subset Fqn is
d as a minimum code of dxy,
166
00:12:01,540 --> 00:12:04,682
and they have to be different,
it's the same as before.
167
00:12:04,682 --> 00:12:08,250
168
00:12:08,250 --> 00:12:10,340
So now if I claim that --
169
00:12:10,340 --> 00:12:15,610
170
00:12:15,610 --> 00:12:21,400
so the minimum distance of a
code is also given by the
171
00:12:21,400 --> 00:12:36,870
minimum between 0 and x in the
code 0 and x, and this is
172
00:12:36,870 --> 00:12:46,260
minimum of the Hamming weight
of x, and you could
173
00:12:46,260 --> 00:12:50,080
do 0 x in the code.
174
00:12:50,080 --> 00:12:51,440
So that's all old stuff.
175
00:12:51,440 --> 00:12:55,850
I just write it down so
we get started here.
176
00:12:55,850 --> 00:13:00,050
Is that clear, from the
group property, why
177
00:13:00,050 --> 00:13:01,680
this would be true?
178
00:13:01,680 --> 00:13:05,990
So if you just take this, we can
add basically x to both x
179
00:13:05,990 --> 00:13:08,890
and y, just translating
the whole relation
180
00:13:08,890 --> 00:13:10,380
to somewhere else.
181
00:13:10,380 --> 00:13:13,380
So in particular, we translate
it here, once we have it here,
182
00:13:13,380 --> 00:13:16,890
than the distance between 0
and x is just the weight.
183
00:13:16,890 --> 00:13:18,660
OK.
184
00:13:18,660 --> 00:13:21,140
So far, so good.
185
00:13:21,140 --> 00:13:22,410
Now what is next?
186
00:13:22,410 --> 00:13:25,060
Generate a matrix.
187
00:13:25,060 --> 00:13:26,710
This is not really
in the notes, but
188
00:13:26,710 --> 00:13:27,960
I think it's important.
189
00:13:27,960 --> 00:13:35,100
190
00:13:35,100 --> 00:13:39,160
So see, the code here
is a subspace.
191
00:13:39,160 --> 00:13:42,190
It's a linear space, so it
has a generator, it has
192
00:13:42,190 --> 00:13:44,210
generators, k generators.
193
00:13:44,210 --> 00:13:59,793
So let g1 be k, write
this off the code.
194
00:13:59,793 --> 00:14:02,430
195
00:14:02,430 --> 00:14:07,080
So as a basis of the vector
space, that this would be a
196
00:14:07,080 --> 00:14:09,610
basis of the vector space, any
basis would be fine here.
197
00:14:09,610 --> 00:14:12,710
198
00:14:12,710 --> 00:14:27,370
Then C may be defined as all
the x in Fqn such that x is
199
00:14:27,370 --> 00:14:29,260
sum over --
200
00:14:29,260 --> 00:14:30,510
what do I call it --
201
00:14:30,510 --> 00:14:32,440
202
00:14:32,440 --> 00:14:47,050
fi gi, where fi is in Fq.
203
00:14:47,050 --> 00:14:56,690
And the reason I introduce
this, we can --
204
00:14:56,690 --> 00:14:58,900
this is just the definition
of a space, right?
205
00:14:58,900 --> 00:14:59,450
That's clear.
206
00:14:59,450 --> 00:15:08,620
So if you have these generators,
you find a
207
00:15:08,620 --> 00:15:13,866
generator matrix, uh-oh,
it already starts.
208
00:15:13,866 --> 00:15:15,116
Let me --
209
00:15:15,116 --> 00:15:24,420
210
00:15:24,420 --> 00:15:42,940
matrix g which contains,
as a m matrix
211
00:15:42,940 --> 00:15:48,680
containing the rows gi.
212
00:15:48,680 --> 00:15:52,320
So the i-th row in the generator
matrix is just gi.
213
00:15:52,320 --> 00:16:12,540
Then you also can write as x
is equal to f times g, f
214
00:16:12,540 --> 00:16:20,040
element Fqk, or just the same
statement as this one, so
215
00:16:20,040 --> 00:16:21,290
nothing has happened.
216
00:16:21,290 --> 00:16:25,280
217
00:16:25,280 --> 00:16:28,260
So basically, the reason I did
that, I wanted to introduce
218
00:16:28,260 --> 00:16:34,350
the term generator matrix, which
is sort of important.
219
00:16:34,350 --> 00:16:52,220
And one more property of this
orthogonal complement
220
00:16:52,220 --> 00:16:54,750
of C, of the code.
221
00:16:54,750 --> 00:16:57,270
So what does that mean?
222
00:16:57,270 --> 00:17:09,030
So the orthogonal complement of
the code you could write as
223
00:17:09,030 --> 00:17:22,140
Fqn such that sum of x_i
y_i is equal to 0.
224
00:17:22,140 --> 00:17:29,210
The sum is obviously over the
field for all y in the code.
225
00:17:29,210 --> 00:17:32,560
226
00:17:32,560 --> 00:17:37,084
What's the dimension of this, of
the orthogonal complement?
227
00:17:37,084 --> 00:17:38,240
AUDIENCE: n minus k.
228
00:17:38,240 --> 00:17:41,400
PROFESSOR: n minus k, clearly,
because we have ambient space
229
00:17:41,400 --> 00:17:46,330
is n dimensional, we impose k
linear constraints on this, by
230
00:17:46,330 --> 00:17:50,830
the k generators, so the k
dimensions of, take note, by
231
00:17:50,830 --> 00:17:52,680
the generators drop out.
232
00:17:52,680 --> 00:17:58,310
So the dimension of
the orthogonal
233
00:17:58,310 --> 00:18:00,910
complement is n minus k.
234
00:18:00,910 --> 00:18:06,100
235
00:18:06,100 --> 00:18:08,400
So what else do we need
to say about this?
236
00:18:08,400 --> 00:18:13,390
237
00:18:13,390 --> 00:18:22,910
C is called the dual code
for this reason.
238
00:18:22,910 --> 00:18:25,510
239
00:18:25,510 --> 00:18:26,470
C is called dual code.
240
00:18:26,470 --> 00:18:29,300
In particular, it's a code
that's a linear space.
241
00:18:29,300 --> 00:18:32,160
It's a subspace of Fqn
again, it's a code.
242
00:18:32,160 --> 00:18:36,130
So it's just as nice
a code as C at this
243
00:18:36,130 --> 00:18:37,380
point in time at least.
244
00:18:37,380 --> 00:18:42,740
245
00:18:42,740 --> 00:18:45,110
So it's called a dual code.
246
00:18:45,110 --> 00:18:48,785
To C, if it is a code, it
has a generator matrix.
247
00:18:48,785 --> 00:18:51,400
248
00:18:51,400 --> 00:19:04,630
Let h be a generator
matrix for C dual.
249
00:19:04,630 --> 00:19:07,470
So in particular, we could
define C dual now, for
250
00:19:07,470 --> 00:19:10,810
example, by the equivalent
of this relation here.
251
00:19:10,810 --> 00:19:17,700
But because it's a dual code,
we now also can define the
252
00:19:17,700 --> 00:19:34,420
original code in an equivalent
way such that x times h
253
00:19:34,420 --> 00:19:37,990
transpose is 0.
254
00:19:37,990 --> 00:19:43,330
We could define our original
code C either as the image of
255
00:19:43,330 --> 00:19:49,580
a matrix g, of a generator
matrix g, or as a kernel of a
256
00:19:49,580 --> 00:19:52,470
parity-check matrix h.
257
00:19:52,470 --> 00:19:55,440
So h is a ...WRITING
ON BOARD...
258
00:19:55,440 --> 00:20:03,790
259
00:20:03,790 --> 00:20:07,960
for C. So that's all pretty much
straight linear algebra,
260
00:20:07,960 --> 00:20:12,340
and I'm sure you've seen that
in many different places.
261
00:20:12,340 --> 00:20:13,660
Any questions about
any of this?
262
00:20:13,660 --> 00:20:17,962
263
00:20:17,962 --> 00:20:19,396
AUDIENCE: So the addition
of the dual
264
00:20:19,396 --> 00:20:20,840
[UNINTELLIGIBLE PHRASE]
265
00:20:20,840 --> 00:20:22,430
the summation
[UNINTELLIGIBLE PHRASE]
266
00:20:22,430 --> 00:20:24,994
equals 0 for all [INAUDIBLE]
other than x, right?
267
00:20:24,994 --> 00:20:26,260
[UNINTELLIGIBLE PHRASE]
268
00:20:26,260 --> 00:20:28,170
PROFESSOR: Oh no, no, no,
it doesn't have to be
269
00:20:28,170 --> 00:20:29,740
different from x.
270
00:20:29,740 --> 00:20:35,460
If y is in the code, if y is
in C, then x has to be
271
00:20:35,460 --> 00:20:37,020
orthogonal to it.
272
00:20:37,020 --> 00:20:38,850
They can be the same vector,
in particular, if you have
273
00:20:38,850 --> 00:20:41,630
binary vectors, an even
made binary vector is
274
00:20:41,630 --> 00:20:43,560
orthogonal to itself.
275
00:20:43,560 --> 00:20:45,820
It's a little bit odd,
but that's the
276
00:20:45,820 --> 00:20:47,070
magic of finite fields.
277
00:20:47,070 --> 00:20:51,100
278
00:20:51,100 --> 00:20:51,275
OK.
279
00:20:51,275 --> 00:20:51,830
Good.
280
00:20:51,830 --> 00:20:53,430
So these are codes,
now we could stop.
281
00:20:53,430 --> 00:20:57,100
We have defined the object,
and obviously it exists,
282
00:20:57,100 --> 00:21:02,720
because we could just write
something down and it exists.
283
00:21:02,720 --> 00:21:06,050
So once we have defined it,
the next question is, what
284
00:21:06,050 --> 00:21:09,120
sort of codes do exist?
285
00:21:09,120 --> 00:21:13,140
So that's what we're
going to do next.
286
00:21:13,140 --> 00:21:16,650
287
00:21:16,650 --> 00:21:17,900
First, question one.
288
00:21:17,900 --> 00:21:30,000
289
00:21:30,000 --> 00:21:36,880
Codes do, what type
of codes do exist?
290
00:21:36,880 --> 00:21:40,130
So which codes do you know?
291
00:21:40,130 --> 00:21:42,920
AUDIENCE: [INAUDIBLE]
292
00:21:42,920 --> 00:21:45,800
PROFESSOR: You know Reed-Muller
codes, you know
293
00:21:45,800 --> 00:21:49,330
probably sporadic binary codes
that are out there.
294
00:21:49,330 --> 00:21:51,900
295
00:21:51,900 --> 00:21:53,430
These are all binary codes.
296
00:21:53,430 --> 00:21:58,220
So what type of codes exist
over larger fields?
297
00:21:58,220 --> 00:22:01,600
298
00:22:01,600 --> 00:22:03,070
Many, many classes.
299
00:22:03,070 --> 00:22:07,400
There exists the equivalent of
the Reed-Muller codes, there
300
00:22:07,400 --> 00:22:10,910
exist QRE Reed-Muller codes,
and there exist generalized
301
00:22:10,910 --> 00:22:13,690
Reed-Muller codes, and,
and, and, and, and.
302
00:22:13,690 --> 00:22:19,700
But we are interested in a
very special class today,
303
00:22:19,700 --> 00:22:23,080
which is MDS codes.
304
00:22:23,080 --> 00:22:35,475
It stands for Maximum
Distance Separable.
305
00:22:35,475 --> 00:22:38,520
306
00:22:38,520 --> 00:22:40,020
It's a strange name.
307
00:22:40,020 --> 00:22:42,350
There's no particular
reason for MDS.
308
00:22:42,350 --> 00:22:51,440
309
00:22:51,440 --> 00:22:52,690
But, let's see what we
can do with that.
310
00:22:52,690 --> 00:22:55,400
311
00:22:55,400 --> 00:22:58,420
What type of codes do exist?
312
00:22:58,420 --> 00:23:04,640
So we have parameters of codes
-- oh, I think you write the
313
00:23:04,640 --> 00:23:06,740
curly bracket, right --
314
00:23:06,740 --> 00:23:09,730
n, k and d.
315
00:23:09,730 --> 00:23:14,040
So that would mean a code of
length n, dimension k, and
316
00:23:14,040 --> 00:23:16,140
distance d.
317
00:23:16,140 --> 00:23:20,450
And let me add something to it
a q, if you want to emphasize
318
00:23:20,450 --> 00:23:21,925
that this is a query field.
319
00:23:21,925 --> 00:23:24,810
320
00:23:24,810 --> 00:23:29,140
So are all numbers
possible here?
321
00:23:29,140 --> 00:23:36,345
What do we have, a 20,
19, 17 code over, I
322
00:23:36,345 --> 00:23:39,390
don't know, over F8.
323
00:23:39,390 --> 00:23:41,580
Is this possible?
324
00:23:41,580 --> 00:23:42,830
What would you think?
325
00:23:42,830 --> 00:23:49,750
326
00:23:49,750 --> 00:23:51,482
No?
327
00:23:51,482 --> 00:23:52,430
AUDIENCE: [INAUDIBLE]
328
00:23:52,430 --> 00:23:54,100
PROFESSOR: It's not possible.
329
00:23:54,100 --> 00:23:56,160
It doesn't seem likely.
330
00:23:56,160 --> 00:24:01,430
What conflicts here, is the
dimension and the distance.
331
00:24:01,430 --> 00:24:03,900
If you get a large dimension,
in particular, if we would
332
00:24:03,900 --> 00:24:07,170
make this 20, what
would that mean?
333
00:24:07,170 --> 00:24:09,840
It would mean we have to
take the entire space.
334
00:24:09,840 --> 00:24:11,990
If you take the entire space,
then the minimum
335
00:24:11,990 --> 00:24:13,780
weight word is 1.
336
00:24:13,780 --> 00:24:16,870
So this is possible.
337
00:24:16,870 --> 00:24:17,940
You know this is possible.
338
00:24:17,940 --> 00:24:22,240
If you drop this by 1, that
seems very unlikely that we
339
00:24:22,240 --> 00:24:25,920
would get a 17 here.
340
00:24:25,920 --> 00:24:29,080
But what do we get here?
341
00:24:29,080 --> 00:24:30,330
2.
342
00:24:30,330 --> 00:24:31,920
343
00:24:31,920 --> 00:24:34,910
You get a 2 because that's what
we can achieve with a
344
00:24:34,910 --> 00:24:37,290
single parity-check code.
345
00:24:37,290 --> 00:24:38,810
The parity-check code
doesn't have to be
346
00:24:38,810 --> 00:24:41,820
restrained to binary.
347
00:24:41,820 --> 00:24:42,130
Why?
348
00:24:42,130 --> 00:24:43,520
Why would it be restrained
to binary?
349
00:24:43,520 --> 00:24:46,850
350
00:24:46,850 --> 00:24:56,430
You could just, the set of all
vectors let's define the
351
00:24:56,430 --> 00:25:03,900
single parity-check codes s
p c, q, as the set of all
352
00:25:03,900 --> 00:25:11,165
vectors such that sum of
the x_i is equal to 0.
353
00:25:11,165 --> 00:25:14,320
354
00:25:14,320 --> 00:25:18,380
So could we have a word
of weight 1 in here?
355
00:25:18,380 --> 00:25:19,270
Obviously not, right?
356
00:25:19,270 --> 00:25:22,970
If it has a weight 1, how
would it add up to 0?
357
00:25:22,970 --> 00:25:25,745
Because one position
would never cancel
358
00:25:25,745 --> 00:25:27,070
with any other position.
359
00:25:27,070 --> 00:25:29,820
So the minimum weight is
2 here, and we get
360
00:25:29,820 --> 00:25:30,600
a distance of 2.
361
00:25:30,600 --> 00:25:32,940
So what's the next one?
362
00:25:32,940 --> 00:25:37,250
363
00:25:37,250 --> 00:25:41,740
It's tempting to say 3, right?
364
00:25:41,740 --> 00:25:47,190
3, but this is very much
a question, now.
365
00:25:47,190 --> 00:25:50,710
Because this is not as easy
to come by as a single
366
00:25:50,710 --> 00:25:53,060
parity-check.
367
00:25:53,060 --> 00:25:56,320
And that's what we're
going to do next.
368
00:25:56,320 --> 00:26:04,010
We're going to define bounds on
the maximum distance that a
369
00:26:04,010 --> 00:26:07,310
code can have altogether.
370
00:26:07,310 --> 00:26:12,210
OK, so let's do the following.
371
00:26:12,210 --> 00:26:14,750
372
00:26:14,750 --> 00:26:27,720
Which parameter is possible?
373
00:26:27,720 --> 00:26:28,970
OK.
374
00:26:28,970 --> 00:26:30,690
375
00:26:30,690 --> 00:26:34,270
So let's assume you have a code,
an n,k,d code, and now
376
00:26:34,270 --> 00:26:38,720
we want to find a relation, a
bound between n, k and d.
377
00:26:38,720 --> 00:26:39,970
How do we do this?
378
00:26:39,970 --> 00:26:43,220
379
00:26:43,220 --> 00:26:44,470
Any ideas?
380
00:26:44,470 --> 00:26:46,720
381
00:26:46,720 --> 00:26:48,740
Let a computer run for
eternity and find
382
00:26:48,740 --> 00:26:49,790
all possible codes?
383
00:26:49,790 --> 00:26:51,190
No, no, no, no.
384
00:26:51,190 --> 00:26:53,256
We don't do this.
385
00:26:53,256 --> 00:26:56,770
We wouldn't get far.
386
00:26:56,770 --> 00:27:07,190
Let's assume we have
an n,k,d code.
387
00:27:07,190 --> 00:27:10,910
388
00:27:10,910 --> 00:27:12,990
What does that mean?
389
00:27:12,990 --> 00:27:15,660
Well, let's write the code
words all down in a huge
390
00:27:15,660 --> 00:27:26,790
matrix, so each row
in this matrix
391
00:27:26,790 --> 00:27:28,240
corresponds to one code word.
392
00:27:28,240 --> 00:27:36,330
So this has a length n, this is
q to the k, q is whatever
393
00:27:36,330 --> 00:27:41,200
the alphabet is of the code in
question, and now we say it's
394
00:27:41,200 --> 00:27:43,640
an n,k,d code.
395
00:27:43,640 --> 00:27:49,660
What that means, it means, among
other things is, say we
396
00:27:49,660 --> 00:27:57,850
delete, just punch out,
d minus 1 positions.
397
00:27:57,850 --> 00:28:01,100
398
00:28:01,100 --> 00:28:04,450
We punch out d minus 1 positions
of all code words
399
00:28:04,450 --> 00:28:07,020
and we look at the code
that remains.
400
00:28:07,020 --> 00:28:08,350
You guys don't have colored
chalk here, huh?
401
00:28:08,350 --> 00:28:12,020
402
00:28:12,020 --> 00:28:14,270
We look at the code that
remains, it means we look at
403
00:28:14,270 --> 00:28:19,875
this part of the matrix.
404
00:28:19,875 --> 00:28:22,930
405
00:28:22,930 --> 00:28:24,180
Is that clear, what
I'm doing here?
406
00:28:24,180 --> 00:28:26,440
407
00:28:26,440 --> 00:28:32,470
So if the code indeed had
distance d, can there be any
408
00:28:32,470 --> 00:28:36,110
two rows equal in this part?
409
00:28:36,110 --> 00:28:38,150
Remember, we punch out
all d minus 1.
410
00:28:38,150 --> 00:28:40,790
411
00:28:40,790 --> 00:28:45,160
Can there be any rows in this
part that are equal?
412
00:28:45,160 --> 00:28:46,130
No, right?
413
00:28:46,130 --> 00:28:47,290
Couldn't be.
414
00:28:47,290 --> 00:28:50,360
They all have to be different.
415
00:28:50,360 --> 00:28:51,560
What does that mean?
416
00:28:51,560 --> 00:28:58,070
They all have to be different,
but how many different tuples
417
00:28:58,070 --> 00:29:00,650
can we have in this part?
418
00:29:00,650 --> 00:29:10,200
Well, we have at most q to
the n minus d minus 1.
419
00:29:10,200 --> 00:29:11,450
That's the length here.
420
00:29:11,450 --> 00:29:14,200
421
00:29:14,200 --> 00:29:17,650
This n minus d minus 1.
422
00:29:17,650 --> 00:29:18,900
Different tuples.
423
00:29:18,900 --> 00:29:32,350
424
00:29:32,350 --> 00:29:35,130
So how can we patch that
together into a relation on
425
00:29:35,130 --> 00:29:36,380
the parameters?
426
00:29:36,380 --> 00:29:42,840
427
00:29:42,840 --> 00:29:46,820
It basically says, q,
this is q to the k.
428
00:29:46,820 --> 00:29:51,460
q to the k is upper
bounded by this.
429
00:29:51,460 --> 00:29:56,790
430
00:29:56,790 --> 00:29:58,040
It's upper bounded by this.
431
00:29:58,040 --> 00:30:00,740
432
00:30:00,740 --> 00:30:04,660
And let me take the
logarithm on here,
433
00:30:04,660 --> 00:30:11,600
and we get this relation.
434
00:30:11,600 --> 00:30:14,265
That's a first incarnation of
the tension that we get on
435
00:30:14,265 --> 00:30:16,950
code construction, on codes.
436
00:30:16,950 --> 00:30:18,270
And bound on this, at least.
437
00:30:18,270 --> 00:30:22,730
If you choose d large,
the distance large,
438
00:30:22,730 --> 00:30:25,930
then k has to go.
439
00:30:25,930 --> 00:30:30,700
If you choose k large, the
distance cannot be very large.
440
00:30:30,700 --> 00:30:34,620
So this is where we, for the
first time, see this tension.
441
00:30:34,620 --> 00:30:38,210
And it's also important, I'm
sorry that I run around like
442
00:30:38,210 --> 00:30:47,605
this here, n has to be
at least k plus d, k
443
00:30:47,605 --> 00:30:48,960
plus d minus 1.
444
00:30:48,960 --> 00:30:51,590
445
00:30:51,590 --> 00:31:02,040
So here, you see this 28 in 3,
it would just satisfy this.
446
00:31:02,040 --> 00:31:05,580
It would just satisfy this.
447
00:31:05,580 --> 00:31:06,830
So do we know it exists?
448
00:31:06,830 --> 00:31:09,180
449
00:31:09,180 --> 00:31:10,430
No.
450
00:31:10,430 --> 00:31:11,690
No, why would it?
451
00:31:11,690 --> 00:31:18,710
So far, we only have looked at
this here, and so, if it would
452
00:31:18,710 --> 00:31:20,420
exist, it would have
to satisfy that.
453
00:31:20,420 --> 00:31:23,110
But there's no reason
to assume it exists.
454
00:31:23,110 --> 00:31:26,250
At the moment, at least.
455
00:31:26,250 --> 00:31:27,680
OK.
456
00:31:27,680 --> 00:31:29,180
This is called the
Singleton bound.
457
00:31:29,180 --> 00:31:44,290
458
00:31:44,290 --> 00:32:02,570
Any code over any field, phi
n, this relationship on the
459
00:32:02,570 --> 00:32:03,440
parameters.
460
00:32:03,440 --> 00:32:04,690
Good.
461
00:32:04,690 --> 00:32:11,040
462
00:32:11,040 --> 00:32:36,740
Any code satisfying and bound
with equality is called MDS.
463
00:32:36,740 --> 00:32:39,750
So we have an MDS code if and
only if it satisfies the
464
00:32:39,750 --> 00:32:42,070
Singleton bound with equality.
465
00:32:42,070 --> 00:32:44,670
That's the definition
of MDS codes.
466
00:32:44,670 --> 00:32:48,280
And now it makes maybe a little
bit more sense to talk
467
00:32:48,280 --> 00:32:51,530
about Maximum Distance Separable
codes, well, in a
468
00:32:51,530 --> 00:32:55,020
sense, they have the maximum
distance among all codes.
469
00:32:55,020 --> 00:32:58,300
You find all codes with the
given n and k, if they're MDS,
470
00:32:58,300 --> 00:33:01,780
they have the maximum
distance.
471
00:33:01,780 --> 00:33:05,720
OK, let's think about
this a little here.
472
00:33:05,720 --> 00:33:09,080
473
00:33:09,080 --> 00:33:11,000
AUDIENCE: [INAUDIBLE]
474
00:33:11,000 --> 00:33:13,925
dependence on q
[UNINTELLIGIBLE]?
475
00:33:13,925 --> 00:33:15,850
PROFESSOR: Yeah, there's a very
strong dependence on q.
476
00:33:15,850 --> 00:33:17,800
The bound, not.
477
00:33:17,800 --> 00:33:20,610
The bound has no dependence
on q.
478
00:33:20,610 --> 00:33:23,800
If the guys exist or not,
that's very much
479
00:33:23,800 --> 00:33:24,680
dependent on q.
480
00:33:24,680 --> 00:33:26,185
We'll get to that.
481
00:33:26,185 --> 00:33:28,000
AUDIENCE: [INAUDIBLE] when the
q is large, we have more
482
00:33:28,000 --> 00:33:30,090
options to [UNINTELLIGIBLE]?
483
00:33:30,090 --> 00:33:30,445
PROFESSOR: Absolutely.
484
00:33:30,445 --> 00:33:31,190
Absolutely.
485
00:33:31,190 --> 00:33:39,180
For binary, there's a very
simple argument to show that
486
00:33:39,180 --> 00:33:43,200
there are no binary MDS codes
except for the parity-check
487
00:33:43,200 --> 00:33:45,790
codes and the repetition
codes and trivial code.
488
00:33:45,790 --> 00:33:53,290
489
00:33:53,290 --> 00:34:09,380
So say we have a binary code,
a binary n,k,d code with a
490
00:34:09,380 --> 00:34:12,150
generator matrix.
491
00:34:12,150 --> 00:34:13,929
So what could the generator
matrix be?
492
00:34:13,929 --> 00:34:22,159
There will be an identity part,
and then there will be
493
00:34:22,159 --> 00:34:26,270
the rest of the generator
matrix, and how could we
494
00:34:26,270 --> 00:34:30,489
possibly fill that in, in
order to make it MDS?
495
00:34:30,489 --> 00:34:39,460
Because this is n, this is k,
and we see in order to make it
496
00:34:39,460 --> 00:34:46,730
MDS, every single row has to
have all entries equal to 1.
497
00:34:46,730 --> 00:34:50,250
Because if not all entries are
equal to 1, here, then we
498
00:34:50,250 --> 00:34:53,780
immediately have exhibited a
code word with a weight less
499
00:34:53,780 --> 00:34:57,560
than n minus k plus 1.
500
00:34:57,560 --> 00:35:01,420
So OK, we know the first row
has to have all 1's.
501
00:35:01,420 --> 00:35:05,110
Because now, the weight of this
row is exactly on the MDS
502
00:35:05,110 --> 00:35:05,950
[INAUDIBLE].
503
00:35:05,950 --> 00:35:07,200
What about the next one?
504
00:35:07,200 --> 00:35:09,490
505
00:35:09,490 --> 00:35:11,080
The next one, same thing.
506
00:35:11,080 --> 00:35:13,880
507
00:35:13,880 --> 00:35:16,030
All entries have to be 1.
508
00:35:16,030 --> 00:35:17,500
But now we see the
problem, right?
509
00:35:17,500 --> 00:35:20,050
Now we add those two guys, it
should again be a code word,
510
00:35:20,050 --> 00:35:22,220
and we have a grade
two code word.
511
00:35:22,220 --> 00:35:29,370
So this is, in a nutshell, to
prove that there are no binary
512
00:35:29,370 --> 00:35:32,410
MDS codes except the
trivial ones.
513
00:35:32,410 --> 00:35:43,390
So the trivial ones are n, n1,
n, n minus 1, 2 and n1, n.
514
00:35:43,390 --> 00:35:45,920
These are the trivial ones.
515
00:35:45,920 --> 00:35:48,355
The space itself, so it's in a
parity-check code, and the
516
00:35:48,355 --> 00:35:50,890
repetition code.
517
00:35:50,890 --> 00:35:53,510
These are the only
binary MDS codes.
518
00:35:53,510 --> 00:35:57,270
And the argument is
roughly there.
519
00:35:57,270 --> 00:35:58,520
OK, where was I?
520
00:35:58,520 --> 00:36:03,170
521
00:36:03,170 --> 00:36:05,000
Yeah, let's think about this
a little bit more.
522
00:36:05,000 --> 00:36:07,370
And we are getting to exactly
your question about the
523
00:36:07,370 --> 00:36:08,620
[UNINTELLIGIBLE].
524
00:36:08,620 --> 00:36:11,380
525
00:36:11,380 --> 00:36:17,380
This here has to hold, this
argument has to hold
526
00:36:17,380 --> 00:36:23,300
regardless of which d minus
1 positions we punch out.
527
00:36:23,300 --> 00:36:25,520
This argument has
always to hold.
528
00:36:25,520 --> 00:36:29,100
Which means, think about it,
it's an enormously strong
529
00:36:29,100 --> 00:36:32,030
combinatorial condition
on the code.
530
00:36:32,030 --> 00:36:39,620
So you have a code, that means
you have a code, you write it
531
00:36:39,620 --> 00:36:42,990
in a matrix like this,
all the code words.
532
00:36:42,990 --> 00:36:49,000
You punch out an arbitrary
collection of d minus 1one
533
00:36:49,000 --> 00:36:53,920
positions, and the rest, the
remaining positions, have to
534
00:36:53,920 --> 00:36:58,100
make up the entire space here.
535
00:36:58,100 --> 00:37:02,910
The entire space Fqn minus
to that right exponent.
536
00:37:02,910 --> 00:37:04,340
This is a very --
537
00:37:04,340 --> 00:37:07,830
think about it, I mean, just
writing down this is an
538
00:37:07,830 --> 00:37:11,290
enormously strong combinatorial
condition.
539
00:37:11,290 --> 00:37:17,870
So that will actually lead to
the codes existing only for a
540
00:37:17,870 --> 00:37:22,820
very, very special, for a
subset of field sizes.
541
00:37:22,820 --> 00:37:26,360
In particular, like you said, we
have to have enough freedom
542
00:37:26,360 --> 00:37:31,800
in the field size to fill up
this matrix to satisfy this.
543
00:37:31,800 --> 00:37:34,360
544
00:37:34,360 --> 00:37:42,230
OK, before we get to that,
before I say a word about the
545
00:37:42,230 --> 00:37:50,880
field size, let me formalize
what I just said here, namely,
546
00:37:50,880 --> 00:37:54,850
that all of the other positions
have to be exactly
547
00:37:54,850 --> 00:37:59,450
the q to the n minus d minus
1, different tuples.
548
00:37:59,450 --> 00:38:17,610
And the definition, let the
code with q to the k, code
549
00:38:17,610 --> 00:38:30,190
words over alphabet Fq.
550
00:38:30,190 --> 00:38:35,410
551
00:38:35,410 --> 00:39:02,990
Let subset of the positions in
C, i is called an information
552
00:39:02,990 --> 00:39:30,970
set if C constrained to i runs
exactly through all the q to
553
00:39:30,970 --> 00:39:37,395
the k, runs through all
the q to the k.
554
00:39:37,395 --> 00:39:46,060
555
00:39:46,060 --> 00:39:48,570
Fqk.
556
00:39:48,570 --> 00:39:53,300
So what it means is, you have a
code, and you have a subset
557
00:39:53,300 --> 00:39:58,960
of positions, maybe this one,
this one, this one, this one.
558
00:39:58,960 --> 00:40:03,090
This is a subset of positions
if the code words.
559
00:40:03,090 --> 00:40:06,230
So if the matrix that remains
after you take out the
560
00:40:06,230 --> 00:40:10,470
punctured columns, runs through
all the q to the k
561
00:40:10,470 --> 00:40:14,298
elements of Fqk, then this
is an information search.
562
00:40:14,298 --> 00:40:15,548
AUDIENCE: [INAUDIBLE]
563
00:40:15,548 --> 00:40:18,040
564
00:40:18,040 --> 00:40:21,870
PROFESSOR: Constrained to
i, because i has size k.
565
00:40:21,870 --> 00:40:25,275
i is just -- its just about
enough to describe every code
566
00:40:25,275 --> 00:40:31,830
word, if the restraint of C to
the set would indeed be giving
567
00:40:31,830 --> 00:40:35,700
a unique vector for
each code word.
568
00:40:35,700 --> 00:40:38,210
The reason to call it -- so
this is the definition of
569
00:40:38,210 --> 00:40:40,220
information set.
570
00:40:40,220 --> 00:40:44,910
The reason to call it an
information set, it's pretty
571
00:40:44,910 --> 00:40:45,620
straight, right?
572
00:40:45,620 --> 00:40:47,685
Why is it called an
information set?
573
00:40:47,685 --> 00:40:53,740
574
00:40:53,740 --> 00:40:55,500
Because it's enough, right?
575
00:40:55,500 --> 00:40:56,250
Because it's enough.
576
00:40:56,250 --> 00:40:59,320
If you know exactly the value
of a code word in these
577
00:40:59,320 --> 00:41:02,820
positions, then it is
enough to recover
578
00:41:02,820 --> 00:41:04,706
the entire code word.
579
00:41:04,706 --> 00:41:07,210
When some genie tells you, gives
you a code world which
580
00:41:07,210 --> 00:41:10,110
was corrupted by noise or
something, but tells you,
581
00:41:10,110 --> 00:41:12,930
these k positions are OK.
582
00:41:12,930 --> 00:41:14,030
That's enough, that's
all you need.
583
00:41:14,030 --> 00:41:15,050
That's an information set.
584
00:41:15,050 --> 00:41:17,420
You can recover the information
from them.
585
00:41:17,420 --> 00:41:21,130
Actually, it is an application
that pops up sometimes.
586
00:41:21,130 --> 00:41:23,870
That somehow, you get side
information about some
587
00:41:23,870 --> 00:41:26,230
positions in the code word
indeed being correct, and
588
00:41:26,230 --> 00:41:27,560
others not.
589
00:41:27,560 --> 00:41:30,340
And others you don't
know about.
590
00:41:30,340 --> 00:41:35,755
So that's the information set,
with respect to our MDS code.
591
00:41:35,755 --> 00:41:38,300
592
00:41:38,300 --> 00:41:45,180
So with respect to our MDS code,
a corollary of the thing
593
00:41:45,180 --> 00:42:14,410
that involved any k positions
in an MDS code,
594
00:42:14,410 --> 00:42:15,660
an information set.
595
00:42:15,660 --> 00:42:22,170
596
00:42:22,170 --> 00:42:25,380
So any k positions on
information set.
597
00:42:25,380 --> 00:42:26,910
It's a really strong property.
598
00:42:26,910 --> 00:42:30,000
Really strong combinatorial
property.
599
00:42:30,000 --> 00:42:33,420
OK, so far, so good.
600
00:42:33,420 --> 00:42:42,470
601
00:42:42,470 --> 00:42:46,970
This is so strong, this
property, that we can say
602
00:42:46,970 --> 00:42:50,700
something about these codes even
without even knowing if
603
00:42:50,700 --> 00:42:51,950
they exist.
604
00:42:51,950 --> 00:42:54,210
605
00:42:54,210 --> 00:42:57,320
So, so far, we have talked about
these codes as if we
606
00:42:57,320 --> 00:42:59,080
knew they existed.
607
00:42:59,080 --> 00:43:01,890
Well, it's not entirely trivial,
since we know those
608
00:43:01,890 --> 00:43:03,400
guys here exist.
609
00:43:03,400 --> 00:43:05,710
So it's not entirely empty,
we're not out in
610
00:43:05,710 --> 00:43:08,250
cuckoo space, here.
611
00:43:08,250 --> 00:43:12,750
But do any other one exist,
except for those?
612
00:43:12,750 --> 00:43:15,480
That's the question.
613
00:43:15,480 --> 00:43:16,470
We don't know that yet.
614
00:43:16,470 --> 00:43:19,920
We will show in a little while
that they do, but we don't
615
00:43:19,920 --> 00:43:21,170
know that yet.
616
00:43:21,170 --> 00:43:23,960
617
00:43:23,960 --> 00:43:26,160
But the interesting part is that
we can derive properties
618
00:43:26,160 --> 00:43:29,870
of those codes without even
knowing they exist.
619
00:43:29,870 --> 00:43:31,470
And how do we do that?
620
00:43:31,470 --> 00:43:40,470
For example, we want to derive
the following property, how
621
00:43:40,470 --> 00:44:00,620
many words of weight d exists
in linear MDS code?
622
00:44:00,620 --> 00:44:01,960
One could ask that, right?
623
00:44:01,960 --> 00:44:05,280
If they exist, they're nice,
and if they exist, we also
624
00:44:05,280 --> 00:44:09,820
want to know how many words do
exist at minimum distance.
625
00:44:09,820 --> 00:44:12,460
Because that translates, again,
directly into union
626
00:44:12,460 --> 00:44:17,060
bound arguments later on, and
probability of error.
627
00:44:17,060 --> 00:44:18,210
So that's a good question.
628
00:44:18,210 --> 00:44:23,300
How many words of weight
d exist in a MDS code?
629
00:44:23,300 --> 00:44:28,630
Let's call this n d, and we
want to know how many.
630
00:44:28,630 --> 00:44:34,550
631
00:44:34,550 --> 00:44:37,000
So I'll let you think about this
for a sec while I erase
632
00:44:37,000 --> 00:44:40,130
the board, and then somebody
will tell me the answer.
633
00:44:40,130 --> 00:44:52,058
634
00:44:52,058 --> 00:44:53,984
So how can we think
about this?
635
00:44:53,984 --> 00:45:00,290
636
00:45:00,290 --> 00:45:04,030
Let's try to do a similar
argument as this one.
637
00:45:04,030 --> 00:45:18,780
Let's look at a single word,
and let's assume that d
638
00:45:18,780 --> 00:45:23,600
positions, we ask the questions
does there exist a
639
00:45:23,600 --> 00:45:27,730
code word within the
first d positions?
640
00:45:27,730 --> 00:45:29,970
It is equivalent to the
question, does there exist a
641
00:45:29,970 --> 00:45:35,410
code word that covers exactly
all d positions?
642
00:45:35,410 --> 00:45:37,140
Any set of d positions.
643
00:45:37,140 --> 00:45:43,522
644
00:45:43,522 --> 00:45:44,480
AUDIENCE: 0 everywhere else?
645
00:45:44,480 --> 00:45:47,400
PROFESSOR: And 0 everywhere
else.
646
00:45:47,400 --> 00:45:48,660
Why is that nice?
647
00:45:48,660 --> 00:45:51,880
If you could prove that, that
there exists a word for all d
648
00:45:51,880 --> 00:45:57,800
positions, because then, we
pretty much know what happens.
649
00:45:57,800 --> 00:46:02,837
Then we know that, well, if this
is true, then there are n
650
00:46:02,837 --> 00:46:04,700
choose d ways to choose
those d positions.
651
00:46:04,700 --> 00:46:08,210
652
00:46:08,210 --> 00:46:11,810
And then within those d
positions, and since it's a
653
00:46:11,810 --> 00:46:16,690
linear code, we can multiply
with the q minus 1 on the
654
00:46:16,690 --> 00:46:17,940
repeated element.
655
00:46:17,940 --> 00:46:20,200
656
00:46:20,200 --> 00:46:23,540
So if we can choose our d
positions arbitrarily, then
657
00:46:23,540 --> 00:46:28,540
this is the number over
words at distance d.
658
00:46:28,540 --> 00:46:31,540
So let's look at a word, and
let's, without loss of
659
00:46:31,540 --> 00:46:33,515
generality, assume it's
a first d positions.
660
00:46:33,515 --> 00:46:36,960
661
00:46:36,960 --> 00:46:39,770
So the first d positions.
662
00:46:39,770 --> 00:46:43,710
So in particular, these would
be the first d minus 1
663
00:46:43,710 --> 00:46:58,200
positions, which would mean
that this have length k.
664
00:46:58,200 --> 00:47:04,450
So if we have an MDS code, this
is an information set.
665
00:47:04,450 --> 00:47:08,020
So if this is an information
set, then we can fill up this
666
00:47:08,020 --> 00:47:11,880
thing with just about
anything we want.
667
00:47:11,880 --> 00:47:18,330
So we choose this information
set to be equal to 1.
668
00:47:18,330 --> 00:47:21,360
This is how we choose this
information set, and by the
669
00:47:21,360 --> 00:47:24,840
property of MDS code, we are
guaranteed that there exists a
670
00:47:24,840 --> 00:47:27,576
code word which is this part
in the information set.
671
00:47:27,576 --> 00:47:30,590
672
00:47:30,590 --> 00:47:34,070
But we also are guaranteed it's
a weight d word, right?
673
00:47:34,070 --> 00:47:37,240
The minimum distance is d,
that means all of these m
674
00:47:37,240 --> 00:47:44,680
entries here, they must all
be non-zero, in this part.
675
00:47:44,680 --> 00:47:47,370
Otherwise, it wouldn't
have weight d.
676
00:47:47,370 --> 00:47:48,330
OK?
677
00:47:48,330 --> 00:47:49,435
And there we have it.
678
00:47:49,435 --> 00:47:51,180
That was all we needed to show.
679
00:47:51,180 --> 00:47:51,910
Right?
680
00:47:51,910 --> 00:47:56,160
Because now we have shown that
there exists a word of weight
681
00:47:56,160 --> 00:47:59,840
d in the first d positions.
682
00:47:59,840 --> 00:48:01,090
Is that clear?
683
00:48:01,090 --> 00:48:14,650
684
00:48:14,650 --> 00:48:15,250
Let's try again.
685
00:48:15,250 --> 00:48:16,270
I will say the same words.
686
00:48:16,270 --> 00:48:17,575
Maybe it becomes clearer
by that.
687
00:48:17,575 --> 00:48:22,710
688
00:48:22,710 --> 00:48:24,130
Let's look at a code word.
689
00:48:24,130 --> 00:48:27,790
This is a generic code word,
at first, and we want to
690
00:48:27,790 --> 00:48:31,410
answer the question, does
there exist a code word
691
00:48:31,410 --> 00:48:34,240
within, which has support only
in the first d positions?
692
00:48:34,240 --> 00:48:36,890
693
00:48:36,890 --> 00:48:42,026
So does there exist a code word
which is non-zero here,
694
00:48:42,026 --> 00:48:46,626
up to d, and which is zero
everywhere else?
695
00:48:46,626 --> 00:48:49,225
That's the question
we want to answer.
696
00:48:49,225 --> 00:48:55,575
697
00:48:55,575 --> 00:48:55,817
OK.
698
00:48:55,817 --> 00:48:57,930
Now here's what we do.
699
00:48:57,930 --> 00:48:59,700
We look at this road and
say, you know what?
700
00:48:59,700 --> 00:49:04,890
Let's look at the last k
positions, which have an
701
00:49:04,890 --> 00:49:08,380
overlap of 1 with
this word here,
702
00:49:08,380 --> 00:49:10,130
because it's an MDS property.
703
00:49:10,130 --> 00:49:14,160
So we have this relation
between n, k, and d.
704
00:49:14,160 --> 00:49:17,760
And since any k positions in
the word are in information
705
00:49:17,760 --> 00:49:24,210
set, so we can choose whatever
we want in this part, and this
706
00:49:24,210 --> 00:49:25,460
is what we choose.
707
00:49:25,460 --> 00:49:28,030
708
00:49:28,030 --> 00:49:33,050
By the property of MDS codes,
this corollary, we are
709
00:49:33,050 --> 00:49:37,600
guaranteed there exists the code
word which in the second
710
00:49:37,600 --> 00:49:41,180
half of the code word
looks like this.
711
00:49:41,180 --> 00:49:44,570
And in the first half,
it looks different.
712
00:49:44,570 --> 00:49:48,110
There's something else here, and
I say, well it cannot have
713
00:49:48,110 --> 00:49:51,640
any 0 in here, because
then it would have
714
00:49:51,640 --> 00:49:53,610
weighed less that d.
715
00:49:53,610 --> 00:49:56,330
So it has non-zeros here.
716
00:49:56,330 --> 00:50:00,780
So indeed, we have shown the
existence of a code word which
717
00:50:00,780 --> 00:50:05,630
has non-zeroes in the
first d positions.
718
00:50:05,630 --> 00:50:07,870
Very simple.
719
00:50:07,870 --> 00:50:10,320
And that was without
loss of generality.
720
00:50:10,320 --> 00:50:14,310
You could make the same argument
for any d positions.
721
00:50:14,310 --> 00:50:15,250
What have we shown?
722
00:50:15,250 --> 00:50:19,480
We have shown that indeed, we
can choose any d positions in
723
00:50:19,480 --> 00:50:22,325
the code to support the
minimum weight code
724
00:50:22,325 --> 00:50:25,540
word of weight d.
725
00:50:25,540 --> 00:50:29,440
This is how many ways we can
choose this, then we have to
726
00:50:29,440 --> 00:50:32,840
multiply it with q minus 1.
727
00:50:32,840 --> 00:50:34,970
All non-zero field elements.
728
00:50:34,970 --> 00:50:37,550
The reason is, we might have
chosen this, or we might have
729
00:50:37,550 --> 00:50:40,920
chosen omega or omega squared
here, or just the multiples,
730
00:50:40,920 --> 00:50:42,812
the scalar multiples of it.
731
00:50:42,812 --> 00:50:47,320
732
00:50:47,320 --> 00:50:48,740
Interesting, right?
733
00:50:48,740 --> 00:50:54,230
This property, that any k
positions on information set
734
00:50:54,230 --> 00:50:56,520
is really strong enough
to prove the --
735
00:50:56,520 --> 00:50:58,570
actually, it's strong enough
to prove the entire weight
736
00:50:58,570 --> 00:51:00,980
distribution of an MDS code.
737
00:51:00,980 --> 00:51:03,175
AUDIENCE: [INAUDIBLE]
738
00:51:03,175 --> 00:51:04,425
[UNINTELLIGIBLE]?
739
00:51:04,425 --> 00:51:06,870
740
00:51:06,870 --> 00:51:10,030
PROFESSOR: No, no, no,
why no, no, no, no.
741
00:51:10,030 --> 00:51:13,660
742
00:51:13,660 --> 00:51:16,190
So then you would
get too much.
743
00:51:16,190 --> 00:51:21,000
If you write q minus 1 to the
d, then you would want to
744
00:51:21,000 --> 00:51:22,920
multiply each position with
a different value.
745
00:51:22,920 --> 00:51:26,590
746
00:51:26,590 --> 00:51:31,440
That would imply that there's
more than one code word in the
747
00:51:31,440 --> 00:51:33,380
first d positions.
748
00:51:33,380 --> 00:51:36,790
More than one code word so
that they are not scalar
749
00:51:36,790 --> 00:51:38,900
multiples of each other.
750
00:51:38,900 --> 00:51:42,060
If that would be true, then
you could find a linear
751
00:51:42,060 --> 00:51:45,940
combination which is still
0 in this part, but has
752
00:51:45,940 --> 00:51:48,470
additional 0 here somewhere.
753
00:51:48,470 --> 00:51:50,470
But if that is true,
then we don't have
754
00:51:50,470 --> 00:51:51,730
enough distance anymore.
755
00:51:51,730 --> 00:51:52,980
Then it's not an MDS code.
756
00:51:52,980 --> 00:51:56,630
757
00:51:56,630 --> 00:51:59,850
All right, so it's
indeed q minus 1.
758
00:51:59,850 --> 00:52:02,870
Within each d positions, we have
one dimensional space.
759
00:52:02,870 --> 00:52:06,184
It's just one dimension.
760
00:52:06,184 --> 00:52:07,434
AUDIENCE: [INAUDIBLE]
761
00:52:07,434 --> 00:52:09,940
762
00:52:09,940 --> 00:52:11,250
far off minimum weight
code words?
763
00:52:11,250 --> 00:52:14,230
764
00:52:14,230 --> 00:52:15,690
PROFESSOR: Yeah, yeah,
definitely.
765
00:52:15,690 --> 00:52:18,440
766
00:52:18,440 --> 00:52:20,665
Any other code must have less,
so it would have less.
767
00:52:20,665 --> 00:52:23,950
768
00:52:23,950 --> 00:52:25,310
But every other code
would have a
769
00:52:25,310 --> 00:52:26,560
smaller minimum distance.
770
00:52:26,560 --> 00:52:31,803
771
00:52:31,803 --> 00:52:34,629
AUDIENCE: [INAUDIBLE].
772
00:52:34,629 --> 00:52:40,172
Suppose we let the last k minus
1 position zero, and the
773
00:52:40,172 --> 00:52:43,670
one before that,
[UNINTELLIGIBLE PHRASE].
774
00:52:43,670 --> 00:52:47,540
And you said that we can do it
for any of [UNINTELLIGIBLE]
775
00:52:47,540 --> 00:52:48,490
total field?
776
00:52:48,490 --> 00:52:49,786
PROFESSOR: Sure.
777
00:52:49,786 --> 00:52:55,770
AUDIENCE: Since it's a linear
code, some of those code words
778
00:52:55,770 --> 00:52:59,220
should be in the linear
code, right?
779
00:52:59,220 --> 00:53:01,240
PROFESSOR: Sure.
780
00:53:01,240 --> 00:53:05,030
AUDIENCE: So because it's a
field, also we are going to
781
00:53:05,030 --> 00:53:06,700
[INAUDIBLE]
782
00:53:06,700 --> 00:53:12,830
there exists an inverse
[UNINTELLIGIBLE]?
783
00:53:12,830 --> 00:53:14,230
PROFESSOR: Absolutely.
784
00:53:14,230 --> 00:53:20,740
AUDIENCE: So if we add those two
code words, we should have
785
00:53:20,740 --> 00:53:22,510
all zero, [UNINTELLIGIBLE]
786
00:53:22,510 --> 00:53:27,590
k minus 1, and have inverse
at the one before.
787
00:53:27,590 --> 00:53:34,910
We get that code word which has
a minimum weight, which is
788
00:53:34,910 --> 00:53:37,862
less that the one
we have here?
789
00:53:37,862 --> 00:53:39,030
PROFESSOR: Good question.
790
00:53:39,030 --> 00:53:40,670
There is a trick out.
791
00:53:40,670 --> 00:53:43,640
There's a way out of this.
792
00:53:43,640 --> 00:53:45,610
Great argument.
793
00:53:45,610 --> 00:53:46,895
But there's a trick out.
794
00:53:46,895 --> 00:53:47,990
AUDIENCE: There's gotta
be an upper bound
795
00:53:47,990 --> 00:53:49,390
PROFESSOR: No, no, there
is a trick, there
796
00:53:49,390 --> 00:53:51,220
is a way out here.
797
00:53:51,220 --> 00:53:55,580
Namely, so let's put
it like this.
798
00:53:55,580 --> 00:53:59,880
Right here we put in a 1, just
for simplicity, let's assume
799
00:53:59,880 --> 00:54:01,550
all the other positions
are also 1.
800
00:54:01,550 --> 00:54:04,160
801
00:54:04,160 --> 00:54:07,540
And then you say, this would
be another code word, which
802
00:54:07,540 --> 00:54:11,300
has here an omega.
803
00:54:11,300 --> 00:54:12,300
I say, you know what?
804
00:54:12,300 --> 00:54:12,850
What's going to happen?
805
00:54:12,850 --> 00:54:15,330
All the other positions are
going to be omega 2.
806
00:54:15,330 --> 00:54:18,110
807
00:54:18,110 --> 00:54:20,790
There's no way to combine
these two guys to get an
808
00:54:20,790 --> 00:54:23,650
additional 0, unless
you get all 0's.
809
00:54:23,650 --> 00:54:24,800
Unless you get to 0.
810
00:54:24,800 --> 00:54:26,250
That's what I said, it's
a one-dimensional
811
00:54:26,250 --> 00:54:28,030
space in these positions.
812
00:54:28,030 --> 00:54:29,280
When it's a soft code .
813
00:54:29,280 --> 00:54:31,889
814
00:54:31,889 --> 00:54:33,139
AUDIENCE:
[UNINTELLIGIBLE PHRASE]
815
00:54:33,139 --> 00:54:39,710
816
00:54:39,710 --> 00:54:43,870
PROFESSOR: It tells you that if
you write down the minimum
817
00:54:43,870 --> 00:54:49,520
weight code words in the q minus
1 times d matrix, that
818
00:54:49,520 --> 00:54:53,620
is, you have a Latin
square, basically.
819
00:54:53,620 --> 00:54:54,790
That's what it tells you.
820
00:54:54,790 --> 00:54:58,460
There's in no position, if you
have anywhere in here an
821
00:54:58,460 --> 00:55:03,390
element alpha and element omega,
the same omega pops up
822
00:55:03,390 --> 00:55:04,640
nowhere else.
823
00:55:04,640 --> 00:55:07,770
824
00:55:07,770 --> 00:55:12,410
There's ramifications of MDS
codes in combinatorics left
825
00:55:12,410 --> 00:55:14,060
and right, so this would
be a Latin square.
826
00:55:14,060 --> 00:55:18,390
827
00:55:18,390 --> 00:55:21,320
You know, you can learn a lot
a lot about MDS codes if you
828
00:55:21,320 --> 00:55:23,750
think a little bit about that,
and about combinatorics
829
00:55:23,750 --> 00:55:25,220
altogether.
830
00:55:25,220 --> 00:55:26,760
OK, where was I?
831
00:55:26,760 --> 00:55:29,608
So we know that's fun.
832
00:55:29,608 --> 00:55:38,250
And actually, in the homework,
you going to do n d plus 1.
833
00:55:38,250 --> 00:55:40,900
834
00:55:40,900 --> 00:55:42,900
So the next one.
835
00:55:42,900 --> 00:55:49,100
But once you do n d plus
1, do all of them.
836
00:55:49,100 --> 00:55:52,370
In a sense it's just inclusion
and exclusion from then on.
837
00:55:52,370 --> 00:55:54,430
The first one is sort of the
toughest one, the rest is
838
00:55:54,430 --> 00:55:56,140
inclusion exclusion.
839
00:55:56,140 --> 00:56:00,610
And just for the heck of it,
when you go home and do the
840
00:56:00,610 --> 00:56:02,950
homework, write them all out.
841
00:56:02,950 --> 00:56:06,910
It's a pretty looking
formula, in the end.
842
00:56:06,910 --> 00:56:10,030
OK, so far, so good.
843
00:56:10,030 --> 00:56:12,955
So we have still talked about
MDS codes without knowing if
844
00:56:12,955 --> 00:56:15,840
they exist.
845
00:56:15,840 --> 00:56:17,430
Except for the trivial
ones here.
846
00:56:17,430 --> 00:56:21,250
847
00:56:21,250 --> 00:56:31,310
And the existence of MDS codes
is actually not known for
848
00:56:31,310 --> 00:56:33,830
which parameters they exist.
849
00:56:33,830 --> 00:56:39,270
So I give you a research
problem.
850
00:56:39,270 --> 00:56:56,580
The research problem is the main
conjecture on MDS codes.
851
00:56:56,580 --> 00:57:00,320
852
00:57:00,320 --> 00:57:01,520
And it's always sort
of tricky.
853
00:57:01,520 --> 00:57:04,330
When a research problem
has a name, then
854
00:57:04,330 --> 00:57:07,590
that signifies danger.
855
00:57:07,590 --> 00:57:12,340
Then it means that
it's not trivial.
856
00:57:12,340 --> 00:57:20,790
The question is, for which k
d and q, for which sets of
857
00:57:20,790 --> 00:57:28,000
parameters n k d q, do
MDS codes exist?
858
00:57:28,000 --> 00:57:44,250
And the conjecture this is that
n k q, because in MDS
859
00:57:44,250 --> 00:57:46,930
code we can actually get
rid of the d here.
860
00:57:46,930 --> 00:58:03,022
e, the longest length
of an MDS code.
861
00:58:03,022 --> 00:58:17,150
The longest length of an MDS
code, I mentioned k over an
862
00:58:17,150 --> 00:58:26,582
alphabet of size q.
863
00:58:26,582 --> 00:58:45,530
The conjecture is that n, k, d
is less than q plus 1 for k at
864
00:58:45,530 --> 00:58:48,360
least 2, unless --
865
00:58:48,360 --> 00:58:50,310
I always have to
look that up --
866
00:58:50,310 --> 00:58:53,394
867
00:58:53,394 --> 00:58:54,644
I think 2q.
868
00:58:54,644 --> 00:58:58,010
869
00:58:58,010 --> 00:59:07,480
And k plus 1 for k
greater than q.
870
00:59:07,480 --> 00:59:11,350
871
00:59:11,350 --> 00:59:22,680
We talk about it in a second,
except that n, there's
872
00:59:22,680 --> 00:59:29,380
a 3, 2 to the s.
873
00:59:29,380 --> 00:59:33,830
So if the alphabet is a power
of 2, alphabet size an
874
00:59:33,830 --> 00:59:35,185
extension field of
2, basically.
875
00:59:35,185 --> 00:59:38,460
876
00:59:38,460 --> 00:59:49,640
q plus 2 and q minus
1 q to the s.
877
00:59:49,640 --> 00:59:52,740
q plus 2.
878
00:59:52,740 --> 00:59:55,680
OK, so this is the main
conjecture on MDS codes.
879
00:59:55,680 --> 00:59:58,700
880
00:59:58,700 --> 01:00:03,970
Basically, it says that the
length can essentially be as
881
01:00:03,970 --> 01:00:08,212
large as the alphabet size,
but not larger.
882
01:00:08,212 --> 01:00:10,190
AUDIENCE: [INAUDIBLE]
883
01:00:10,190 --> 01:00:12,410
PROFESSOR: This q, yeah?
884
01:00:12,410 --> 01:00:13,490
Oh, yeah, n k q, sorry.
885
01:00:13,490 --> 01:00:15,650
It doesn't make sense
otherwise.
886
01:00:15,650 --> 01:00:18,780
887
01:00:18,780 --> 01:00:22,210
So the lengths can be in the
same order of magnitude as the
888
01:00:22,210 --> 01:00:23,140
alphabet size.
889
01:00:23,140 --> 01:00:27,700
That gives enough room, enough
choices, to fill up this
890
01:00:27,700 --> 01:00:31,390
matrix with the information set,
with the MDS property on
891
01:00:31,390 --> 01:00:34,240
the information sets.
892
01:00:34,240 --> 01:00:39,390
This is the parity-check code,
this row is just taken out as
893
01:00:39,390 --> 01:00:40,640
a trivial code.
894
01:00:40,640 --> 01:00:43,970
895
01:00:43,970 --> 01:00:49,750
And then, the demon of
mathematics conspired that
896
01:00:49,750 --> 01:00:51,940
this would also be true.
897
01:00:51,940 --> 01:00:55,810
So if you have an extension
field of 2, and you want to
898
01:00:55,810 --> 01:01:00,720
give it a dimension three, MDS
code, they exist for q plus 2.
899
01:01:00,720 --> 01:01:05,130
900
01:01:05,130 --> 01:01:05,241
Right.
901
01:01:05,241 --> 01:01:08,130
There are, of course, reasons
for this, but they go pretty
902
01:01:08,130 --> 01:01:11,400
deep, why they exist for those
parameters, and this is just
903
01:01:11,400 --> 01:01:14,150
mysterious.
904
01:01:14,150 --> 01:01:15,830
One can give reasons,
so on another hand,
905
01:01:15,830 --> 01:01:17,080
it's just so, right.
906
01:01:17,080 --> 01:01:20,610
907
01:01:20,610 --> 01:01:22,870
There are exceptionally enough
that they have names.
908
01:01:22,870 --> 01:01:29,370
The first one is the Hexacode,
it's something with a
909
01:01:29,370 --> 01:01:42,230
generator matrix, and
this goes over F4.
910
01:01:42,230 --> 01:01:49,110
So that's an MDS code of length
six, so this is a n6,
911
01:01:49,110 --> 01:01:54,380
3, 4, MDS code over
alphabet size 4.
912
01:01:54,380 --> 01:01:58,650
That's the first one, in
that sequence here.
913
01:01:58,650 --> 01:02:00,790
Anyway.
914
01:02:00,790 --> 01:02:02,180
Otherwise, we have
this conjecture.
915
01:02:02,180 --> 01:02:05,850
If you solve this, you are going
to be rich and famous,
916
01:02:05,850 --> 01:02:10,660
you're going to live
in Hollywood, and
917
01:02:10,660 --> 01:02:11,760
maybe, maybe not.
918
01:02:11,760 --> 01:02:15,530
But you're going to be probably
not rich, you're
919
01:02:15,530 --> 01:02:17,730
going to be famous about a
couple of hundred people who
920
01:02:17,730 --> 01:02:23,160
know about this MDS conjecture,
but very smart
921
01:02:23,160 --> 01:02:25,475
people have been looking for
this for a long, long time.
922
01:02:25,475 --> 01:02:26,725
OK.
923
01:02:26,725 --> 01:02:29,356
924
01:02:29,356 --> 01:02:31,260
All right, 20 minutes left.
925
01:02:31,260 --> 01:02:34,840
So it's better we define, we
make sure those codes exist.
926
01:02:34,840 --> 01:02:36,964
Do we have any question about
this MDS conjecture?
927
01:02:36,964 --> 01:02:46,220
928
01:02:46,220 --> 01:02:50,070
OK, last 20 minutes, let's
at least make sure
929
01:02:50,070 --> 01:02:51,410
those things exist.
930
01:02:51,410 --> 01:02:52,660
Reed-Solomon codes.
931
01:02:52,660 --> 01:03:00,550
932
01:03:00,550 --> 01:03:07,210
So Reed-Solomon codes
cover this case.
933
01:03:07,210 --> 01:03:09,970
They are examples of codes
which lie, which
934
01:03:09,970 --> 01:03:12,690
satisfy this equality.
935
01:03:12,690 --> 01:03:14,840
OK, so how do we define
Reed-Solomon codes?
936
01:03:14,840 --> 01:03:20,780
937
01:03:20,780 --> 01:03:24,590
Now, just in a true
mathematician spirit, write
938
01:03:24,590 --> 01:03:27,010
down consider the following.
939
01:03:27,010 --> 01:03:31,593
Consider the following code.
940
01:03:31,593 --> 01:03:37,510
941
01:03:37,510 --> 01:03:38,760
See?
942
01:03:38,760 --> 01:03:49,100
943
01:03:49,100 --> 01:04:01,135
Beta 0 beta q minus 1.
944
01:04:01,135 --> 01:04:09,808
945
01:04:09,808 --> 01:04:14,520
The beta i are the distinct
field elements, the distinct
946
01:04:14,520 --> 01:04:17,310
elements in the finite field.
947
01:04:17,310 --> 01:04:21,075
f is a polynomial.
948
01:04:21,075 --> 01:04:24,590
949
01:04:24,590 --> 01:04:33,810
f is a polynomial, and the
degree is less than k.
950
01:04:33,810 --> 01:04:34,950
OK, good.
951
01:04:34,950 --> 01:04:37,280
So we have defined a code.
952
01:04:37,280 --> 01:04:40,540
So what that means is we
start from polynomials.
953
01:04:40,540 --> 01:04:46,496
The set of all polynomials
of degree at most k.
954
01:04:46,496 --> 01:04:48,160
So what do we know
about that set?
955
01:04:48,160 --> 01:04:50,520
It's a vector space, right?
956
01:04:50,520 --> 01:04:53,000
The set of all polynomials
of degree at most k.
957
01:04:53,000 --> 01:04:55,640
We can add them to get a
polynomial of degree at most
958
01:04:55,640 --> 01:04:58,320
k, we can multiply them with a
scalar to get a polynomial
959
01:04:58,320 --> 01:05:00,390
with degree at most k.
960
01:05:00,390 --> 01:05:03,790
It's a vector space.
961
01:05:03,790 --> 01:05:10,820
So we take this vector space and
evaluate for any element
962
01:05:10,820 --> 01:05:12,090
in that vector space.
963
01:05:12,090 --> 01:05:21,740
This element in all non-zero
elements of the field and we
964
01:05:21,740 --> 01:05:22,270
get a code.
965
01:05:22,270 --> 01:05:31,160
We get a set of vectors,
so we get a set of
966
01:05:31,160 --> 01:05:33,830
vectors, and that --
967
01:05:33,830 --> 01:05:35,830
AUDIENCE: [INAUDIBLE]
968
01:05:35,830 --> 01:05:37,180
PROFESSOR: Yeah, I took
all elements.
969
01:05:37,180 --> 01:05:38,310
Why not?
970
01:05:38,310 --> 01:05:40,260
Why not all elements?
971
01:05:40,260 --> 01:05:43,910
Strictly speaking, I should have
taken one more in order
972
01:05:43,910 --> 01:05:46,070
to get the one here.
973
01:05:46,070 --> 01:05:47,800
We can talk about
that in a sec.
974
01:05:47,800 --> 01:05:49,420
But this one more element
would be --
975
01:05:49,420 --> 01:05:53,980
976
01:05:53,980 --> 01:05:55,170
so it's a code.
977
01:05:55,170 --> 01:05:56,130
First of all, it's a code.
978
01:05:56,130 --> 01:05:56,370
Right?
979
01:05:56,370 --> 01:05:58,410
We all see it's a code.
980
01:05:58,410 --> 01:06:01,200
And once you see it's a code,
we ask, what are the
981
01:06:01,200 --> 01:06:02,450
parameters?
982
01:06:02,450 --> 01:06:09,310
983
01:06:09,310 --> 01:06:10,560
The parameters.
984
01:06:10,560 --> 01:06:16,640
985
01:06:16,640 --> 01:06:22,440
So length, length
is the easy one.
986
01:06:22,440 --> 01:06:25,450
Well, it's q.
987
01:06:25,450 --> 01:06:26,700
What is dimension?
988
01:06:26,700 --> 01:06:29,810
989
01:06:29,810 --> 01:06:33,660
Dimension of C. What's
the dimension?
990
01:06:33,660 --> 01:06:38,810
991
01:06:38,810 --> 01:06:40,480
It's a little bit tricky,
that question.
992
01:06:40,480 --> 01:06:45,020
993
01:06:45,020 --> 01:06:48,550
I actually, at Illinois,
we have to
994
01:06:48,550 --> 01:06:51,560
take a class on teaching.
995
01:06:51,560 --> 01:06:53,840
How to become an effective
teacher.
996
01:06:53,840 --> 01:06:58,070
And one of the things they told
us is that if you ask a
997
01:06:58,070 --> 01:07:03,610
question, you have to wait for
12 seconds to get an answer.
998
01:07:03,610 --> 01:07:05,760
So what's the dimension?
999
01:07:05,760 --> 01:07:07,010
There you go.
1000
01:07:07,010 --> 01:07:12,560
1001
01:07:12,560 --> 01:07:16,700
This mapping, this mapping
from a vector space to a
1002
01:07:16,700 --> 01:07:17,950
vector space.
1003
01:07:17,950 --> 01:07:20,170
1004
01:07:20,170 --> 01:07:22,070
This mapping, also
called evaluation
1005
01:07:22,070 --> 01:07:25,300
map, is a linear map.
1006
01:07:25,300 --> 01:07:32,810
It's a linear map, meaning
that, well, let's start
1007
01:07:32,810 --> 01:07:34,140
differently.
1008
01:07:34,140 --> 01:07:35,750
Let's start differently.
1009
01:07:35,750 --> 01:07:38,900
Do any two polynomials map
to the same code word?
1010
01:07:38,900 --> 01:07:42,430
1011
01:07:42,430 --> 01:07:43,140
That you know.
1012
01:07:43,140 --> 01:07:46,760
That you cannot.
1013
01:07:46,760 --> 01:07:54,860
Are there any two codes, two
polynomials, so are there f of
1014
01:07:54,860 --> 01:08:10,740
x, g of x, such that f of beta
0, so that they coincide in
1015
01:08:10,740 --> 01:08:13,520
all positions?
1016
01:08:13,520 --> 01:08:15,370
No, then they would be
the same, right?
1017
01:08:15,370 --> 01:08:18,700
And the reason is because if
there would be something like
1018
01:08:18,700 --> 01:08:25,880
that, then you could just look
at h is f of x minus g of x,
1019
01:08:25,880 --> 01:08:29,130
which is just another polynomial
of degree k.
1020
01:08:29,130 --> 01:08:33,100
And this would have to vanish
in all positions.
1021
01:08:33,100 --> 01:08:40,950
If k is less than q, it could
not possibly vanish in all
1022
01:08:40,950 --> 01:08:44,399
positions, because then the
polynomial of degree k would
1023
01:08:44,399 --> 01:08:47,200
vanish in more than
k positions.
1024
01:08:47,200 --> 01:08:48,720
Fundamental theorem
of algebra.
1025
01:08:48,720 --> 01:08:50,430
The very beginning.
1026
01:08:50,430 --> 01:08:54,859
So the dimension of C is indeed
k, the same as the
1027
01:08:54,859 --> 01:08:58,740
dimension of this
vector space.
1028
01:08:58,740 --> 01:09:00,850
The dimension of the vector
space of polynomials of the
1029
01:09:00,850 --> 01:09:03,000
degree k minus 1.
1030
01:09:03,000 --> 01:09:05,950
And the distance, if k
is less than q, the
1031
01:09:05,950 --> 01:09:11,072
distance is equal to q.
1032
01:09:11,072 --> 01:09:18,920
The distance, what is it?
1033
01:09:18,920 --> 01:09:21,770
Same argument, roughly
the same argument.
1034
01:09:21,770 --> 01:09:27,120
I think that's a linear code, so
if it's a linear code, the
1035
01:09:27,120 --> 01:09:29,490
minimum distance of the code
is the same as the minimum
1036
01:09:29,490 --> 01:09:32,580
weight of a non-zero word.
1037
01:09:32,580 --> 01:09:34,580
What's the minimum weight
of a non-zero word?
1038
01:09:34,580 --> 01:09:37,200
1039
01:09:37,200 --> 01:09:41,319
These are polynomials
of degree k minus 1.
1040
01:09:41,319 --> 01:09:43,920
What's the minimum weight
of a non-zero word?
1041
01:09:43,920 --> 01:09:46,910
Well, we start out with the
weight 1, and whenever the
1042
01:09:46,910 --> 01:09:52,300
polynomial evaluates to 0, one
of the weights drops out.
1043
01:09:52,300 --> 01:09:55,470
So I claim the minimum distance
as the minimum
1044
01:09:55,470 --> 01:10:12,100
weight, weight of the non-zero
word, and this is n minus,
1045
01:10:12,100 --> 01:10:16,090
well, if any of these
polynomials vanishes in all,
1046
01:10:16,090 --> 01:10:19,990
it vanishes in at most,
k minus 1 positions.
1047
01:10:19,990 --> 01:10:25,210
At most, k minus 1 of these
vectors here, of these
1048
01:10:25,210 --> 01:10:27,220
entries, is equal to 0.
1049
01:10:27,220 --> 01:10:31,970
So it drops by, at
most, k minus 1.
1050
01:10:31,970 --> 01:10:35,260
Drops by at most, k minus 1.
1051
01:10:35,260 --> 01:10:36,510
And there we have it.
1052
01:10:36,510 --> 01:10:39,454
1053
01:10:39,454 --> 01:10:40,380
There we have it.
1054
01:10:40,380 --> 01:10:43,114
There we have, oh, this is q.
1055
01:10:43,114 --> 01:10:46,150
1056
01:10:46,150 --> 01:10:46,960
There we have it.
1057
01:10:46,960 --> 01:10:53,280
There we have that the minimum
distance of the code satisfies
1058
01:10:53,280 --> 01:10:54,598
this equation.
1059
01:10:54,598 --> 01:10:56,470
AUDIENCE: [INAUDIBLE]
1060
01:10:56,470 --> 01:10:57,640
PROFESSOR: What?
1061
01:10:57,640 --> 01:10:58,230
AUDIENCE: The dimension?
1062
01:10:58,230 --> 01:10:58,760
PROFESSOR: The dimension.
1063
01:10:58,760 --> 01:11:01,900
So, it's the same argument,
roughly.
1064
01:11:01,900 --> 01:11:07,410
So I say, the dimension, so
let's just say the size of the
1065
01:11:07,410 --> 01:11:11,710
code is q to the k.
1066
01:11:11,710 --> 01:11:15,730
When is the size of the q to the
k, if no two elements in
1067
01:11:15,730 --> 01:11:18,540
the space evaluate to
the same code word?
1068
01:11:18,540 --> 01:11:21,432
But if two of them would
evaluate to the same code
1069
01:11:21,432 --> 01:11:26,930
word, then we would less size
than the vector space had.
1070
01:11:26,930 --> 01:11:30,320
But if two of them evaluate to
the same code word, that means
1071
01:11:30,320 --> 01:11:37,070
this is true for all
four positions.
1072
01:11:37,070 --> 01:11:41,160
Then we could define a
polynomial h of the
1073
01:11:41,160 --> 01:11:42,990
degree k minus 1.
1074
01:11:42,990 --> 01:11:49,645
which disappears in more than
k minus 1 positions.
1075
01:11:49,645 --> 01:11:52,510
I mean, all positions.
1076
01:11:52,510 --> 01:11:56,230
Cannot be, hence the size of
the code is q to the k, so
1077
01:11:56,230 --> 01:11:58,370
this is a linear map,
dimension is k.
1078
01:11:58,370 --> 01:12:01,660
1079
01:12:01,660 --> 01:12:03,120
OK.
1080
01:12:03,120 --> 01:12:03,660
So cool.
1081
01:12:03,660 --> 01:12:04,600
So we have it, right?
1082
01:12:04,600 --> 01:12:06,050
We have our MDS codes.
1083
01:12:06,050 --> 01:12:07,195
They exist.
1084
01:12:07,195 --> 01:12:08,080
Here they are.
1085
01:12:08,080 --> 01:12:10,610
They are Reed-Solomon codes.
1086
01:12:10,610 --> 01:12:15,840
Not all MDS codes are
Reed-Solomon codes, but the
1087
01:12:15,840 --> 01:12:19,220
ones we are interested
in, they are.
1088
01:12:19,220 --> 01:12:21,400
AUDIENCE: [INAUDIBLE]
1089
01:12:21,400 --> 01:12:25,000
PROFESSOR: Well, the distance is
at least this, but the MDS
1090
01:12:25,000 --> 01:12:30,270
bounds is at most this,
so it's equal to this.
1091
01:12:30,270 --> 01:12:43,720
But the MDS bounds, so the
MDS bound has this is.
1092
01:12:43,720 --> 01:12:46,760
So with that.
1093
01:12:46,760 --> 01:12:52,140
So it's indeed, they lie exactly
bang on to this.
1094
01:12:52,140 --> 01:12:54,840
There are MDS codes,
Reed-Solomon codes.
1095
01:12:54,840 --> 01:12:55,910
So that is good.
1096
01:12:55,910 --> 01:12:57,612
So we know what they are.
1097
01:12:57,612 --> 01:13:03,280
So incidentally, where do you
think this one more point is
1098
01:13:03,280 --> 01:13:04,980
that you would evaluate
our polynomials in?
1099
01:13:04,980 --> 01:13:11,120
1100
01:13:11,120 --> 01:13:12,400
You've heard about projective
geometries?
1101
01:13:12,400 --> 01:13:16,840
1102
01:13:16,840 --> 01:13:20,400
There's one more point,
it's infinity.
1103
01:13:20,400 --> 01:13:24,940
You have, basically, if you look
at the numbers, in order
1104
01:13:24,940 --> 01:13:27,410
to close it up, you want to
add infinity to that, too.
1105
01:13:27,410 --> 01:13:32,020
1106
01:13:32,020 --> 01:13:37,690
In order to get this one more,
this one addition in length,
1107
01:13:37,690 --> 01:13:40,450
you want to evaluate this
also at infinity.
1108
01:13:40,450 --> 01:13:43,440
You will have opportunity to
do that in the homework.
1109
01:13:43,440 --> 01:13:45,420
I looked at the homework and
I was pleased to see this
1110
01:13:45,420 --> 01:13:47,220
problem there.
1111
01:13:47,220 --> 01:13:50,190
I hope you will be
pleased, too.
1112
01:13:50,190 --> 01:13:53,780
OK, all right.
1113
01:13:53,780 --> 01:13:55,030
Any questions about this?
1114
01:13:55,030 --> 01:13:58,616
1115
01:13:58,616 --> 01:14:00,200
Let's see what else
I wanted to say.
1116
01:14:00,200 --> 01:14:10,300
1117
01:14:10,300 --> 01:14:14,240
Because it just gives me a few
minutes to talk about a few
1118
01:14:14,240 --> 01:14:28,570
properties of Reed-Solomon
codes, a few properties of
1119
01:14:28,570 --> 01:14:29,820
Reed-Solomon codes.
1120
01:14:29,820 --> 01:14:31,890
1121
01:14:31,890 --> 01:14:34,035
And what did I want
to say there?
1122
01:14:34,035 --> 01:14:46,430
1123
01:14:46,430 --> 01:14:59,620
On nested codes, so an RS code
with parameters n k, maybe we
1124
01:14:59,620 --> 01:15:00,945
define them [UNINTELLIGIBLE]
like this.
1125
01:15:00,945 --> 01:15:09,140
1126
01:15:09,140 --> 01:15:21,965
q is properly contained,
k minus 1, minus 1.
1127
01:15:21,965 --> 01:15:26,730
1128
01:15:26,730 --> 01:15:31,840
This is pretty straight from
the definition of RS codes.
1129
01:15:31,840 --> 01:15:34,850
1130
01:15:34,850 --> 01:15:39,480
The set of polynomials of
degree at most k minus 1
1131
01:15:39,480 --> 01:15:41,990
contains the set of polynomials
of degree at
1132
01:15:41,990 --> 01:15:44,170
most k minus 1.
1133
01:15:44,170 --> 01:15:51,270
So they are nested codes,
property one.
1134
01:15:51,270 --> 01:15:54,030
1135
01:15:54,030 --> 01:15:55,960
You will see this is important,
that they are
1136
01:15:55,960 --> 01:15:59,170
nested codes, for various
constructions where
1137
01:15:59,170 --> 01:16:01,160
Reed-Solomon codes take
part in later on.
1138
01:16:01,160 --> 01:16:04,000
1139
01:16:04,000 --> 01:16:15,990
A punctured RS code is
again an MDS code.
1140
01:16:15,990 --> 01:16:20,050
1141
01:16:20,050 --> 01:16:22,460
Why is that so?
1142
01:16:22,460 --> 01:16:23,710
Why is that so?
1143
01:16:23,710 --> 01:16:27,610
1144
01:16:27,610 --> 01:16:30,510
Well, you see it?
1145
01:16:30,510 --> 01:16:33,670
1146
01:16:33,670 --> 01:16:36,490
Say if you puncture a
Reed-Solomon code.
1147
01:16:36,490 --> 01:16:41,640
That means we just choose to not
evaluate our code in this,
1148
01:16:41,640 --> 01:16:43,450
this position.
1149
01:16:43,450 --> 01:16:44,610
And this field element.
1150
01:16:44,610 --> 01:16:45,960
Well, we just drop
that coordinate.
1151
01:16:45,960 --> 01:16:48,830
1152
01:16:48,830 --> 01:16:51,810
Does anything change in the
arguments we have made?
1153
01:16:51,810 --> 01:16:58,450
Well, the length is now 1 less,
the dimension, well, the
1154
01:16:58,450 --> 01:17:03,630
dimension is still the same,
as long as k is not larger
1155
01:17:03,630 --> 01:17:05,680
than the length of the code.
1156
01:17:05,680 --> 01:17:10,650
The distance, still the same as
the length, the distance is
1157
01:17:10,650 --> 01:17:16,240
at least the length minus
the number of 0's.
1158
01:17:16,240 --> 01:17:18,630
So that equation still holds.
1159
01:17:18,630 --> 01:17:20,390
Well, but that's
all we needed.
1160
01:17:20,390 --> 01:17:22,360
Still MDS code.
1161
01:17:22,360 --> 01:17:25,460
So there was really no --
it was not important.
1162
01:17:25,460 --> 01:17:28,640
It was not important if you took
all field elements, or a
1163
01:17:28,640 --> 01:17:31,870
subset of the field elements
with MDS property.
1164
01:17:31,870 --> 01:17:34,230
That has nothing
to do with it.
1165
01:17:34,230 --> 01:17:36,220
In particular, we often
in the end, we often
1166
01:17:36,220 --> 01:17:40,240
will drop the 0 element.
1167
01:17:40,240 --> 01:17:44,590
We often choose not to evaluate
these polynomials in
1168
01:17:44,590 --> 01:17:48,270
the 0 of the field.
1169
01:17:48,270 --> 01:17:51,775
A punctured Reed-Solomon
code is an MDS code.
1170
01:17:51,775 --> 01:17:56,030
1171
01:17:56,030 --> 01:17:58,350
So what else did I want
to say about this?
1172
01:17:58,350 --> 01:18:01,801
1173
01:18:01,801 --> 01:18:04,270
What else did I want
to say about this?
1174
01:18:04,270 --> 01:18:09,610
1175
01:18:09,610 --> 01:18:10,860
A generator matrix.
1176
01:18:10,860 --> 01:18:17,180
1177
01:18:17,180 --> 01:18:19,060
How would a generator
matrix look like?
1178
01:18:19,060 --> 01:18:28,389
1179
01:18:28,389 --> 01:18:30,160
Yeah, how would it look like?
1180
01:18:30,160 --> 01:18:32,850
1181
01:18:32,850 --> 01:18:34,260
Basically, we can come
from here, right?
1182
01:18:34,260 --> 01:18:37,085
We can take the generators
of that space.
1183
01:18:37,085 --> 01:18:40,360
1184
01:18:40,360 --> 01:18:43,820
So basically, we
say that one --
1185
01:18:43,820 --> 01:18:52,070
1186
01:18:52,070 --> 01:19:02,780
generate the set of polynomials,
that vector space
1187
01:19:02,780 --> 01:19:04,800
of polynomials with --
1188
01:19:04,800 --> 01:19:09,910
1189
01:19:09,910 --> 01:19:12,060
so this is the basis of
that vector space.
1190
01:19:12,060 --> 01:19:14,610
1191
01:19:14,610 --> 01:19:20,840
So if we map that basis, then we
get a basis of the image of
1192
01:19:20,840 --> 01:19:22,090
the mapping.
1193
01:19:22,090 --> 01:19:24,200
1194
01:19:24,200 --> 01:19:27,610
And the mapping of that
basis would give this.
1195
01:19:27,610 --> 01:19:32,080
So we evaluate the function
1 in all field elements --
1196
01:19:32,080 --> 01:19:35,960
gives us 1.
1197
01:19:35,960 --> 01:19:39,410
We evaluate the function x
in all field elements.
1198
01:19:39,410 --> 01:19:41,610
This gives us the next
generator of the
1199
01:19:41,610 --> 01:19:43,220
Reed-Solomon code.
1200
01:19:43,220 --> 01:19:54,880
Well, 0 gives 0, 1 gives, oh,
let's write like this.
1201
01:19:54,880 --> 01:19:57,160
We evaluate it in all
field elements.
1202
01:19:57,160 --> 01:20:01,740
1203
01:20:01,740 --> 01:20:04,910
These are all the
field elements.
1204
01:20:04,910 --> 01:20:23,103
The next one, and this
goes up to beta --
1205
01:20:23,103 --> 01:20:28,630
1206
01:20:28,630 --> 01:20:30,745
OK, so this would be
a generator matrix.
1207
01:20:30,745 --> 01:20:34,150
1208
01:20:34,150 --> 01:20:35,320
That's fine.
1209
01:20:35,320 --> 01:20:45,030
So now, in order to make things
a bit more interesting,
1210
01:20:45,030 --> 01:20:46,645
do you have to stop five
minutes early?
1211
01:20:46,645 --> 01:20:47,895
We just started five
minutes late?
1212
01:20:47,895 --> 01:20:50,980
1213
01:20:50,980 --> 01:20:54,435
OK then, I think that's over.
1214
01:20:54,435 --> 01:20:56,550
I think it's over.
1215
01:20:56,550 --> 01:21:02,150
One more thing for you guys to
think about until you reach
1216
01:21:02,150 --> 01:21:06,250
home, then the rest
you do next time.
1217
01:21:06,250 --> 01:21:18,820
So let beta 0 be equal to 0
beta 1, or beta i equal to
1218
01:21:18,820 --> 01:21:21,745
omega i minus 1 where omega
is primitive in the field.
1219
01:21:21,745 --> 01:21:32,690
1220
01:21:32,690 --> 01:21:41,885
Then we can write the
matrix v of omega.
1221
01:21:41,885 --> 01:22:03,310
1222
01:22:03,310 --> 01:22:07,410
I tend to see that the first k
columns, the first k rows of
1223
01:22:07,410 --> 01:22:09,860
this matrix would be a
generator matrix of a
1224
01:22:09,860 --> 01:22:10,830
Reed-Solomon code.
1225
01:22:10,830 --> 01:22:13,330
Of course it's the same
as [UNINTELLIGIBLE].
1226
01:22:13,330 --> 01:22:20,390
If we now delete the first
position, we erase the first,
1227
01:22:20,390 --> 01:22:23,580
we puncture the first position
all out, and we look at the
1228
01:22:23,580 --> 01:22:25,500
rest of the matrix.
1229
01:22:25,500 --> 01:22:26,370
This factor of the matrix.
1230
01:22:26,370 --> 01:22:30,180
Does this remind anybody
of anything?
1231
01:22:30,180 --> 01:22:32,740
It's a DFT, it's a Fourier
transform.
1232
01:22:32,740 --> 01:22:35,980
And that's what we start
with next time.
1233
01:22:35,980 --> 01:22:41,780
So think about why this is
a Fourier transform.
1234
01:22:41,780 --> 01:22:45,890
And maybe that's
a nice analogy.
1235
01:22:45,890 --> 01:22:47,790
So we get the distance.
1236
01:22:47,790 --> 01:22:51,560
The distance is at least
something, which means it's
1237
01:22:51,560 --> 01:22:52,360
not impulsive.
1238
01:22:52,360 --> 01:22:54,000
It's not a single 1 somewhere.
1239
01:22:54,000 --> 01:22:56,400
The vector that we get
is not impulsive.
1240
01:22:56,400 --> 01:22:59,680
Maybe it has something to do
with the bandwidth constraint
1241
01:22:59,680 --> 01:23:02,380
and the frequency domain.
1242
01:23:02,380 --> 01:23:03,990
That's what you have to
think about on the way
1243
01:23:03,990 --> 01:23:05,690
home, and that's it.
1244
01:23:05,690 --> 01:23:06,940
Thanks so much.
1245
01:23:06,940 --> 01:23:15,114
| 677.169 | 1 |
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I dont get linear algebra
I had a linear algebra course for my 1st year civil engineering curriculum, and I passed with a 3.2 GPA but I only conceptually understood about 10% of what was taught to me.
I don't know what an eigenvalue/eigenvector is, what the hell is a subspace, nullspace, imagespace. What the hell is a linear transformation, what the hell is a determinant of an nxn matrix, what the hell is a matrix.
How the hell was I able to get a decent mark in a subject I know nothing about?
Facepalm.
I found calculus 1 (single variable) way easier to understand than this stuff.
I can't explain the whole linear algebra curriculum in a short post, but it is a fundamental part of mathematics. It sounds like you basically know nothing about the subject but you managed to pass with a decent grade. Good job?
1. Solving systems of linear equations is part of linear algebra, and this is probably the part that is used the most by the most people.
2. Finish calculus and differential equations and then revisit linear algebra. Conceptually it might make more sense then.
tahayassen
#3
Dec24-12, 12:10 PM
P: 273Stephen Tashi
#4
Dec24-12, 01:08 PM
Sci Advisor
P: 3,245
I dont get linear algebraIf you want to understand linear algebra intuitively, you'll get better advice by asking for an explanation of one concept at a time rather than by asking for an explanation of the entire subject. When it comes to intuition, people have a wide variety of ways of looking at mathematical concepts. You can get 5 different ways of looking at one simple idea.
johnqwertyful
#5
Dec24-12, 01:16 PM
P: 331
The key with linear algebra is mathematical maturity. You need to understand that definitions are just definitions. There's nothing deeper.
An eigenvalue, λ, and eigenvector, x of a matrix A are such that Ax=λx. That's IT. There's absolutely nothing more to it than that. That's all that it means. Why you care, how it's used is a completely different question. But that's all it is.
A matrix is just an array of numbers. That's ALL. Nothing more. That's all there is to it. Nothing deeper, nothing more. An array of numbers. Don't try to pull things out of it that simply aren't there. Yes, you can do cool things with it. Yes, you apply it in weird places. But that's ALL IT IS. An array of numbers.
The one thing I think that's taught poorly is vector spaces. Why they give you an example of an algebraic structure before you understand what an algebraic structure IS, is completely past me.
An algebraic structure is a SET with ONE OR MORE operations defined on it. In a VECTOR SPACE, the set is the set of vectors. The operations are scalar multiplication and vector addition.
An algebraic structure IS math. It's such a confusing, deep subject if you don't really understand what's going on. But when you get it, it's pretty cool. Anything you do in math is in an algebraic structure (most the time, you're dealing with Euclidean space. Euclidean space is the "normal" space with "normal" rules).
A much better example of a structure is what's called a FIELD. (NOT a vector field, when you get to multivariate calculus. This is extremely important) A FIELD is a structure with elements that has two operations, + and * defined over it. It has a list of axioms; closure, 4 additive ones, 4 multiplicative ones, one associative one. An axiom is a DEFINITION.
See, in the real world we don't have wild 2s running around. "2" does NOT exist in nature. You always have 2 something. 2 rocks, 2 buildings, 2 blades of grass, 2 whatever. But "2" does NOT exist. We CREATE "2" to describe the real world. To describe the world, we create these ALGEBRAIC STRUCTURES.
A field IS numbers. When you ask your friend what 2+2 equals, you're working in a FIELD, namely R (the real numbers). Make sense? An algebraic structure IS math. Whatever you do in math is a structure. A vector space is ANOTHER example of a structure. Just one that's studied extensively in linear algebra. Anything you want to know about the operations (EXCEPT WHAT THE OPERATIONS ARE ACTUALLY DOING!), you can derive from the axioms. In a vector space, you can derive ALL you want to know about scalar multiplication or vector addition from the axioms. BUT the one thing you CAN'T derive is WHAT YOU'RE ACTUALLY DOING when you add vectors. YOU must define that.
So long story short, you probably DO understand it. You're just looking for something that's not there. Definitions are just definitions.
Vargo
#6
Dec24-12, 01:41 PM
P: 350johnqwertyful
#7
Dec24-12, 01:48 PM
P: 331
Quote by VargoLinear algebra isn't meaningless at all, when did I ever say anything like that? You're just learning things rigorously, without much if any physical intuition.
Like with eigenvalues/eigenvectors. There really isn't a physical intuition behind it (maybe there is? I just never heard of any). It just is. That doesn't mean that it's "meaningless". It's used to solve differential equations later, which renders them super useful.
Not everything in math has some physical significance.
Vargo
#8
Dec24-12, 02:36 PM
P: 350
I apologize for misinterpreting your comment.
I amend my comment to say that without context, an abstract system of rules and definitions such as linear algebra can be hard to hold onto.
Quote by johnqwertyful
Not everything in math has some physical significance.
I know at least one other poster here who would agree with this. I tend to disagree though.
In the case of an eigenvector, its physical significance is that it represents a subspace that is invariant under a linear transformation. The eigenvalue is the scaling factor applied to that invariant subspace. (geometry is physical enough for me )
johnqwertyful
#9
Dec24-12, 04:32 PM
P: 331
Quote by Vargo
I amend my comment to say that without context, an abstract system of rules and definitions such as linear algebra can be hard to hold onto.
I'll agree with that for sure. It takes awhile to get used to definitions. Trying to find context is its own task. Applied linear algebra is a huge (fascinating) field.
Studiot
#10
Dec24-12, 04:42 PM
P: 5,462
I'm sorry to to a civil engineer with such an attitude.
Linear alegbra is vitally important to some aspects of civil engineering such as the ability to solve (very) large sets of simultaneous equations for structural or hydraulic purposes.
Knowledge of eigenvectors help prevent such disasters as Tacoma Narrows.
If concrete is important to you then so is the knowledge of the Bogue equations, which form a stochiometric linear algebra.
I can honestly say that the only major non linear analysis I was involved in, during my time in engineering maths in civils, concerned pressure fluctuations in major gas pipelines.
chiro
#11
Dec24-12, 05:12 PM
P: 4,572
It just extends the idea of linearity to multi-dimensional objects.
It's not intuitive because you represent an object that is multi-dimensional but is treated like a single object instead of a collection of other objects.
To see this, you should look at the multi-variable and manifold calculus and look at the analogues between the one-dimensional and the multi-dimensional forms.
Linearity is the simplest kind of object and the algebra also provides techniques of decomposition and re-composition of general vectors, matrices, and even tensors.
symbolipoint
#12
Dec25-12, 12:41 AM
HW Helper
PF Gold
P: 2,786Since you, according to one of your recent posts, are currently studying Algebra 1, you are not yet ready to understand much of Linear Algebra. Give yourself about 2 more years.
KingKai
#13
Dec25-12, 06:15 AM
P: 31
I'm sorry to to a civil engineer with such an attitude.
I'm not a civil engineer yet, I'm an undergrad student.
Attitude? I was simply expressing my confusion over this newly (I can't stress the word newly enough) learned subject. It is my sole intention to strengthen my intuition with the subject in the same way I am intuitive with calculus and geometry.
I don't hate linear algebra, it not as though I want to attack it with a light saber, I just find it more abstract than any other branch of math I have been exposed to.
Perhaps soil mechanics is your bag
As I am a 1st year undergrad student with no exposure whatsoever to the specialties of civil engineering, drawing such conclusions based on the limited info and limited time of exposure I have had with linear algebra (3 months) is a little too extreme.
You just fueled me only to pursue linear algebra further.
Merry Christmas :)
BraneChild
#14
Apr3-13, 06:38 PM
P: 5
Funny,
I thought Linear Algebra was easier to grasp than Calculus. I guess it's because it's hard for me to visualize a mathematical concept (it took me a while to understand what a derivative is from a geometric point of view). With linear algebra you just take a system of linear equations, strip the constants and coefficients from it and viola, you have a matrice! And from there you can apply elementary row operations on it to get a solution, find it's inverse, it's determinant, etc...
To be fair though I learned Linear Algebra independently (which probably made it easier), and I've only gotten the basics (I haven't learned about eigenvalues or linear transformations yet).
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Learn Multimedia Algebra in Win95
08/01/96
Algebra in Simplest Terms is a 24-lesson CD-ROM series that reviews fundamental and teaches advanced algebra concepts, using an electronic textbook format. Based on the successful PBS College Algebra series produced by COMAP's Sol Garfunkel, Ph.D., the course combines professionally produced video and text narration with an online dictionary and graphing calculator, management system, placement tests and more. More than 2,000 instructional screens as well as 2,200 tests and exercise questions are integrated. The program starts with a lesson on the Language of Algebra and concludes with a Probabilities lesson. It also provides feedback for questions and exercises, reinforcing key concepts. Liafail, Inc., Minneapolis, MN, (612) 925-3727.W
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MSRI-UP 2007 Description
MSRI-UP 2007 research topic: Computational Science and Mathematics.
Prerequisites: Participating students should have already taken the Calculus sequence and a course in Linear Algebra. A course in numerical analysis would be helpful. For maximum benefit, it would be useful if the students have taken a course in physics, chemistry or biology.
Overview of the summer program: The MSRI-UP summer program is designed for undergraduate students who have completed two years of university-level mathematics courses and would like to conduct research in the mathematical sciences.
During the summer, each of the 12 students participants will
participate in the mathematics research program under the direction of Dr. Meza
complete a research project done in collaboration with other MSRI-UP students
give a presentation and write a technical report on his/her research project
attend a series of colloquium talks given by leading researches in their field
attend workshops aimed at developing skills and techniques needed for research careers in the mathematical sciences; and
learn techniques that will maximize a student's likelihood of admissions to graduate programs as well as the likelihood of winning fellowships
After the summer, each student will:
have an opportunity to to attend a national mathematics or science conference where students will present their research.
be part of a network of mentors that will provide continuous advice in the long term as the student makes progress in his/her studies.
be contacted regarding future research opportunities
Topic description: Computational science is now widely considered to be the third pillar of science alongside experiments and theory. The purpose of this research program is to give students a brief introduction to the most widely used mathematical techniques for solving some of the most challenging scientific problems today. We will draw from current problems being worked on at Lawrence Berkeley National Laboratory including the search for dark energy, climate modeling, nanoscience, and biology. The program will describe the mathematical underpinnings of the scientific applications and discuss the necessary mathematical algorithms needed to solve the problems. For each topic, the lectures will be structured so that a domain scientist from LBNL will discuss the problem and outline the main scientific questions. This will be followed by a set of lectures that describe how to formulate the problem mathematically and what numerical algorithms are needed to solve the problem. We will then address open areas where further research is needed to be able to address the future needs of the scientists.
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With this applet you can explore the impact on a graph of a standard function when you change the parameters. You can also...
see more
With this applet you can explore the impact on a graph of a standard function when you change the parameters. You can also change the graph (using the so called hotspots) and see the impact on the parameters. Also, you can take a look at the effects that operations have on one function or on two fucntions.
This Formula Solver program walks you through the steps for solving a system of two linear equations in two variables, using...
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This Formula Solver program walks you through the steps for solving a system of two linear equations in two variables, using the graphing method. You can use your own values, and it will draw graphs and plot the solution.
This site is designed to help pre-algebra, algebra, and trigonometry students better understand algebraic and trigonometric...
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This site is designed to help pre-algebra, algebra, and trigonometry students better understand algebraic and trigonometric functions. The function is introduced and defined, while the domain and range are demonstrated with graphics and java applets.
This site includes more than 40 tutorials in Intermediate Algebra topics with practice tests and answer keys. The site is...
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This site includes more than 40 tutorials in Intermediate Algebra topics with practice tests and answer keys. The site is designed to assist the user in preparing for math placement tests and the math portion of the GRE.
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Engineering Analysis/Linear Independence and Basis
Before reading this chapter, students should know how to take the transpose of a matrix, and the determinant of a matrix. Students should also know what the inverse of a matrix is, and how to calculate it. These topics are covered in Linear Algebra.
Contents
A set of vectors are said to be linearly dependant on one another if any vector v from the set can be constructed from a linear combination of the other vectors in the set. Given the following linear equation:
The set of vectors V is linearly independent only if all the a coefficients are zero. If we combine the v vectors together into a single row vector:
And we combine all the a coefficients into a single column vector:
We have the following linear equation:
We can show that this equation can only be satisifed for , the matrix must be invertable:
Remember that for the matrix to be invertable, the determinate must be non-zero.
The rank of a matrix is the largest number of linearly independent rows or columns in the matrix.
To determine the Rank, typically the matrix is reduced to row-echelon form. From the reduced form, the number of non-zero rows, or the number of non-zero columns (whichever is smaller) is the rank of the matrix.
Frequently, it is useful to change the basis vectors to a different set of vectors that span the set, but have different properties. If we have a space V, with basis vectors and a vector in V called x, we can use the new basis vectors to represent x:
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What if the most expensive and ambitious structure ever assembled, the International Space Station, was threatened. CIA agent and former NASA candidate Peter Novak, son of a U.S. moonwalker, is... More > safely back on Earth, ready to settle down and begin a family. But the Chinese will not soon forget how he spoiled their plans for space race glory—they want revenge on America for losing face.
A commercial space corporation in America is the unwitting vector of a deeply-rooted plot by the Chinese to sabotage forty years of hard work and multinational cooperation. Shortly after his celebrated trip to the moon and back, Peter rockets into orbit on a harrowing rescue mission. Cosmonauts are trapped aboard a crippled ship, and the International Space Station is in danger.
Peter must risk his life to save the stranded space men, once more flying by the seat of his pants—he has a 1970s Skylab spacesuit and a flashlight against sophisticated incendiary devices.< Less
MATHEMATICS: GCSE: Shape, Space, Statistics and Probability (164 pages) provides a detailed study of shapes, their properties and measurement, statistics and probability . The book contains a wealth... More > of worked examples and exercises accompanied by worked answers. The content material is appropriate for study from GCSE level (and upwards) in mathematics and statisticsLinear Algebra I is a book for university students of any university branch of science. You will find summaries of theory and exercises solved, of the following topics: Matrices, Resolution of Linear... More > Systems Equations, Vector Spaces, Linear Transformations, Diagonalization of endomorphism, and Bilinear and Quadratic Forms.
I have 20 years of experience teaching mathematics at the university level. And, as a teacher of Algebra, Calculus, Statistics, etc., of university students, and, as a result of the needs that I have seen in my students, I have written this book.
This book is characterized by being practical and didactic. It is also useful as a guide for the student.
I hope it will be useful to you, above all.< Less
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Graphs and Digraphs - 5th edition
Summary: Written for advanced undergraduate and beginning graduate students, the fifth edition of this best-selling book provides a wide range of new examples along with historical discussions of mathematicians, problems, and conjectures. It features new and expanded coverage of such topics as toughness, graph minors, perfect graphs, list colorings, nowhere zero flows, list edge colorings, the road coloring problem, and the rainbow number of a graph. Additional applications, exercises, and ex...show moreamples illustrate the concepts and theorems. A solutions manual is available for qualifying instructors7375.86 +$3.99 s/h
New
EuroBooks Horcott Rd, Fairford,
New Book. Shipped from UK within 4 to 14 business days. Established seller since 2000.
$78.04 +$3.99 s/h
Good
TextbookBarn Woodland Hills, CA
1439826277
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books.google.com - This... Linear Algebra
Numerical Linear Algebra
This important topics in the field are covered with a fresh perspective, including iterative methods for systems of equations and eigenvalue problems and the underlying principles of conditioning and stability. Presentation is in the form of 40 lectures, which each focus on one or two central ideas. The unity between topics is emphasized throughout, with no risk of getting lost in details and technicalities. The book breaks with tradition by beginning with the QR factorization - an important and fresh idea for students, and the thread that connects most of the algorithms of numerical linear algebra.
User ratings
This is an excellent book on numerical linear algebra, a very good textbook for a senior undergraduate course. I like the writing style and have been enjoying the reading. Very often it explains "why", not just gives "what" and "how". Highly recommended.
Review: Numerical Linear Algebra
User Review - Darin - Goodreads
The strength of this book is in the conceptual discussions. This isn't the book to use to learn the mechanics of the methods described. It's one of my three favorite numerical linear algebra books.Read full review
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Santa Fe Springs Calculus also read aloud to students to practice the above mentioned strategies to enrich their reading skills. Trigonometry involves functions of sine, cosine, and tangent. These functions can be best represented by graphs.
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Developmental Arithmetic (3-0-3). Credit not applicable toward degrees. Required of students whose ACT Mathematics Main score is less than 15 or COMPASS Math score of 30 or less. Fundamental topics in arithmetic, geometry, and pre-algebra.
099 Developmental Algebra (3-0-3). Credit not applicable toward degrees. Required of students whose ACT Mathematics Main score is at least 15 but less than 19 or COMPASS Math score of 31 to 58. Fundamental topics in algebra for students with insufficient knowledge of high school level mathematics. PR: ACT Mathematics Main score of 15 or grade of "S" in MATH 098.
109 Algebra (3-0-3). Real numbers, exponents, roots and radicals; polynomials, first and second degree equations and inequalities; functions and graphs. PR: ACT Mathematics main score of 19 or grade of "S" in MATH 099.
211 Informal Geometry (3-0-3). Theorems are motivated by using experiences with physical objects or pictures and most of them are stated without proof. Point approach is used with space as the set of all points; review elementary geometry, measurement, observation, intuition and inductive reasoning, distance, coordinate systems, convexitivity, separation, angles, and polygons. No field credit for math majors/minors. PR: MATH 101 or higher.
220 Calculus I (4-0-4). A study of elements of plane analytical geometry, including polar coordinates, the derivative of a function with applications, integrals and applications, differentiation of transcendental functions, and methods of integration. PR: MATH 109 and MATH 110, or GNET 116, or ACT Mathematics main score of 26 or COMPASS Trigonometry score of 46 or above.
250 Discrete Mathematics (3-0-3). Treats a variety of themes in discrete mathematics: logic and proof, to develop students' ability to think abstractly; induction and recursion, the use of smaller cases to solve larger cases of problems; combinatorics, mathematics of counting and arranging objects; algorithms and their analysis, the sequence of instructions; discrete structures, e.g., graphs, trees, sets; and mathematical models, applying one theory to many different problems. PR: MATH 109 and MATH 110 or GNET 116.
290 Topics in Mathematics (1-4 hours credit). Formal course in diverse areas of mathematics. Course may be repeated for different topics. Specific topics will be announced and indicated by subtitle on the student transcript. PR: Consent of instructor.
400 Introduction to Topology (3-0-3). A study of set theory; topological spaces, cartesian products, connectedness; separation axioms; convergences; compactness. Special attention will be given to the interpretation of the above ideas in terms of the real line and other metric spaces. PR: MATH 240.
490 Topics in Mathematics (1-4 hours credit per semester). Advanced formal courses in diverse areas of mathematics. Courses may be repeated for different topics. Specific topics will be announced and indicated by subtitle on transcript. PR: Consent of instructor.
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College math textbooks are expensive, but professors continue to adopt them. One reason is that the textbooks are bundled with ancillary materials that make life easier for the professor, if not always the student. These materials include test generators, homework assignments, online homework systems, solution manuals, PowerPoint slide presentations, instructor manuals, and so on. Adjunct instructors must often teach at many different schools, and often have limited time to prepare, so they rely on these supplementary materials.
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Description by Tapeworm Video:
Calculus 1 and 2 Tutor
The Calculus 1 & 2 Tutor is the easiest way to improve your grades! How does a baby learn to speak' By being immersed in everyday conversation. What is the best way to learn Calculus' By being immersed in it! During this course the instructor will work out hundreds of examples with each step fully narrated so no one gets lost! See why thousands have discovered that the easiest way to higher grades is to learn by examples!This course covers the essential material necessary to prepare a student for a course in Calculus 1 & 2
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Algebraic Sudoku: Book 2Algebraic Sudoku follows the traditional algebra curriculum, while challenging students' minds with fun puzzles that develop logic, reasoning skills, concentration, and confidence. Each Sudoku puzzle is like a mini-lesson, with background, discussion, strategy, and demonstration for solving each problem. After completing the algebra exercises, students are given enough data that will allow them to reason their way through the remaining cells of the Sudoku puzzle that follows. Each activity is presented on a ready-to-use, reproducible master that can be easily photocopied or reproduced as a transparency for full-class instruction and discussion. With more than 30 activities in each book, Algebraic Sudoku is a must-have resource for all students enrolled in or preparing to take algebra, or for anyone who wants to keep their algebraic skills sharp.
| 677.169 | 1 |
Geometric Algebra for Comp.Science - 2nd edition
Summary: Until geometr...show moreic programming. Geometric Algebra for Computer Science presents a compelling alternative to the limitations of linear algebra. Geometric algebra, or GA, is a compact, time-effective, and performance-enhancing way to represent the geometry of 3D objects in computer programs. In this book you will find an introduction to GA that will give you a strong grasp of its relationship to linear algebra and its significance for your work. You will learn how to use GA to represent objects and perform geometric operations on them. And you will begin mastering proven techniques for making GA an integral part of your applications in a way that simplifies your code without slowing it down.*Explains GA as a natural extension of linear algebra and conveys its significance for 3D programming of geometry in graphics, vision, and robotics. *Systematically explores the concepts and techniques that are key to representing elementary objects and geometric operators using GA. *Covers in detail the conformal model, a convenient way to implement 3D geometry using a 5D representation space. *Presents effective approaches to making GA an integral part of your programming. *Includes numerous drills and programming exercises helpful for both students and practitioners. *Companion web site includes links to GAViewer, a program that will allow you to interact with many of the 3D figures in the book, and Gaigen 2, the platform for the instructive programming exercises that conclude each chapter
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Effective Ways of Studying Math
This information was adapted from materials by Paul Nolting.
Goals for Learning Math
*To build a good foundation for
acquiring more advanced math skills *To learn problem-solving skills in math
that can be applied to real-life situations *To increase confidence in yourself to
allow you to study and learn math in a more enjoyable way *To feel confident in taking math tests
without "going blank," or "freezing up" *To eliminate test anxiety
How to Achieve the Goals
Before Attending Your First Class
*Have a positive attitude toward the class
and make a firm commitment to perform well in your math class.
*Communicate with your instructor and
develop a good rapport with him/her.
*Discover your weak areas and ask your
instructor to review these areas with you the first week of class.
*Develop a study schedule that will
allow time to work on math every day (at least one hour or more of math per
day). Continual reinforcement of newly acquired math skills will help you
remember the material.
*Read sections of the textbook that
will be covered in the next class. In math courses, information is presented
in sequence. What you learn in one chapter is essential to your understanding
of information in the next chapter. The key to understanding mathematics is to
thoroughly comprehend how or why an operation works. To solve problems you
need to reason through them. Simply memorizing steps,
rules, and procedures without understanding them is a strategy that will not
work in math.
During Class Time *Attend class regularly. Since information
in math is sequential, if you miss a class, you miss a step in the sequence
that you will need to catch up on, or you will fall behind. Sit in the front
of the class if you have trouble with math.
*Have a notebook for Math
ONLY. In note taking for math, be sure to write
down all formulas as well as all examples and steps.
*Label all class notes and homework
with date, page, and section number.
*Write sentences and math problems down
the page--not straight across.
*Write a one-word description in the
margin of your notes indicating definitions, formulas, specific examples, etc.
Develop a note taking system. Ask questions when you do not understand.
*Write down questions for your
instructor or tutor while you are completing your math homework.
After Class *If possible, try to schedule a study time
soon after your math class.
*Review your notes, preferably within
three hours but at least on the same day.
*Work all of the examples that the instructor worked in class. Make note
cards of these examples. For additional practice, work all the example
problems in the text. Work until you have mastered the concepts.
*Get help before the next class on homework problems you cannot work
correctly. Talk to your instructor or get assistance from a tutor in the lab.
*Keep up-to-date with all of your assignments. If you miss a class, call
your instructor or a classmate before the
next class. Your instructor may allow you to attend the same course that is
taught at a different time or day. Remember that you
are responsible for material covered in class that you have missed.
How to Survive Math Tests
Before the Test *Study for the test EVERYDAY. Constant review will help you
retain the information in your long-term memory. Carry note cards of formulas
and sample problems with you, so that you can study between classes or while
waiting for an appointment.
*Review definitions and formulas from
note cards. Review homework assignments: read over your notes and work the
examples.
*Work all the problems in the chapter
review, marking the difficult ones.
*If your textbook has a practice test,
take the test and check your results. Or, make up your own test and exchange
it with a classmate. Simulating the testing situation can help lessen your
anxiety when it is time to take the actual exam.
*Get help in the math lab with problems
you still do not understand.
*Take full advantage of ALL the helpful
resources in the math lab. Tutors are on hand for individual assistance. For
difficult topics, review the videotapes in the lab before going to class.
Bring a blank tape and make your own copy of the videotapes.
Taking the Test *If you studied correctly, you should feel
confident. If you are experiencing anxiety, take a few deep breaths and relax
for a minute.
*Write down formulas or definitions as
soon as you receive the test paper.
*Skim through the test and answer the
easiest questions first. Be aware of the time allotted.
*If you get partial credit for
problems, do not leave any questions blank.
After the Test *Work the problems that you missed as soon
as possible after the test.
*Do not discard your old tests, you can
use them to study for tests you take later.
*Keep up with your test grades.
Studying for the Final Exam
*Review all of your previous tests.
*Redo chapter reviews and chapter tests
in your textbooks.
*Work any worksheets given to you by
your instructor.
*Watch the video (or make your own
copy) of the final exam review in the math lab.
*Take the
practice exam found on
the MTSU Developmental Math web site.
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Calibrated Peer Review (CPR) is a program, for networked computers, that enables frequent writing assignments without any...
see more
Calibrated Peer Review (CPR) is a program, for networked computers, that enables frequent writing assignments without any increase in instructor work. In fact, CPR can reduce the time an instructor now spends reading and assessing student writing. CPR offers instructors the choice of creating their own writing assignments or using the rapidly expanding assignment library. If you believe in constructivist learning, writing is the most important tool that you have. But if you have a class of 300 students, grading essays challenges even the true believer. Calibrated Peer Review (CPR)can be used in classes of any size. CPR is based on the model of peer review in science. The student reads a document, either on-line or hard copy, then writes about it. When the student has demonstrated competence as a reviewer, the program delivers three peer documents on for review. The student answers content and style questions and assigns scores. Finally, the student does a self-review. The student grade comes from writing and reviewing. Even though the program is only in its third year, approximately 100,000 students have used it. Although CPR was designed for use in large chemistry classes, experience has shown that it can serve in many other disciplines, as well. Currently, business, chemistry, economics, English, and life science instructors are using CPR in college, graduate and professional, high schools and middle schools. CPR was developed in the Chemistry Department at U.C.L.A. with funding provided by the National Science Foundation and Howard Hughes Medical Institute handbook prepared by the AICPA and National Business Education Association and the National Teachers of...
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This handbook prepared by the AICPA and National Business Education Association and the National Teachers of Mathematicscontains 15 lesson plans designed to teach various aspects of accounting to high school students. Each lesson includes objectives, instruction procedures, a learning overview, activities and solutions. Topics include: financial statement analysis, budgeting and forecasting, break-even analysis, cost/benefit analysis, time value of money, financial planning, writing a business plan among others.
Finite Math for Windows is a software package that enables students to easily solve problems and/or check their work in...
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Finite Math for Windows is a software package that enables students to easily solve problems and/or check their work in Finite Mathematics. It consists of 17 modules that correspond to the chapters in most Finite Mathematics books. Use of the software is intended to move the students from the calculator world into the more business-like world of software to help prepare them for their careersThis applet demonstrates the Central Limit Theorem using histograms. It is primarily an interactive teaching tool, but could...
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This applet demonstrates the Central Limit Theorem using histograms. It is primarily an interactive teaching tool, but could potentially be used as part of learning material. The web page should meet most accessibility requirements, but the nature of the applet means that is probably will not.
״Prepares college students for the mathematics they need in the social sciences, computer science, business, economics, and...
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״Prepares college students for the mathematics they need in the social sciences, computer science, business, economics, and the physical sciences up to the pre-Calculus level. It is also intended to serve a course that has as its objective an introduction to, or review of, what is currently called "pre-Calculus" topics. Some of the topics that are amplified in modern discrete mathematics and finite mathematics courses are introduced
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George David Birkhoff
analysisA branch of mathematics that deals with continuous change and with certain general types of processes that have emerged from the study of continuous change, such as limits, differentiation, and integration....
mathematicsThe science of structure, order, and relation that has evolved from elemental practices of counting, measuring, and describing the shapes of objects. It deals with logical reasoning and quantitative calculation,...
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Description:New. 1559538953 New book may have school stamps or class set...New. 1559538953 New book may have school stamps or class set numbers on the side but was not issued to a student. 100% guaranteed fast shipping! !
Description:New. 1559538961 New book may have school stamps or class set...New. 1559538961 New book may have school stamps or class set numbers on the side but was not issued to a student. 100% guaranteed fast shipping! !
This is an update of Serra's classic textbook on geometry using the discovery method of hands on exercises to develop and refine geometry skills.
It takes time for students to adapt to this. They've been under rote and lecture for years. But with time and patience home schoolers can thrive and students in traditional schools have a tool to unravel some of the more difficult topics like parallel lines and proofs.
The hands on constructions in the textbook can go a long way to 'getting it.' And some of the mind puzzles help train the kids to see the patterns that can unlock the mysteries of doing proofs.
The newest versions are not as good as the older ones. The content is the same, but current textbook authors have to put in many pictures and a variety of color to meet current text standards.
Sadly, the pictures and color are more of a distraction than a help. Special needs kids and kids frustrated with math don't need this kind of stimulus. They need content which nurtures them and Serra does it head and shoulders above all others.
As a math teacher and collector/reviewer of math textbooks. I rate all other geometry texts no better than 3 stars, with most at one or two.
This Sierra book is a 4 only because of too many pictures he had to put in. The older ones are 5 star because they have the best mix of picture
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Units of a log of a physical quantity You reasoning is flawed. Different units can be multiplied/divided but they cannot be added/subtracted. For example, you can divide meters by seconds and get meters per seconds but you cannot add seconds+meters. That is why taking the log of a unit doesn't make sense because log(1+x) = x-x^2/2+x^3/3... So how can you add sec-sec^2+sec^3...?
Math resources for electrical engineering? @Pacerier Because reviews are personal opinions and some people are bound to genuinely dislike something no matter what others may think of it. In this case, some people think that Kresyszig is bad for first-time-learning-something but excellent for later reference/quick review and hence the "bad reviews". In my opinion, it is excellent for both because I have used it for both. An excellent introduction to basic ODEs/PDEs, linear algebra, numerical analysis, Fourier analysis, Numerical PDEs. It has helped me with graduate courses and exams too later on.
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new. We distribute directly for the publisher. How...New. Brand new. We distribute directly for the publisher. How many dimensions does our universe require for a comprehensive physical description? In 1905, Poincaré argued philosophically about the necessity of the three familiar dimensions, while recent research is based on 11 dimensions or even 23 dimensions. The notion of dimension itself presented a basic problem to the pioneers of topology. Cantor asked if dimension was a topological feature of Euclidean space. To answer this question, some important topological ideas were introduced by Brouwer, giving shape to a subject whose development dominated the twentieth century. The basic notions in topology are varied and a comprehensive grounding in point-set topology, the definition and use of the fundamental group, and the beginnings of homology theory requires considerable time. The goal of this book is a focused introduction through these classical topics, aiming throughout at the classical result of the Invariance of Dimension. This text is based on the author's course given at Vassar College and is intended for advanced undergraduate students. It is suitable for a semester-long course on topology for students who have studied real analysis and linear algebra. It is also a good choice for a capstone course, senior seminar, or independent study.
Description:Good. A First Course in Topology: Continuity and Dimension ...Good. A First Course in Topology: Continuity and Dimension (Student
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Prealgebra (5th Edition)
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Contents
Lecture 0: An Introduction to Calculus
So How does this Course Work, Anyway?
Learning calculus takes determination that goes far beyond simply reading text. Make sure to always have paper and pencils ready to draw diagrams or make quick calculations when solving a problem; using a calculator can only get you so far in Calculus.
Some students will start learning calculus by trying the problems and will only resort to reading the text if they get stuck on a problem or do not know how to proceed. A better plan is to make sure that you know and fully understand the material before beginning work on problems. A link to problems relevant to the material in each lecture is located at the bottom of each page. The problems are separated into three levels. When first going through this course, it is recommended that you be able to complete all of the Level 1 problems and most of Level 2 before continuing to the next lecture. The Level 3 problems are for those who feel they have mastered this calculus course and still want a challenge. Separate links are included that show the answers to each problem and the worked out solutions to a few problems from each level.
A huge factor that will influence your understanding of calculus is your ability to think logically. These lectures will emphasize the importance of writing solutions to problems as a set of ordered steps rather than a hodgepodge of random formulas. You will find that many calculus problems become easier when the steps are written down on paper instead of constrained in your mind. And if you understand the material but still get stuck on a problem: start over. Throw away your work and get a clean sheet of paper with no writing on it. Ofttimes when starting a problem anew, an entirely different solution will come to light that you did not see before. If all else fails: come back later. That's right, you heard me: go outside, talk with a friend, watch television, get some rest, have fun. The God of Calculus will not smite you for leaving problems undone.
The authors of these lectures hope that you will not only be opened to the infinite applications that Calculus has to the modern world, but will find an intrinsic beauty that goes beyond what we as mere mortals can comprehend.
What is Calculus?
To define what calculus is, we should start with the origin of the name. The word calculus in Latin means "pebble". This is quite fitting because calculus deals with the very small. So inconceivably small, in fact, that it is ofttimes hard to differentiate the numbers we deal with from zero. In all, Calculus is the study of change. This is especially true with regards to the four major topics in calculus, which we will cover in this order:
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Palisades Park Ge use fractions and intro to variables. Learn problem solving skills that are needed for the real world in math. Learn how to solve advanced math problems with ease to prepare a student for calculus
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Find a Sunol PrecalculusNot being ready, students can't study calculus successfully. Therefore, in Precalculus, students will be introduced to the important and basic mathematical concepts inquired before in algebra with deeper and higher details. They comprise, but not limited in, inequalities, equations, absolute values, and graphs of lines and circles
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Elementary algebra encompasses some of the basic concepts of algebra, one of the main branches of mathematics. It is typically taught to secondary school students and builds on their understanding of arithmetic. Whereas arithmetic deals with specified numbers,[1] algebra introduces quantities without fixed values, known as variables.[2]A coefficient is a numerical value which multiplies a variable (the operator is omitted). A term is an addend or a summand, a group of coefficients, variables, constants and exponents that may be separated from the other terms by the plus and minus operators.[3] Letters represent variables and constants. By convention, letters at the beginning of the alphabet (e.g. ) are typically used to represent constants, and those toward the end of the alphabet (e.g. and ) are used to represent variables.[4] They are usually written in italics.[5]
Usually terms with the highest power (exponent), are written on the left, for example, is written to the left of . When a coefficient is one, it is usually omitted (e.g. is written ).[9] Likewise when the exponent (power) is one, (e.g. is written ).[10] When the exponent is zero, the result is always 1 (e.g. is always rewritten to ).[11] However , being undefined, should not appear in an expression, and care should be taken in simplifying expressions in which variables may appear in exponents.
Other types of notation are used in algebraic expressions when the required formatting is not available, or can not be implied, such as where only letters and symbols are available. For example, exponents are usually formatted using superscripts, e.g. . In plain text, and in the TeX mark-up language, the caret symbol "^" represents exponents, so is written as "x^2".[12][13] In programming languages such as Ada,[14]Fortran,[15]Perl,[16]Python[17] and Ruby,[18] a double asterisk is used, so is written as "x**2". Many programming languages and calculators use a single asterisk to represent the multiplication symbol,[19] and it must be explicitly used, for example, is written "3*x".
Example of variables showing the relationship between a circle's diameter and its circumference. For any circle, its circumference, divided by its diameter, is equal to the constant pi, (approximately 3.14).
Elementary algebra builds on and extends arithmetic[20] by introducing letters called variables to represent general (non-specified) numbers. This is useful for several reasons.
Variables may represent numbers whose values are not yet known. For example, if the temperature today, T, is 20 degrees higher than the temperature yesterday, Y, then the problem can be described algebraically as .[21]
Variables allow one to describe general problems,[22] without specifying the values of the quantities that are involved. For example, it can be stated specifically that 5 minutes is equivalent to seconds. A more general (algebraic) description may state that the number of seconds, , where m is the number of minutes.
Variables allow one to describe mathematical relationships between quantities that may vary.[23] For example, the relationship between the circumference, c, and diameter, d, of a circle is described by .
Variables allow one to describe some mathematical properties. For example, a basic property of addition is commutativity which states that the order of numbers being added together does not matter. Commutativity is stated algebraically as .[24]
An equation states that two expressions are equal using the symbol for equality, (the equals sign).[26] One of the most well-known equations describes Pythagoras' law relating the length of the sides of a right angle triangle:[27]
This equation states that , representing the square of the length of the side that is the hypotenuse (the side opposite the right angle), is equal to the sum (addition) of the squares of the other two sides whose lengths are represented by and .
An equation is the claim that two expressions have the same value and are equal. Some equations are true for all values of the involved variables (such as ); such equations are called identities. Conditional equations are true for only some values of the involved variables, e.g. is true only for and . The values of the variables which make the equation true are the solutions of the equation and can be found through equation solving.
Another type of equation is an inequality. Inequalities are used to show that one side of the equation is greater, or less, than the other. The symbols used for this are: where represents 'greater than', and where represents 'less than'. Just like standard equality equations, numbers can be added, subtracted, multiplied or divided. The only exception is that when multiplying or dividing by a negative number, the inequality symbol must be flipped.
By definition, equality is an equivalence relation, meaning it has the properties (a) reflexive (i.e. ), (b) symmetric (i.e. if then ) (c) transitive (i.e. if and then ).[28] It also satisfies the important property that if two symbols are used for equal things, then one symbol can be substituted for the other in any true statement about the first and the statement will remain true. This implies the following properties:
Substitution is replacing the terms in an expression to create a new expression. Substituting 3 for a in the expression a*5 makes a new expression 3*5 with meaning 15. Substituting the terms of a statement makes a new statement. When the original statement is true independent of the values of the terms, the statement created by substitutions is also true. Hence definitions can be made in symbolic terms and interpreted through substitution: if , where := means "is defined to equal", substituting 3 for informs the reader of this statement that means 3*3=9. Often it's not known whether the statement is true independent of the values of the terms, and substitution allows one to derive restrictions on the possible values, or show what conditions the statement holds under. For example, taking the statement x+1=0, if x is substituted with 1, this imples 1+1=2=0, which is false, which implies that if x+1=0 then x can't be 1.
If x and yintegers, rationals, or real numbers, then xy=0 implies x=0 or y=0. Suppose abc=0. Then, substituting a for x and bc for y, we learn a=0 or bc=0. Then we can substitute again, letting x=b and y=c, to show that if bc=0 then b=0 or c=0. Therefore if abc=0, then a=0 or (b=0 or c=0), so abc=0 implies a=0 or b=0 or c=0.
Consider if the original fact were stated as "ab=0 implies a=0 or b=0." Then when we say "suppose abc=0," we have a conflict of terms when we substitute. Yet the above logic is still valid to show that if abc=0 then a=0 or b=0 or c=0 if instead of letting a=a and b=bc we substitute a for a and b for bc (and with bc=0, substituting b for a and c for b). This shows that substituting for the terms in a statement isn't always the same as letting the terms from the statement equal the substituted terms. In this situation it's clear that if we substitute an expression a into the a term of the original equation, the a substituted does not refer to the a in the statement "ab=0 implies a=0 or b=0."
Linear equations are so-called, because when they are plotted, they describe a straight line (hence linear). The simplest equations to solve are linear equations that have only one variable. They contain only constant numbers and a single variable without an exponent. As an example, consider:
Problem in words: If you double my son's age and add 4, the resulting answer is 12. How old is my son?
Equivalent equation: where represent my son's age
To solve this kind of equation, the technique is add, subtract, multiply, or divide both sides of the equation by the same number in order to isolate the variable on one side of the equation. Once the variable is isolated, the other side of the equation is the value of the variable.[31] This problem and its solution are as follows:
1. Equation to solve:
2. Subtract 4 from both sides:
3. This simplifies to:
4. Divide both sides by 2:
5. Simplifies to the solution:
The general form of a linear equation with one variable, can be written as:
Following the same procedure (i.e. subtract from both sides, and then divide by ), the general solution is given by
Solving two linear equations with a unique solution at the point that they intersect.
A linear equation with two variables has many (i.e. an infinite number of) solutions.[32] For example:
Problem in words: I am 22 years older than my son. How old are we?
Equivalent equation: where is my age, is my son's age.
This can not be worked out by itself. If I told you my son's age, then there would no longer be two unknowns (variables), and the problem becomes a linear equation with just one variable, that can be solved as described above.
To solve a linear equation with two variables (unknowns), requires two related equations. For example, if I also revealed that:
Problem in words:
In 10 years time, I will be twice as old as my son.
Equivalent equation:
Subtract 10 from both sides:
Multiple out brackets:
Simplify:
Now there are two related linear equations, each with two unknowns, which lets us produce a linear equation with just one variable, by subtracting one from the other (called the elimination method):[33]
Second equation
First equation
Subtract the first equation from
the second in order to remove
Simplify
Add 12 to both sides
Rearrange
In other words, my son is aged 12, and as I am 22 years older, I must be 34. In 10 years time, my son will 22, and I will be twice his age, 44. This problem is illustrated on the associated plot of the equations.
Quadratic equation plot of showing its roots at and , and that the quadratic can be rewritten as
A quadratic equation is one which includes a term with an exponent of 2, for example, ,[34] and no term with higher exponent. The name derives from the Latin quadrus, meaning square.[35] In general, a quadratic equation can be expressed in the form ,[36] where is not zero (if it were zero, then the equation would not be quadratic but linear). Because of this a quadratic equation must contain the term , which is known as the quadratic term. Hence , and so we may divide by and rearrange the equation into the standard form
It follows from the zero-product property that either or are the solutions, since precisely one of the factors must be equal to zero. All quadratic equations will have two solutions in the complex number system, but need not have any in the real number system. For example,
has no real number solution since no real number squared equals −1. Sometimes a quadratic equation has a root of multiplicity 2, such as:
For this equation, −1 is a root of multiplicity 2. This means −1 appears two times.
The graph of the logarithm to base 2 crosses the x axis (horizontal axis) at 1 and passes through the points with coordinates(2, 1), (4, 2), and (8, 3). For example, log2(8) = 3, because 23 = 8. The graph gets arbitrarily close to the y axis, but does not meet or intersect it.
An exponential equation is one which has the form for ,[37] which has solution
when . Elementary algebraic techniques are used to rewrite a given equation in the above way before arriving at the solution. For example, if
then, by subtracting 1 from both sides of the equation, and then dividing both sides by 3 we obtain
whence
or
A logarithmic equation is an equation of the form for , which has solution
For example, if
then, by adding 2 to both sides of the equation, followed by dividing both sides by 4, we get
A radical equation is one that includes a radical sign, which includes square roots, , cube roots, , and nth roots, . Recall that an nth root can be rewritten in exponential format, so that is equivalent to . Combined with regular exponents (powers), then (the square root of cubed), can be rewritten as .[38] So a common form of a radical equation is (equivalent to ) where and are integers. It has solution:
The equations and are parallel and cannot intersect, and is unsolvable.
In the above example, a solution exists. However, there are also systems of equations which do not have any solution. Such a system is called inconsistent. An obvious example is
As 0≠2, the second equation in the system has no solution. Therefore, the system has no solution. However, not all inconsistent systems are recognized at first sight. As an example, let us consider the system
Multiplying by 2 both sides of the second equation, and adding it to the first one results in
Plot of a quadratic equation (red) and a linear equation (blue) that do not intersect, and consequently for which there is no common solution.
There are also systems which have infinitely many solutions, in contrast to a system with a unique solution (meaning, a unique pair of values for and ) For example:
Isolating in the second equation:
And using this value in the first equation in the system:
The equality is true, but it does not provide a value for . Indeed, one can easily verify (by just filling in some values of ) that for any there is a solution as long as . There is an infinite number of solutions for this system.
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This series of videos, created by Salman Khan of the Khan Academics, introduces students to basic probability. Anyone using these videos should a reasonable grounding in basic algebra before viewing. This collection...
This set of videos, created by Salman Kahn of the Kahn Academy, focus on trigonometry. Watch the "Geometry" playlist first if you have trouble understanding the topics covered here, it will certainly be a useful aid. ...
Astronomy is a science that asks fundamental questions about the very fundament of things, the universe. How big and how far away are the planets and stars? How did they form and when? How do they move and why? Finding...
This series of lectures, created by Salman Khan of the Khan Academy, focuses on topics covered in a first year course in differential equations. A basic understanding of differentiation and integration from Calculus...
This is a series of lectures, authored by Chris Tisdell of the University of New South Wales, for MATH2111 "Higher Several Variable Calculus" and "Vector Calculus", which is a 2nd-year mathematics subject taught at UNSW...
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#6 Understanding and Teaching Primary Mathematics
Understanding and Teaching Primary Mathematics is a full colour, core textbook for anyone training to teach Maths at primary level. A self-audit is provided at the outset to assess your prior knowledge and highlight areas for your personal development. The book will then help you to develop your mathematical understanding by encouraging you to think mathematically. You will be asked to create an action plan to aid your development and this will also allow you to build up a portfolio of evidence to help you achieve QTS (Qualified Teacher Status). Rather than treating subject knowledge and pedagogy as two separate entities, the book sets maths subject knowledge in context, showing you how to apply it in the classroom, through models and cases of best practice. Misconception boxes appear throughout each chapter to highlight common misconceptions children have in maths and help you to make connections between different areas of mathematics so that your pupils will see mathematics as a whole rather than a series of separate and often disparate ideas. Case studies in each chapter also provide you with an exemplar lesson plan annotated by a new teacher which shows you how to deal with a common misconception. This book will help you to deliver Maths at primary level in a clear and confident manner.
#7 Engineering Mathematics Pocket Book, 4 Edition (with Solutions)
This compendium of essential formulae, definitions, tables and general information provides the mathematical information required by students, technicians, scientists and engineers in day-to-day engineering practice. A practical and versatile reference source, now in its fourth edition, the layout has been changed and the book has been streamlined to ensure the information is even more quickly and readily available - making it a handy companion on-site, in the office as well as for academic study. It also acts as a practical revision guide for those undertaking BTEC Nationals, Higher Nationals and NVQs, where engineering mathematics is an underpinning requirement of the course.
#10 Organic Chemistry by John McMurry + Solutions Manual
30-09-2010, 11:58
Organic Chemistry by John McMurry + Solutions Manual | 520MB
Succeed in your course with the help of this proven best-seller! John McMurry's Organic Chemistry is consistently praised as the most clearly written book available for the course. In John McMurry's words: "I wrote this book because I love writing. I get great pleasure and satisfaction from taking a complicated subject, turning it around until I see it clearly from a new angle, and then explaining it in simple words." Through his lucid writing and ability to show the beauty and logic of organic chemistry, McMurry makes learning enjoyable. The highest compliment that can be given to a chemistry book applies to McMurry: It works!
No other book on the market today can match the 30-year success of Halliday, Resnick and Walker's Fundamentals of Physics! In a breezy, easy-to-understand the book offers a solid understanding of fundamental physics concepts, and helps readers apply this conceptual understanding to quantitative problem solving. This book offers a unique combination of authoritative content and stimulating applications.
#17 Crook (2010) DVDRip XviD-[DDR].
Crook (Crook - Its Good To Be Bad) movie is based on racial attacks on Indian students in Australia. Neha Sharma, leading protagonist of the film, plays the character of a girl who is born and brought up in Australia. Neha Sharma plays a student who tries to mend the cultural gap between Indians and Australians.
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If you have any doubts about legality of content or you have another suspicions,
feel free to Contact Us.
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IB Mathematics Higher Level
Mrs. Amy Leonard Room 5 - 345
2009 - 2010
Welcome to IB Math HL! You are now enrolled in a higher level International
Baccalaureate course designed for you to earn college credit and fulfill your IB Diploma
requirements. This is a rigorous class and attendance is extremely important if you
want to pass the IB exam.
You are expected to be in the classroom when the bell rings. If you are
late, it will affect your employability skills grade.
We will be using the textbook Mathematics HL & SL by Peter Smythe. We
will be covering 24 chapters plus additional material that appears on the
IB exam. In addition to the core requirements, you will each need to
choose a higher level option for paper 3 – either Statistics or Analysis.
Homework will be assigned daily whether the topic is covered in class or
not. It will be due during the following class.
If you are absent, it is your responsibility to find out what work you have
missed and to make it up. You will have one day for every day you were
absent to make up the homework. You will be able to access assignments
on my teacher web page at:
Any test missed must be taken after school within 5 days following your
return to school. It will be done AFTER school from 2:20 P.M. to 3:30
P.M. by appointment.
Grades will be weighted 80% for tests and quizzes and 20% for
employability skills. Employability points consist of 15% toward homework
and 5% toward attendance and the code of conduct. An unexcused
absence, tardy, dress code violation, or misconduct will result in a loss of 5
points. Semester one will include a midterm exam worth 20% of the first
semester grade, and semester two will include a "mock" exam worth 20%
of the second semester grade.
Cheating in any way will not be tolerated. Cheating on a test will result in a
grade of zero. If you are caught copying another student's paper, both
will be confiscated. Working together is encouraged - copying another's
work is not.
A graphing calculator (TI-83 or TI-84) is required for this class. If this
is a problem, see me immediately. The TI-89 is NOT permitted by the
IBO.
There will be NO food or drinks allowed in the classroom. Cell phone and
laptop usage during class is also not permitted.
Two papers of the mock IB Math HL exam will be administered on
Saturday April 17, 2008 from 7:30A.M. until approximately noon. The
third paper will be given during a regular school day to be announced at a
later date. This test is mandatory. Please do not schedule any other
activities for this Saturday morning.
Please don't hesitate to ask questions as needed. I will be available after school most
days to help with any concepts you don't understand. This course is fast passed and
rigorous. I am always willing to help. I am looking forward to an enjoyable and
rewarding year together!
Mrs. Amy Leonard
Riverview High School
One Ram Way
Sarasota, FL 34231
(941) 923-1484 Ext. 64149
e-mail: amy_leonard@sarasota.k12.fl.us
IB Math HL
Your first homework assignment is to complete this page, signifying that you have
read and understand the expectations for this class. You must also have a parent or
guardian read and sign this form. Keep the first two pages of this letter for your
information throughout the year and return this page tomorrow.
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JEP is a Java package for parsing and evaluating mathematical expressions. It supports user defined variables, constants, and...
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JEP is a Java package for parsing and evaluating mathematical expressions. It supports user defined variables, constants, and functions. A number of common mathematical functions and constants are included.
This is a collection of 339 videos that work out typical exercises that first, second and third semester calculus students...
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This is a collection of 339 videos that work out typical exercises that first, second and third semester calculus students are asked to solve. The lengths of the videos range from a couple of minutes to up to seven minute depending on the complexity of the exercise. They are all closed captioned, and graphs and other diagrams accompany the words and equations when applicable.
Funded by the National Science Foundation, the Learning Matrix collection provides access to about a 1,000 online resources...
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Funded by the National Science Foundation, the Learning Matrix collection provides access to about a 1,000 online resources useful to faculty teaching introductory science and mathematics courses as well as providing instructional and pedagogical trainingEducation.com is a great resource for busy parents who wish to understand better their children's progress in education and...
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Education.com is a great resource for busy parents who wish to understand better their children's progress in education and development of social skills. This valuable information empowers parents to improve and contribute to their kids' success. My favorite area of the website is the activities page with fun and educational ideas for parents and teachers! I also find it very useful that Education.com shares expert advice with parents while also enabling them to have own discussions.
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Summary: The fundamental goal in Tussy and Gustafson's BASIC MATHEMATICS FOR COLLEGE STUDENTS, Third Edition is to teach students to read, write, and think about mathematics through building a conceptual foundation in the language of mathematics. The book blends instructional approaches that include vocabulary, practice, and well-defined pedagogy, along with an emphasis on reasoning, modeling, communication, and technology skills. Also students planning to take an introductor...show morey algebra course in the future can use this text to build the mathematical foundation they will need.
Tussy and Gustafson understand the challenges of teaching developmental students and this book reflects a holistic approach to teaching mathematics that includes developing study skills, problem solving, and critical thinking alongside mathematical concepts. New features in this edition include a pretest for students to gauge their understanding of prerequisite concepts, problems that make correlations between student life and the mathematical concepts, and study skills information designed to give students the best chance to succeed in the course. Additionally, the text's widely acclaimed Study Sets at the end of every section are tailored to improve students' ability to read, write, and communicate mathematical ideas.
New to the Edition
Check Your Knowledge: Pretests, situated at the beginning of every chapter, have been added to this edition as a way to gauge a student's knowledge base for the upcoming chapter. An instructor may assign the pretest to see how well prepared their students are to understanding the chapter; thereby, allowing the instructor to teach accordingly to their students' abilities from the results of the pretest. Students may also take the pretest by themselves and check their answers at the back of the book, which gives them the opportunity to identify what they already know and on what concepts they need to concentrate.
Study Skills Workshop: At the beginning of each chapter is a one-page study skills guide. This complete mini-course in math study skills provides extra help for developmental students who may have weak study skills, as well as additional assistance and direction for any student. These workshops provide a guide for students to successfully pass the course. For example, students learn how to use a calendar to schedule study times, how to take organized notes, best practices for study groups, and how to effectively study for tests. This helpful reference can be used in the classroom or assigned as homework and is sequenced to match the needs of students as they move through the semester.
Think It Through: Each chapter contains either one or two problems that make the connection between mathematics and student life. These problems are student-relevant and require mathematics skills from the chapter to be applied to a real-life situation. Topics include tuition costs, statistics about college life and many more topics directly connected to the student experience.
New Chapter Openers with TLE Labs: TLE (The Learning Equation) is interactive courseware that uses a guided inquiry approach to teaching developmental math concepts. Each chapter opens with a lab that has students construct their own understanding of the concept to build their problem-solving skills. Each lab addresses a real-world application, with the instruction progressing the student through the concepts and skills necessary for solving the problem. TLE enhances the learning process and is perfect for any instructor wanting to teach via a hybrid course.
ThomsonNOW with HOMEWORK FUNCTIONALITY. Assigned from the instructor, the enhanced iLrn functionality provides direct tutorial assistance to students solving specified questions pulled from the textbook's Problem Sets. This effective and beneficial assistance gives students opportunity to try similar, algorithmically-generated problems, detailed tutorial help, the ability to solve the problem in steps and helpful hints in solving the problem.
iLrn/MathNOW a personalized online learning companion that helps students gauge their unique study needs and makes the most of their study time by building focused personalized learning plans that reinforce key concepts. Completely tailored to the Tussy/Gustafson text, this new resource will help your students diagnose their concept weaknesses and focus their studies to make their efforts efficient and effective. Pre-Tests give students an initial assessment of their knowledge. Personalized Learning Plans, based upon the students' performance on the pre-test quiz, outline key learning needs and organize materials specific to those needs. Post-Tests assess student mastery of core chapter concepts; results can be emailed to the instructor!
Features
STUDY SETS are found at the end of every section and feature a unique organization, tailored to improve students' ability to read, write, and communicate mathematical ideas; thereby, approaching topics from a variety of perspectives. Each comprehensive STUDY SET is divided into six parts: VOCABULARY, CONCEPTS, NOTATION, PRACTICE, APPLICATIONS, and REVIEW.
VOCABULARY, NOTATION, and WRITING problems help students improve their ability to read, write, and communicate mathematical ideas.
The CONCEPT problems section in the STUDY SETS reinforces major ideas through exploration and foster independent thinking and the ability to interpret graphs and data.
PRACTICE problems in the STUDY SETS provide the necessary drill for mastery while the APPLICATIONS provide opportunities for students to deal with real-life situations. Each STUDY SET concludes with a REVIEW section that consists of problems randomly selected from previous sections.
SELF CHECK problems, adjacent to most worked examples, reinforce concepts and build confidence. The answer to each Self Check is printed adjacent to the problem to give students instant feedback.
The KEY CONCEPT section is a one-page review found at the end of each chapter that reinforces important concepts.
REAL-LIFE APPLICATIONS are presented from a number of disciplines, including science, business, economics, manufacturing, entertainment, history, art, music, and mathematics.
ACCENT ON TECHNOLOGY sections introduce keystrokes and show how scientific calculators can be used to solve application problems, for instructors who wish to integrate calculators into their course.
CUMULATIVE REVIEW EXERCISES at the end of Chapters 2, 4, 6, 8 and 10 help students retain what they have learned in prior chapters.
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GED Math Workbook
(back cover) A solid review of math basics emphasizes topics that appear most frequently on the GED: number operations and number sense; ...Show synopsis(back cover) A solid review of math basics emphasizes topics that appear most frequently on the GED: number operations and number sense; measurement; geometry; algebra, functions and patterns; data analysis; statistics and probability Hundreds of exercises with answers A diagnostic test and four practice tests with answers Questions reflect math questions on the actual GED in format and degree of difficultyHide synopsis
Description:New. This self-teaching workbook offers extensive preparation...New. This self-teaching workbook offers extensive preparation and brush-up in math for all who plan to take the GED High School Equivalency Test. A diagnostic test with answers is presented to help test takers assess their strengths and weaknesses. The ma
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Abstract: Historically,
the calculus has displaced many algebraic methods for solving classical
problems. This article illustrates how classroom use of a Computer Algebra
System can enable teachers to connect these historical methods to the
precalculus curriculum.
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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WolframAlpha
Description
Find out how much vitamin C is in a bowl of ice cream. Learn what European country has the fourth largest population of children. Compute solutions to difficult trig and calculus problems. Balance complex chemical equations. Discover what is overhead as you gaze up at the stars. Finally crack that crossword puzzle.
Whoever you are and whatever you do, Wolfram|Alpha delivers insight and understanding into any facet of your life.
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Synopses & Reviews
Please note that used books may not include additional media (study guides, CDs, DVDs, solutions manuals, etc.) as described in the publisher comments.
Publisher Comments:
The Tobey/Slater series builds essential skills one at a time by breaking the mathematics down into manageable pieces. This practical "building block" organization makes it easy for readers to understand each topic and gain confidence as they move through each section. The authors provide a "How am I Doing?" guide to give readers constant reinforcement and to ensure that they understand each concept before moving on to the next. With Tobey/Slater, readers have a tutor and study companion with them every step of the way. Whole Numbers, Fractions, Decimals, Ratio and Proportion, Percent, Measurement, Geometry, Statistics, Signed Numbers, Introduction to Algebra, Consumer Finance Applications, Tables, Scientific Calculators. For all readers interested in basic college mathematics.
About has served as the mathematics in 2002. Jeff travels around the country speaking on student retention and is also a consultant to the Federal Government. He lives in Marblehead, Mass.
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Math Without Borders by David Chandler
Math Without Borders provides Home Study Companions to supplement some of the best high school math textbooks. These Home Study Companions consist of video lessons and other materials that aid a homeschooled or any student to study the course with the textbook.
As of this writing, the company provides the following Home Study Companions:
Since regular mathematics textbooks are not designed to be used alone, these Home Study Companions "teach" the material in the video lessons, and show how to solve lots of example problems. Chandler is an experienced teacher, and models the thinking skills and strategies that go into analysis and problem solving. He illustrates, demonstrates, and clarifies the underlying concepts, and also points out links between different parts of the subject and applications beyond what is in the text.
Website: Math Without Borders. Prices: $69 for each CD (each course). You will also need to acquire the textbook (can buy used at Amazon or elsewhere), and preferably Geometer's Sketchpad program for the geometry course.
Reviews of Math Without Borders Home Study Companion products
Special review by Maria Miller, author & owner of HomeschoolMath.net
David Chandler has put together A Home Study Companion - Algebra 1, which consists of whiteboard video lessons to accompany every lesson in Foerster's Algebra 1 book. You can fast forward or rewind these lessons at will, to find the exact spot you're looking for.
In these lessons, he usually presents the concept at hand, just like a normal teacher would do in class. He then goes through and explains in detail several examples from the corresponding lesson in Foerster's book.
In essence, you get to listen to an excellent, experienced math teacher explain the complete Algebra 1 class: the concepts and solved examples
Home Study Companion - Geometry (HSC) by David Chandler really makes this book a home run for homeschoolers, because it provides complete, worked out solutions (not just answers) to all problems in the Central (the main worktext) and Project sections of the Geometry: A Guided Inquiry textbook. The book itself contains answers to its various review and self-test sections. With the Home Study Companion, you will always have help available should you get stuck while doing the problems from the worktext or the "projects".
Not only that, but the Home Study Companion includes a collection of nearly 300 interactive demonstrations using The Geometer's Sketchpad. These demonstrations cover most of the main concepts and many additional explorations of the Central and Projects sections of each chapter.
How do these demonstrations work? For example, let's say you're asked to prove that the diagonals of a parallelogram bisect each other. A demonstration of that would have a parallelogram and its diagonals, and then measurements of the two parts of each diagonal (in centimeters or inches). You would then change the shape of the parallelogram in Sketchpad and see those measurements stay equal to each other. That is not a proof, but it is an interactive demo that helps you understand the matter.
Here are some example screenshots. Click to enlarge. Remember the screenshots are static; in reality the demonstrations are dynamic..
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0721665969A thorough review of general mathematics and its applications in radiography as well as a handy reference for basic formulas and calculations used by radiographers. This book starts with basic mathematic principles and moves on to present the reader with the calculations required to produce high quality radiographic images and practice problems which aid in the understanding of these calculations. Areas discussed include mAs conversions, inverse square law, radiographic density and contrast, screens and grid ratios, grid conversions, geometric and magnification unsharpness, and graphs and conversions between conventional and systems international (SI) units of radiation dosimetry. No other book provides such clear instruction for radiographic calculations. A handy Instructor's Manual is also
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Numerical Analysis
Numerical Analysis
This is a free online course offered by the Saylor Foundation.'...
More
This is a free online course offered by the Saylor Foundation.
' approximations of the real numbers. For example, consider marking a ruler at \sqrt{2}. We know that \sqrt{2} \approx 1.4142, but if we put the mark there, we know we are in error for there is an infinite sequence of nonzero digits following the 2. Even more: a number doesn't have any width, yet any mark we make would have a width, and in that width lives an infinite number of real numbers. You may ask yourself: isn't it sufficient to represent \sqrt{2} with 1.414? This is the kind of question that this course will explore. We have been trying to answer such questions for over 2,000 years (it is said that people have given their lives for the idea of \sqrt{2}, and they certainly wouldn't think 1.414 sufficient). Modern computers can perform billions of arithmetic operations per second and trying to predict the path of a tropical storm can require many trillions of operations. How do we carry out such simulations and how do our approximations affect the result? The answer to the first question is certainly colored by the second!
Numerical analysis is a broad and growing discipline with many open questions. This course is designed to be a first look at the discipline. Over the course of this semester, we will survey some of the basic problems and methods needed to simulate the solutions of ordinary differential equations. We will build the methods ourselves, starting with computer arithmetic, so that you will understand all of the pieces and how they fit together in state of the art algorithms. Along the way, we will write programs to solve equations, plot curves, integrate functions, and solve initial value problems. At the end of some chapters we will suggest – in a section called "Of Things Not Covered" – some topics that would have been included if we had more time or other avenues to explore if you are interested in the topics presented in the unit.'
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You may already know that Maple T.A. is a web-based testing and assessment tool designed for courses involving mathematics, but did you know that Maple T.A. has the ability to assess courses outside of mathematics as well? No matter if you're teaching biology, geography, English, or any other course not involving math, Maple T.A. can support your testing and assessment requirements for assignments, homework, drill and mastery.
With more than twice as many sections (making for complete coverage of single-variable calculus) and more than 460 fully worked-out examples, this new version of the Calculus Study Guide is a modern Maple-based interactive ebook. Join this webinar for a look at this Guide.
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...In introductory linear algebra, this is done using the technique of elimination which works for systems of two or three equations. However, when more than three equations need to be solved simultaneously, one needs to begin using arrays and matrices. Thus, central to linear algebra is the study of matrices and how to perform basic operation such as matrix multiplication.
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{"currencyCode":"USD","itemData":[{"priceBreaksMAP":null,"buyingPrice":184.6,"ASIN":"0470432055","isPreorder":0},{"priceBreaksMAP":null,"buyingPrice":56.33,"ASIN":"0470458224","isPreorder":0}],"shippingId":"0470432055::zvcPH78TPcYh6KdrxNsLvt0QL35jB4k%2BMTYfry8mQbXZSBE5jZgrwkfr1KkJ3yKW4lIllJn%2FCwoBWa%2BXrIy4pkSxFHVyI3eQW1qPXCeEeS7xM6%2FJWgYWjw%3D%3D,0470458224::%2BWiNusurfQwi7McIXz5WYkIHWRAa0hgz45co%2B65%2FfYiedjz%2BtYEGixYC%2Flu7DeT6CcSgwKjzXai0RGl05a8IyoX8uVUYApD5atcsLkWhL5fgdUWu2nr5 book is fairly understandable, more so than most new math textbooks. Although it's the Applications Version, there are very few applications. There are many mistakes of the typo variety.
This doesn't sound like a good review but I would still recommend it for its purpose. Having seen so many horrible Algebra, Calculus and Differential Equations textbooks in the last few years, this is an improvement, it could have been much worse. They do need to proofread the text much more thoroughly for the next edition and if they're going to call it the Applications Version, they need to put some more applications in it.
clear as mud, expensive as hell, the examples don't help you solve half the problems, most of the key proofs are missing, while other large chunks or print space is wasted on useless history lessons about mathematicians.
but... you'll buy it anyway... because you need to, as a student you don't have a choice. if you're a professor considering this book, I'd consider something better, and cheaper.
As a student though, I didn't learn one thing from this book or my instructor. This book presented the problems and the need to know something, Google, wikipedia, and youtube provided the education.
Ironic how the people who get my money aren't the ones that give me the knowledge.
It came in a timely fashion, despite me purchasing it after my class had already started...Brand-spanking new condition and the packaging was easy to take apart and fit perfectly around the book itself. AWESOME.
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Basic College Mathematics - With CD - 8th edition
Summary: The-a book written with student success as its top priority, now with an emphasis on study skills growth and an expanded instructor supplements package4.62 +$3.99 s/h
Acceptable
SellBackYourBook Aurora, IL
032155712300 +$3.99 s/h
Acceptable
Seattle Goodwill WA Seattle, WA
2009 Paperback Fair Edge discolored.
$12.50 +$3.99 s/h
Good
kelseyskorner Ferndale, WA
Solid binding. Good clean pages-some in front have a slight wrinkle. Solid clean cover with some moderate wear.Disc is included. Good softcover book.
$14.92 +$3.99 s/h
Acceptable
Better World Books Mishawaka, IN
Former Library book. Binding is slightly damaged and/or book has some loose pages. No missing pages. With CD! 100% Money Back Guarantee. Shipped to over one million happy customers. Your purchase bene...show morefits world literacy! ...show less
$14
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books.google.com - Preconditioning techniques have emerged as an essential part of successful and efficient iterative solutions of matrices. Ke Chen's book offers a comprehensive introduction to these methods. A vast range of explicit and implicit sparse preconditioners are covered, including the conjugate gradient, multi-level... Preconditioning Techniques and Applications
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Algebra 2 with Trigonometry, Version 2.0, provides
teachers with 108 lessons and accompanying homework sets. The text covers the performance indicators for the new New
York State Integrated Algebra 2 with Trigonometry Regents Exam. Lessons explore important concepts and develop critical
and long-lasting skills. Homework reinforces both essential knowlege and important techniques. The course is based
on the concept of families of functions. Graphical, tabular, and algebraic approaches are stressed throughout the course.
Algebra 2 with Trigonometry is given away freely
on this website in PDF form. Each lesson, with its correlated homework, can be accessed individually with any computer
that has a PDF reader. We encourage users to sample the lessons, use them with their classes, and then give us at eMathInstruction
feedback to make the lessons better.
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Analytic Trigonometry with Applications
Featuring rich applications and integrated coverage of graphing utilities, this hands-on trigonometry text guides students step by step, from the ...Show synopsisFeaturing rich applications and integrated coverage of graphing utilities, this hands-on trigonometry text guides students 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 students to be actively involved in the learning process. Key pedagogical elements, such as annotated examples, think boxes, caution warnings, and reviews help students comprehend and retain the material
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Mathematics 571 Introduction to Probability Models, Spring 2012
Calculus and differential equations are used to construct models where there
is no uncertainty. In this course, we will incorporate uncertainty into models
of systems. For example, we can consider the decay of a radioactive substance
as occurring because at random times, individual atoms decay. Instead of
asking for the mass of the substance at a particular time, we can ask for the
probability distribution of the mass at that time. Other examples include
the value of the Standard and Poors 500 stock index, the number of people in
line at a grocery check-out, the number of descendents of a in a branch of a
family tree in a given generation, and the distribution of gas particles in
a box at a given time.
The primary text book for the course is Introduction to Probability
Models, Tenth Edition by Sheldon Ross. We will assume that participants
have a working knowledge of the probability and statistics at the level of the
first three chapters of this text. We will begin with an overview of stochastic processes
before beginning with
the fourth chapter of the text, Markov Chains. If need be, we can always return to these
earlier chapters to fill in any gaps in background. The aim of the text and of
the course is very applied. Our goal is to show how simple probability tools
can be used to describe a large number of systems with two common
characteristics: the systems evolve with time, and uncertainty is a primary
consideration.
Some of the things we will investigate are
Compound Poisson Processes and Insurance Claims.
The Gambler's ruin problem.
Random walks
Branching processes.
Queuing systems.
Pricing stock options.
To be prepared for this course you must have some background in elementary
probability, calculus, linear differential equations, and matrix algebra.
The ability to write simple computer programs or to use MAPLE will be helpful,
but is not essential.
If you have any doubts about your preparation, or any other questions,
contact me .
There will be (mostly) weekly homework which will account for 50% of your grade, a midterm exam that is 20% of your grade (probably right before spring break) and a final exam (see the final exam
schedule for date and time) which will account for 30% of your grade. This is a U/G course, so graduate students will be expected to solve additional and/or more challenging problems.
I will email you your reading assignments, additional lecture notes and homework assignments as the semester goes along.
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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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Prerequisite: Placement through the assessment process or MATH 075 or MATH 075SP or equivalent
Note: MATH-090-1030 is linked to PHYSC-112-1032 as one integrated learning community. Students MUST enroll in person in the Student Services Building, Admissions & Records. Online enrollment is not available. For more information, contact Bruce Koller at (925) 969-2471 or bkoller@dvc.edu.
Course
Section
Days
Time
Room
Units
Start/Stop
Dates
MATH-090
1030
MW
9:30-10:20am
MA-250
5 Units
Full Term
TTH
9:30-10:45am
Catalog Course Description
MATH-090 Elementary Algebra
5 - 5 Units
SC
Not Degree Applicable
Variable Hours
Prerequisite: Placement through the assessment process or MATH-075 or MATH-075SP or equivalent
Note: Formerly MATH-110
Formerly MATH-110
This course is an introduction to the techniques and reasoning of algebra, including linear equations and inequalities, development and use of formulas, algebraic expressions, systems of equations, graphs and introduction to quadratic equations.
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Beginning and Intermediate Algebra - 3rd edition
ISBN13:978-0077350086 ISBN10: 0077350081 This edition has also been released as: ISBN13: 978-0073384214 ISBN10: 0073384216
Summary: Miller/O'Neill/Hyde continues to offer an enlightened approach grounded in the fundamentals of classroom experience in Beginning and Intermediate 2 reflecti...show moreve of the comments and instruction presented to students in the classroom. In this way, the text communicates to students, the very points their instructors are likely to make during lecture, helping to reinforce the concepts and provide instruction that leads students to mastery and success. The authors included in this edition, Problem-Recognition Exercises, that many instructors will likely identify to be similar to worksheets they have personally developed for distribution to students. The intent of the Problem-Recognition exercises, is to help students overcome what is sometimes a natural inclination toward applying problem-sovling algorithms that may not always be appropriate. In addition, the exercise sets have been revised to include even more core exercises than were present in the first, as they do inside class with their instructor. For even more support, students have access to a wealth of supplements, including McGraw-Hill's online homework management system, MathZone008182 +$3.99 s/h
Acceptable
firstclassbooks.com Little Rock, AR
Edition: 3, Hardcover, Fast shipping! Access codes and CDs are not guaranteed with used books!
$54
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Day 5: Rank and determinants
Day 6: Singular and non-singular matrices
To solve Ax = b, one might hope to multiply both sides of this equality by the inverse of A. Accordingly, we defined matrix inverses, when to expect them to exist, and how to compute them (if they exist).
Day 7: MAPLE
Day 8: Eigenvalues and Eigenvectors
We introduced the notion of eigenvectors and eigenvalues. For 2x2 matrices A with distinct eigenvalues l and m with associated eigenvectors v and w (respectively), we noticed the following nice geometric interpretation of the situation: for an arbitrary vector x, Ax is the vector x stretched in the v-ward direction by l and stretched in th w-ward direction by m. So knowing the eigenvalues/eigenvectors of a matrix generally puts you in a better position to anticipate the value of Ax.
Day 22: Non-homogeneous Systems of Linear DE's
Day 26: Non-Linear Autonomous Systems
We introduced non-linear autonomous systems of differential equations and began to describe their phase portraits and equilibrium points. You may find the the following Java applet useful for plotting phase portraits of autonomous systems in addition to MAPLE's capabilities.
Day 27: Stability of Nonlinear Autonomous Systems
We defined the notions of stable, asymptotically stable, and unstable equilibrium solutions to nonlinear autonomous systems. We also had our fifth quiz.
Day 29: Classifying Stability in Linear Autonomous Systems
In a manner analogous to the approximation of linear functions to differential functions, we described how linear systems approximate non-linear autonomous systems (with differentiable components). This approximation allows us to use what we know about stability of linear systems to describe stability in non-linear autonomous systems. Accordingly, we completely characterized the stability of linear systems from the trace and determinant of the associated matrix.
Day 30: Classifying Stability in Nonlinear Autonomous Systems
We discussed what information can be gathered about the stability of equilibrium solutions to non-linear autonomous systems from their linearizations. We also discussed other methods for analysing the stability of non-linear systems, namely, the so-called phase-plane method.
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Enter a set of data points and a function or multiple functions, then manipulate those functions to fit those points. Manipulate the function on a coordinate plane using slider bars. Learn how each co... More: lessons, discussions, ratings, reviews,...
This game explores functions in a different way: a and b = f(a) are drawn in a unique numerical line. When the user changes a, b = f(a) changes following a rule. The objective of the game is to discov... More: lessons, discussions, ratings, reviews,...
Highlight the language of domain and range, and the ideas of continuity and discontinuity, with this tool that links symbolic and graphic representations of each interval of a piecewise linear functio... More: lessons, discussions, ratings, reviews,...
Simplesim is suited for modelling of non-analytic relations in systems which are causal in the sense that different courses of events interact in a way that is difficult to see and understand. Exam... More: lessons, discussions, ratings, reviews,...
Plomplex is a complex function plotter using domain coloring. You can compose a function with a complex variable z, and generate a domain coloring plot of it. You can choose the plot range as well as ... More: lessons, discussions, ratings, reviews,...
What's the reliability of cancer tests, diabetes tests, and pregnancy tests? This brief discussion shows some functions to be used on your graphing calculator to visualize a graph of the accuracy of a... More: lessons, discussions, ratings, reviews,...
This physics-exploration applet allows the user to experiment with different roller coaster track designs, then test. Friction and mass are modeled. Includes hot links to explanations of various phys... More: lessons, discussions, ratings, reviews,...
Windows software which allows the display of 2D and 3D diagrams both on one, and on different screens. Display 2D diagrams in the Cartesian and polar systems of coordinates. Display 3D diagrams in t... More: lessons, discussions, ratings, reviews,...
Commercial site with one free access a day. Students use mapping diagram to create a relation, then they can check if it is a function from the mapping diagram, ordered pairs and graph. After studen
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Elementary Statistics-With CD and Formula Card - 7th edition
Summary: ELEMENTARY STATISTICS: A STEP BY STEP APPROACH is for general beginning 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. This edition places more emphasis on conceptual understanding and understanding results. This edition also features increased emphasis on Excel, MINITAB, and the TI-83 Plus and TI-84 Plus graphing cal...show moreculators; computing technologies commonly used in such courses
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Geometry
Geometry introduces students to mathematics as a deductive science. It shows how a few simple assumptions about points, lines, etc., can lead to a substantial body of geometric theorems and the ability to solve problems involving the properties of various two- and three-dimensional geometric figures. For each chapter in the textbook, students start with a preview of the concepts covered and a self-evaluation of the prerequisite skills needed. The students engage study material both in their textbook and online, including games, other activities and animations. Checkpoints are provided each step of the way for students to practice their skills and problem solving with additional practice assigned in the event the student encounters difficulties.
In completing this subject students construct various types of proofs, including two-column proofs; apply postulate and theorems and explain their reasoning in solving geometric problems; and solve various types of problems using geometric and algebraic concepts and principles. Completion of Algebra 1 is a prerequisite for taking Geometry, and access to a hand-held or computer-based scientific or graphing calculator is required.
There are 12 examinations in Geometry.
Larson, Geometry, Holt McDougal.
What others are saying:
The American School is a great program for students who aren't caught up on credits. The program gives students a chance to learn on their own and take in what they read. It puts kids in a place where they have to learn how to plan things and stay on track. — Taylor, Michigan
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Promoting Understanding of Linear Equations with the Median Slope Algorithm
Michael Edwards
February 2005, Volume 98, Issue 6, Page 414
Abstract: An author-invented algorithm, the median slope, to help students understandthe basics of linear equations before and without using the graphing calculator. Unlike calculator-based linear regression techniques, the steps that underlie the median slope algorithm are entirely accessible to first-year algebra students. Step by step examples are given as well as directions for programming the median slope algorithm into the graphing calculator.
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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vision, the geometric laws that relate different views of a scene. Geometry, one of the oldest branches of ... multipleviews of a scene from the perspective of various types of geometries. A key feature is that it ... role incomputer communications. Producers and users of images, in particular three-dimensional images, ...
filtering. The detection and use of interest points incomputervision is presented with applications for ... Exploit the image geometryin order to match different views of a pictured scene Calibrate the camera from ... programming. It can be used as a companion book in university-level computervision courses. It constitutes an ...
an in-depth analysis of the state-of-the-art research inComputerVision. The courses are delivered ... 3 Dynamic Graph Cuts and their Applications inComputerVision. Part 4 Discriminative Graphical Models for ... Technical Computervision is the science and technology of making machines that see. It is ...
a more proper tool for studying the geometry of multipleviews is the so-called rank consideration of the ... | 42 MB This book introduces the geometry of 3-D vision, that is, the reconstruction of 3-D models of ... objects from a collection of 2-D images. It details the classic theory of two viewgeometry and shows that ...
Study ComputerVision for Visual Effects by Richard J. Radke English | ISBN: 0521766877 | 2013 ... and television. The author describes classical computervision algorithms used on a regular basis in ... from the field of computervision, the study of how to automatically understand images. ComputerVision ...
Multiplicity diagrams can be viewed as schemes for describing the phenomenon of "symmetry breaking" ... in quantum physics. The subject of this book is the multiplicity diagrams associated with the ... , hierarchical patterns inmultiplicity diagrams, lacunae, and the multiplicity diagrams of the rank 2 and rank ...
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Basic Training in Mathematics A Fitness Program for Science Students
9780306450365
ISBN:
0306450364
Pub Date: 1995 Publisher: Kluwer Academic Print on Demand
Summary: Based on course material used by the author at Yale University, this practical text addresses the widening gap found between the mathematics required for upper-level courses in the physical sciences and the knowledge of incoming students. This superb book offers students an excellent opportunity to strengthen their mathematical skills by solving various problems in differential calculus. By covering material in its s...implest form, students can look forward to a smooth entry into any course in the physical sciences.
Shankar, Ramamurti is the author of Basic Training in Mathematics A Fitness Program for Science Students, published 1995 under ISBN 9780306450365 and 0306450364. Eight hundred eighteen Basic Training in Mathematics A Fitness Program for Science Students textbooks are available for sale on ValoreBooks.com, one hundred eleven used from the cheapest price of $45.31, or buy new starting at $64.03.[read more [more the author's course at Yale University, this book teaches all the basic mathematical prerequisites for upper-level science study. "Am [more]
This item is printed on demand. Based on the author's course at Yale University, this book teaches all the basic mathematical prerequisites for upper-level science study. "American Journal of Physics" calls it, "substantively solid...written in a par.[less]
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Precalculus: Concepts Through Functions, a Right Triangle Approach to Trigonometry
NOTE: This edition features the exact same content as the traditional text in a convenient, three-hole-punched, loose-leaf version. Books a la Carte ...Show synopsisNOTE Precalculus: Concepts Through Functions, A Right Triangle Approach to Trigonometry, Third Edition focuses on the fundamentals: "preparation" for class, "practice with" homework, and "reviewing "of key concepts. With the Concepts Through Functions series, the Sullivans expose students to functions in the first chapter and maintain a continuous theme of functions throughout the text. This approach ensures students master basic skills and develop the conceptual understanding they need for the course, ultimately preparing students for future math courses as well. This Package includes: 0321925998/ 9780321925992 Precalculus: Concepts Through Functions, A Right Triangle Approach to Trigonometry, Books a la Carte Edition, 3/eHide30095Fine. Hardcover. Instructor Edition: Same as student edition...Fine. Hardcover. Instructor Edition: Same as student edition with additional notes or answers. Almost new condition. SKU: 9780321930095Hardcover. Instructor Edition: Same as student edition with...Hardcover. Instructor Edition: Same as student edition with additional notes or answers. New Condition. SKU: 9780321930095
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Synopses & Reviews
Publisher Comments:
This book introduces the concepts of linear algebra through the careful study of two and three-dimensional Euclidean geometry. This approach makes it possible to start with vectors, linear transformations, and matrices in the context of familiar plane geometry and to move directly to topics such as dot products, determinants, eigenvalues, and quadratic forms. The later chapters deal with n-dimensional Euclidean space and other finite-dimensional vector space.
Book News Annotation:
Revised and expanded undergraduate textbook introduces the concepts of linear algebra through the careful study of two- and three-dimensional Euclidean geometry. Except for the applications in the final chapter, the only prerequisites are high-school geometry, algebra, and introductory trigonometry
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The book's innovative story-line style models the problem-solving process, presents the development of concepts in a natural way, and engages students in meaningful encounters with the material. The updated end-of-chapter investigations provide opportunities to work on many open-ended, non-routine problems and, through a modified "Moore method," to make conjectures from which theorems emerge. The revised end-of-chapter notes provide historical background to the chapter's ideas, introduce standard terminology, and make connections with mainstream mathematics. The final chapter of projects provides ideas for continued research.
Explorations in Topology, Second Edition, enhances upper division courses and is a valuable reference for all levels of students and researchers working in topology.
Students begin to solve substantial problems from the start
Ideas unfold through the context of a storyline, and students become actively involved
The text models the problem-solving process, presents the development of concepts in a natural way, and helps the reader engage with the material
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GeoGebra is a free and multi-platform dynamic mathematics software for education in secondary schools that joins geometry,...
see more
GeoGebra is a free and multi-platform dynamic mathematics software for education in secondary schools that joins geometry, algebra and calculus. On the one hand, GeoGebra is a dynamic geometry system--you can do constructions with points, vectors, segments, lines, conic sections as well as functions and change them dynamically afterwards; on the other hand, equations and coordinates can be entered directly. Thus, GeoGebra has the ability to deal with variables for numbers, vectors and points, 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.
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More About
This Textbook
Overview
The Encyclopedia of Integer Sequences contains more than 5000 integer sequences, arranged for easy reference, with more than half never before catalogued. In addition to having more than double the material of Sloane's A Handbook of Integer Sequences (Academic Press, 1973), this encyclopedia gives the name, mathematical description, and citations to the literature for each sequence. It includes essays on origins, uses, and connections, with interesting diagrams or illustrations. More than a table, this is an introduction to the field, showing readers how to identify and work with sequences. It includes an extensive bibliography of current and classic references. An index to all the sequences in the book is also available separately in Macintosh and IBM formats.
Audience: Those professionals and researchers in all areas of applied and theoretical science and engineering who need to identify and work with integer sequences, including amateurs and recreational mathematics enthusiasts, and scienceand mathematics libraries. Of interest to those working in: number theory, combinatronics, graph theory, discrete mathematics, computer science, algebra, geometry, communications, information theory, physics, chemistry, statistics, botany, and zoology.
Related Subjects
Table of Contents
Description of the Book: Description of a Typical Entry. Arrangement of a Table. Number of Terms Given. References. What Sequences are Included? The Figures. How to Handle a Strange Sequence: How to See if a Sequence Is in the Table. If the Sequence is Not in the Table. Finding the Next Term. Recurrences and Generating Functions. Analysis of Differences. Other Methods for Hand Analysis. Transformations of Sequences. Methods for Computer Investigation of Sequences. The On-line Versions of the Encyclopedia. The Floppy Disk. Further Topics: Applications. History. Differences from the 1973 Book. Future Plans. Acknowledgments. The Table of Sequences. Bibliography
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third edition has a new open format and expanded graphics which enforce concepts. Through an informal, active writing style and the use of numerous pedagogical aids, such as examples with matched problems, annotations, think boxes, chapter summaries and cumulative reviews, the text guides students of varied disciplines to success in college algebra and trigonometry. In response to the increasing popularity of graphing calculators in mathematics, numerous exercises that exploit the capabilities of these calculators have been incorporated into this edition. There is a new chapter on probability; other new features include illustrative examples, new and varied problems, and new and updated applications.
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Graphs: An Introductory Approach - A First Course in Discrete Mathematics
An introduction to discrete mathematics, this new text on graph theory develops a mathematical framework to interrelate and solve different problems. ...Show synopsisAn introduction to discrete mathematics, this new text on graph theory develops a mathematical framework to interrelate and solve different problems. It introduces the concepts of logic, proof and mathematical problem-solving and places an emphasis on algorithms in every chapter is in good to acceptable condition with minor...Fair. Book is in good to acceptable condition with minor blemishes on the cover. Pages have highlighting and minor writing. Ships next business day Graphs: An Introductory Approach--A First Course in...Good. Graphs: An Introductory Approach--A First Course in Discrete Mathematics
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Maths Question Paper for Class 11 CBSE
Knowing Math is important as this subject enables students to choose their preferred job. Poor knowledge in Math restricts students to explore in many fields. To give importance to this subject, the CBSE board has designed each Math syllabus carefully. Under the guidance of several subject experts, CBSE board has prepared Math syllabus for class 11 and also make this available online. Additionally, the board has prepared suitable question papers for Math that evaluate students' expertise at the end of every academic session. Maths question paper for Class 11 CBSE represents the praiseworthy educational pattern of the concerned board. Moreover, CBSE question papers are available year-wise. Hence, students can collect and take adequate help from these question papers at their convenient time.
Maths Question Papers for Class 11 CBSE 2013
Math is a basic subject and it is included in each CBSE syllabus in a requisite manner. Each CBSE syllabus is designed under the strict vigilance of several subject experts associated with the concerned board and hence, CBSE syllabus is specified as a worthwhile study material for students. Moreover, the board also prepares question papers for each academic session and these question papers assess students' expertise in a through manner. The board follows CCE pattern and hence, students' learning problems are detected and consequently, they get suitable assistance to improve their performance in exams. Maths question papers for class 11 CBSE 2013 are designed by following all CBSE guidelines. Therefore, students' expertise is thoroughly assessed by these papers.
CBSE Maths Question Paper for Class 11 2012
CBSE Maths syllabus is undoubtedly a great learning resource for students. By using this syllabus thoroughly, students can get requisite knowledge in a step-by-step manner. Additionally, to assess students' knowledge properly, the board also prepares suitable question papers for each subject. These question papers are designed for each academic session and the board makes these question papers available online for the convenience of students. CBSE Maths question paper for class 11 2012 is good to follow to get requisite knowledge about the original question paper. Additionally, this question paper follows a standard educational pattern in all respects and therefore, it assesses students' knowledge in a proper manner.
Question Paper Maths for Class 11 CBSE 2011
Question papers are designed to evaluate students' knowledge in a right manner. In brief, question paper acts as a measuring tool that assesses students' understanding in each subject. Specific question paper is prepared for specific subject and most importantly, each CBSE question paper is designed under the guidance of several subject experts. Hence, students' knowledge is evaluated thoroughly. Question papers play a vital role in the examination process and based on the performance in exams; students are recognized at the end of each academic session. Therefore, question papers and students' result are quite interlinked. Question paper Maths for class 11 CBSE 2011 is available online and students can use this as a reference.
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Homework Helpers Pre-calculus
9781564149404
1564149404
Summary: The book pays particular attention to strategies for understanding and constructing logical arguments and proofs. Common patterns that reappear in many proofs are highlighted. In addition, geometry problems that review arithmetic and algebraic skills are presented and fully explained. Homework Helpers: Pre-Calculus is a straightforward, understandable, and thorough review of the topics in a typical pre-calculus class..., including: - Linear functions - Polynomials - Rational functions - Exponential functions - Logarithmic functions - Systems of equations
Szecsei, Denise is the author of Homework Helpers Pre-calculus, published 2007 under ISBN 9781564149404 and 1564149404. Two hundred twenty six Homework Helpers Pre-calculus textbooks are available for sale on ValoreBooks.com, one hundred sixteen used from the cheapest price of $0.01, or buy new starting at $10
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Summary: This edition features the exact same content as the traditional text in a convenient, three-hole- punched, loose-leaf version. Books a la Carte also offer a great value-this format costs significantly less than a new textbook.
The first book of theMathematics in Actionseries,Prealgebra Problem Solving,Fourth Editionillustrates how mathematics arises naturally from everyday situations through updated and revised real-life activities and accompa...show morenying practice exercises.
This unique approach helps students increase their knowledge of mathematics, sharpen their problem-solving skills, and raise their overall confidence in their ability to learn. Technology integrated throughout the text helps students interpret real-life data algebraically, numerically, symbolically, and graphically. The active style of this book develops students' mathematical literacy and builds a solid foundation for future study in mathematics and other disciplines. ...show less
12-27-10 other 3$167.62
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Vector Calculus - 4th edition
Summary: Intended for one-semester courses in the calculus of functions of several variables and vector analysis, Vector Calculus is widely used at the sophomore and junior level. Acclaimed authors Jerrold Marsden and Anthony Tromba help students foster computational skills and intuitive understanding with a careful balance of theory, applications, optional materials, and historical notes.
The Geometry of Real-Valued Functions Limits and Continuity Differentiation Introduction to Paths Properties of the Derivative Gradients and Directional Derivatives Some Technical Differentiation Theorems
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including precalculus
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This is an essentially self-contained book on the theory of convex functions and convex optimization in Banach spaces, with a special interest in Orlicz spaces. Approximate algorithms based on the stability principles and the solution of the corresponding nonlinear equationsaredeveloped in this text.A synopsis of the geometry of Banach spaces, aspects... more...
This is the perfect introduction for those who have a lingering fear of maths. If you think that maths is difficult, confusing, dull or just plain scary, then The Maths Handbook is your ideal companion.Covering all the basics including fractions, equations, primes, squares and square roots, geometry and fractals, Dr Richard Elwes will lead you gently... more...
From the author of the highly successful The Complete Idiot's Guide to Calculus comes the perfect book for high school and college students. Following a standard algebra curriculum, it will teach students the basics so that they can make sense of their textbooks and get through algebra class with flying colors.Tips for simplifying tricky operations Get the skills you need to solve problems and equations and be ready for algebra class,... more...
Master calculus from the comfort of home!
Want to "know it ALL" when it comes to calculus? This book gives you the expert, one-on-one instruction you need, whether you're new to calculus or you're looking to ramp up your skills. Providing easy-to-understand concepts and thoroughly explained exercises, math whiz Stan Gibilisco serves as your own... more...
We want to help you succeed on the GMAT math section If math is the hardest part of the GMAT for you, we're here to help. McGraw-Hill's Conquering GMAT Math is packed with strategies for answering every kind of GMAT math question. You'll also get intensive practice with every question type to help you build your test-taking confidence. With... more...
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Synopses & Reviews
Publisher Comments:
Written for junior and senior undergraduates, this remarkably clear and accessible treatment covers set theory, the real number system, metric spaces, continuous functions, Riemann integration, multiple integrals, and more. Rigorous and carefully presented, the text assumes a year of calculus and features problems at the end of each chapter. 1968
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The course introduces elements of linear algebra as needed to manipulate and solve linear, constant coefficient systems of first-order ODEs. In the second part of the course, students are introduced to the most commonly used curvilinear coordinate systems and develop skill in setting up and solving double and triple integrals. The final part of the course introduces the theorems and techniques of vector calculus essential to mathematical physics and further study in real analysis.
Learning Outcomes
Upon completing this course, it is expected that a student will be able to do the following:
Mathematical Foundations
Matrices: Carry out basic operations with matrices and vectors and use the determinant to determine the solvability of a linear system of equations.
Linear Algebraic Systems: Solve a system of linear algebraic equations using matrix methods.
Eigenvalues and Eigenvectors: Derive the eigenvalues and eigenvectors of a square matrix.
Systems of ODEs: Solve systems of linear, constant coefficient, first order ODEs using matrix methods.
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ISBN: 0130909548 / ISBN-13: 9780130909541 ...Show more utilize critical thinking skills, analyze and interpret data, and problem solve using applied situations encountered in daily life. The problem solving strategy, highlighted by *A Mathematics Blueprint for Problem Solving, helps students determine where to begin the problem-solving process, as well as how to plan subsequent problem-solving steps. Chapter organizers help students focus their study on the concepts and examples. Developing Your Study Skills boxes throughout the text give students tips to help them improve their study skills.These features, together with the applications and emphasis on problem solving, help students to become effective and confident problem solvers. *Please note: Intermediate Algebra includes a similar feature called Problem Solving Procedure
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Mathematics A Discrete Introduction
9780534398989
ISBN:
0534398987
Edition: 2 Pub Date: 2005 Publisher: Thomson Learning
Summary: With a wealth of learning aids and a clear presentation, this book teaches students not only how to write proofs, but how to think clearly and present cases logically beyond this course. All the material is directly applicable to computer science and engineering, but it is presented from a mathematician's perspective.
Scheinerman, Edward R. is the author of Mathematics A Discrete Introduction, published 2005... under ISBN 9780534398989 and 0534398987. Four hundred forty eight Mathematics A Discrete Introduction textbooks are available for sale on ValoreBooks.com, one hundred twenty three used from the cheapest price of $13.96, or buy new starting at $55
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books.google.com - The two fields of Geometric Modeling and Algebraic Geometry, though closely related, are traditionally represented by two almost disjoint scientific communities. Both fields deal with objects defined by algebraic equations, but the objects are studied in different ways. This contributed book presents,... Modeling and Algebraic Geometry
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There Are Integrated Programs, and There Are Integrated Programs
by Richard Askey January 2001
There are two types of "integrated programs." There are those like the ones in Singapore, which teach some arithmetic or algebra and some geometry each year with connections among them used. This is something that mathematicians have been proposing for a very long time (Mathematical Association of America 1923). Similar programs exist in many other countries. Singapore is mentioned because textbooks in that country are good, are written in English, and are accessible in the United States. (See
The other is illustrated by some of the National Science Foundation (NSF)-funded programs from the NSF call for new programs in the early 1990s. These have a different focus, trying to teach mathematics in the context of real-world problems. It is this second type of integrated program that has become the focus of controversy in the United States.
To explain some of my concerns, consider an example from Contemporary Mathematics in Context (CMIC), written by the Core Plus staff. This example appears on pages 311–13 in Course 1A, near the end of the first semester of what for most students will be ninth grade. The real-world problem is how to get an estimate for the average time when doing a task many times by doing a sample study:
Compute the average time for each task in the feasibility study project using the formula:
Average time =
best time + 4(most likely time) + worst time.
6
Nothing is written in the textbook about where this formula comes from, but the following information is given on page T313 in the Teacher's Guide.
Background information: This is a commonly used way to compute the expected value of the task time. It assumes a specific type of probability distribution (called a beta probability distribution) and is only an estimate. It is, of course, neither necessary nor desirable to pursue this point technically, but it can be used as some quick justification for students who want to know there the "4" came from. This will be followed up a bit in a later probability unit in Course 2.
Ignoring the fact that the "explanation" says nothing at all about where the 4 comes from, another serious problem is illustrated by this "explanation." The beta distribution is the integrand in the beta function integral.
Notice that there is no 4 anywhere. In fact, the beta distribution has nothing to do with this formula. The formula is just the basic building block of Simpson's rule, which allows one to exactly integrate a cubic polynomial.
Where did the idea that the beta distribution had anything to do with this come from? I had no idea, so I posted a question to one of the authors on the Math Forum discussion group math-teach.‡ The answer that I received said, in part, "In management science, they often use the expected value of a beta distribution to estimate the 'average' task time."
The author then gave a source for the following quotation: "[The equation for the average or expected time is] based on the assumption that the uncertainty in activity times can be described by a beta probability distribution...." So one major problem with real-world applications as a basis for school mathematics education is that the developers do not know enough to get things right. Also, if they copy problems from books in many applied areas, little or no mathematical reason will be given for the use of specific parts of mathematics. A similar formula to the one given above is used in the treatment of volumes in CMIC. This is also done very poorly, with essentially no geometry being used. This appears in the second half of the first course. In particular, no geometric reasons are given for the factor of 1/3 in the formula for the volume of a pyramid.
There are other reasons to be concerned about the type of programs similar to this one. The following quotation is from the introduction to "Applications and Misapplications of Cognitive Psychology to Mathematics Education" (Anderson, Reder, and Simon).
There is a frequent misperception that the move from behaviorism to cognitivism implied an abandonment of the possibilities of decomposing knowledge into its elements for purposes of study and decontextualizing these elements for purposes of instruction.
A program that postpones a proof of the Pythagorean theorem until the third year of high school because students do not learn how to expand until then is unacceptable by world standards, and that is what we have to consider. That is the case for Contemporary Mathematics in Context.
Mathematical Association of America. The Reorganization of Mathematics in Secondary Education: A Report by the National Committee on Mathematical Requirements under the Auspices of the Mathematical Association of America, Inc. Mathematical Association of America, 1923. World Wide Web.
Richard Askey has taught at the University of Wisconsin—Madison since 1963. His research is in special functions, which includes "higher trigonometry," that is, functions on higher-dimensional spheres, and extensions of the gamma and beta functions.
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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Discrete Mathematics Demystified - 08 edition
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 Steven Krantz, Discrete Mathematics Demystified explains this challenging topic in an eff...show moreective and enlightening way. You will learn about logic, proofs, functions, matrices, sequences, series, and much more. Concise explanations, real-world examples, and worked equations make it easy to understand the material, and end-of-chapter exercises and a final exam help reinforce learning.
This fast and easy guide offers:
* Numerous figures to illustrate key concepts * Sample problems with worked solutions * Coverage of set theory, graph theory, and number theory * Chapters on cryptography and Boolean algebra * A time-saving approach to performing better on an exam or at work
Simple enough for a beginner, but challenging enough for an advanced student, Discrete Mathematics Demystified is your integral tool for mastering this complex subject11 +$3.99 s/h
New
EuroBooks Horcott Rd, Fairford,
New Book. Shipped from UK within 4 to 14 business days. Established seller since 2000 354 p. Contains: Illustrations, black & white, Tables, black & white, Figures
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Thorough understanding of the theoretical underpinnings of this powerful tool can be left to the math majors. Those who ask for help in a calculus course are most often taking it as a requirement for a technical field. Here, the practical application of derivatives and integrals are what is important.
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Find a Roosevelt, NY MathLinear Algebra is the study of matrices, more generally linear systems, how we can mathematically understand and solve these huge systems (sometimes infinite), and the application of linear systems to solve problems in science and engineering. During my graduate work, I took a year of linear alg
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This introductory textbook puts forth a clear and focused point of view on the differential geometry of curves and surfaces, emphasizing the global aspects. The excellent collection of examples and exercises (with hints) will help students in learning the material. Advanced undergraduates and graduate students will find this a nice entry point to differential geometry. In order to study the global properties of curves and surfaces, it is necessary to have more sophisticated tools than are usually found in textbooks on the topic.
Game Theory 101: The Complete Textbook is a no-nonsense, games-centered introduction to strategic form (matrix) and extensive form (game tree) games. From the first lesson to the last, this textbook introduces games of increasing complexity and then teaches the game theoretical tools necessary to solve them. Quick, efficient, and to the point, Game Theory 101: The Complete Textbook is perfect for introductory game theory, intermediate microeconomics, and political science.
An Engineer's Guide to Mathematica enables the reader to attain the skills to create Mathematica 9 programs that solve a wide range of engineering problems and that display the results with annotated graphics. This book can be used to learn Mathematica, as a companion to engineering texts, and also as a reference for obtaining numerical and symbolic solutions to a wide range of engineering topics. The material is presented in an engineering context and the creation of interactive graphics is emphasized. The first part of the book introduces Mathematica's syntax and commands useful in solving engineering problems.
A critical overview of the current debate and topical thinking on international comparative investigations in mathematics education. The contributors are all major figures in international comparisons in mathematics. The book highlights strengths and weaknesses in various systems worldwide, allowing teachers, researchers and academics to compare and contrast different approaches. A significant contribution to the international debate on standards in mathematics.
This book collects the work of thirty-five instructors who share their innovations and insights about teaching discrete mathematics. Whether you teach at the college or high school level; whether your students are from mathematics, computer science, or engineering; whether you emphasize logic, proof, counting, graph theory, or applications, you will find resources in this book to supplement your discrete mathematics course.
Science does not exist in a vacuum and, therefore, shouldn't be taught that way. In that spirit, Activities Linking Science With Math, 5-8, is a hands-on guide for preservice and inservice elementary and middle school teachers who want to connect science instruction with other areas of studyincluding visual arts, social sciences, language arts, and especially math.
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Algebra 1 is the foundation to all future math courses and for some it may be necessary for your career of choice. It is important that a student masters success in the areas of problem solving, graphing and solving linear and quadratic functions, exponents, and solving systems of equations. Mo...
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Applied Mathematics with Elementary Education
This major is designed for students who plan to teach mathematics at the elementary school level (grades 1-6). Graduates develop effective instructional strategies and content knowledge in elementary curriculum while specializing in mathematics seminar explores representative fiction, poetry, or drama by major figures in world literature, centering on a theme such as love, tragedy, comedy, immortality, madness, wasteland, quest for knowledge, voyages, or exploration. This is a presentation-intensive course. Prerequisite: Any 200 level English course cultural previousMATH103 -
Introductory Algebra
This is an introduction to algebra beginning with a brief review of operations with real numbers. Topics include: algebraic expressions, solving linear equations and inequalities, rules for exponents, operations with polynomials, factoring techniques, quadratic equations, graphing linear equations, solving systems of equations. Restrictions: not open to students who have completed another mathematics course.
MATH104 -
Intermediate AlgebraMATH106X -
Algebraic Operations
This course is intended to strengthen students' ability in algebra. The course begins with such introductory topics as operations with real numbers, linear equations and inequalities, polynomials and factoring, quadratic equations, and systems of equations. This course also includes an introduction to rational expressions, radicals, and rational exponents.
MATH107 -
College GeometryMATH109 -
Modern Mathematics
This course is an introduction to mathematics developed in the last one-hundred years. The course connects recently-discovered mathematics with current, real-world problems. Aesthetic elements of mathematics are emphasized. Topics may include the mathematics of voting, sharing, touring, games, networks, scheduling, money, symmetry, fractal shapes, descriptive statistics, and probability. The course can be used as a prerequisite for MATH 208-Introduction to Statistics, but will not serve as a prerequisite for MATH 203-Precalculus or MATH 205-Calculus. The course is appropriate for students majoring in Communication, Criminal Justice, English/History/Humanities-with Secondary Ed, English, Environmental Studies, Fashion Design, History, Hospitality and Event Management, Humanities, Human Services, Law and Public Affairs, Legal Studies, Psychology, Sociology, or Sport Management. Prerequisite: MATH 103 with a grade of C or better or through placement testing.
MATH116 -
Merchandising and Financial Mathematics
This course focuses on retail mathematics. Topics include simple and compound interest, the time-value of capital, annuities, amortization, sinking funds, bond and investment, business problem-solving and decision making. Other topics include profit, loss, and break-even analysis, pricing, inventory, and merchandise planning. The course introduces basic theories of statistics. Prerequisite: MATH 103 with a grade of C or better or demonstrated competency through placement testing.
MATH203 -
Precalculus
This course prepares students for the study of calculus, physics and other courses requiring precalculus skills. Included is a review of algebra, coordinate geometry, the solution of systems of equations, and the analysis and graphing of linear, quadratic, inverse, polynomial, and rational functions. There is a thorough treatment of exponential, logarithmic, and trigonometric functions. An important goal is for students to develop a geometric understanding of functions and their properties. Prerequisite: MATH 104 with a grade of C or better or demonstrated competency through placement testing. Restrictions: not open to students who have completed 205, 206, or any 300 level mathematics course successfully.
MATH205 -
Calculus IMATH206 -
Calculus IIMATH207 -
Applied Trigonometry
This course is an in-depth study of trigonometry with attention to theory, proofs, modeling, and history. Trigonometric and related functions are used to model, analyze, and solve real-life problems. Applications are chosen from disciplines such as agriculture, architecture, astronomy, biology, business, chemistry, earth science, engineering, medicine, meteorology, and physics. Topics covered include a review of trigonometric functions, right triangle trigonometry, analytic trigonometry, vectors and dot products, complex number theory, trigonometric forms of complex numbers, exponential, logarithmic and trigonometric models, Gaussian and logistic growth models, conic sections, and polar equations of conics. Prerequisite: MATH 205 with a grade of C or better.
MATH208 -
StatisticsMATH210 -
Math Applied to Science
This course provides a review of fundamental mathematical concepts such as probability and trigonometric, exponential and logarithmic functions and explores the ways that these topics and techniques have been applied to investigations in architecture, calculus, exponential growth and decay, logarithmic scales, earthquake analysis, astronomy, biology, medicine, genetics, radiocarbon dating, chemistry, and Newtonian physics. The course is designed to demonstrate the power and utility of mathematics and explores the development of mathematics during the Middle Ages and the Renaissance, especially in Greek, Hindu and Arabic cultures. Prerequisite: MATH 205 with a grade of C or better.
MATH212 -
Finite MathematicsMATH215X -
Discrete Math
MATH304X -
Mathematics for Educators
This course engages students in mathematical concepts through examples, investigations, and active problem solving explorations. Content is drawn from subject matter knowledge required for elementary and early childhood licensure, with emphasis on number theory and operations. This course is for students seeking elementary or early childhood licensure. Concurrent enrollment in ED 335 is required.
MATH307 -
Calculus III
This course is an introduction to sequences and series, parametric and polar curves, vectors and vector functions, partial derivatives, multiple integration, and vector calculus. Prerequisite: MATH 206 with a grade of C or better.
MATH320 -
Differential EquationsMATH325 -
Linear Algebra
This is an introductory course in linear algebra blending the requirements of theory, problem solving, analytical thinking, computational techniques, and applications. Topics include in-depth treatment of matrix algebra, linear systems, vector spaces, linear transformations, determinants and computer methods, as well as applications and modeling of real phenomena in transportation systems, archaeology, economics, communications, demography, weather prediction, connectivity of networks, graph theory, and fractals. Prerequisite: MATH 206 with a grade of C or better.
MATH328 -
Mathematics Applied to Management
This course explores the art of mathematical modeling of managerial decision problems and the science of developing the solution techniques for these models. Topics include management science techniques used in today's businesses, e.g., break-even analysis, presentation models, linear programming, transportation and assignments problems, decision theory, forecasting and inventory models, Markov analysis, and solution of nonlinear models in business using calculus-based optimization. Prerequisite: MATH 206 with a grade of C or better.
MATH330 -
Mathematical Modeling
This is an application-oriented course on how to solve real word problems from the social, medical and life sciences, business, and economics by setting up a mathematical model of the situation and then developing techniques for analyzing these models and solving them. Topics include the modeling process, linear models, financial models, modeling using proportionality, fitting linear and nonlinear models to data graphically, the least-squared criterion, linear programming models, modeling using the derivative, matrix and probability models, Markov chain models, and modeling interactive dynamic systems. Prerequisite: MATH 206 with a grade of C or better.
MATH338 -
Mathematical Statistics
In this introduction to statistical theory, the roles probability and statistics play in business analysis and decision making are investigated. Topics include probability distributions, statistical inference, sampling distribution theory, and applications. Prerequisite: Math 206 with a grade of C or better.
MATH399 -
Mathematical Applications
In this capstone course, Students investigate mathematics from a variety of fields and choose a topic for a mathematics project in their Field of Application. Mathematical methods for analysis, modeling, prediction, and/or problem solving are discussed. Students demonstrate knowledge of a substantial area of mathematics and present their work at a department seminar or the Connected Learning Symposium.
MATH499 -
Internship
The internship seminar is a work or research experience where students combine theory and practice.
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Marden's Theorem concerns the relative positions of the roots of a cubic polynomial and those of its derivative. This article gives a proof of Marden's theorem, along with dynamic geometry animations and some of the history of the result.
These applets provide interactive student activities to make connections between the graphical and analytical interpretation of "completing the square" and writing the equation of a parabola in general and standard form.
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95% of Introductory Statistics students will never take another Statistics course. What do you want to learn?
Discover the Power of Real Data
Mario Triola remains the market-leading statistics author by engaging readers of each edition with an abundance of real data in the examples, applications, and exercises. Statistics is all around us, and Triola helps readers understand how this course will impact their lives beyond the classroom–as consumers, citizens, and professionals.
Elementary Statistics Using the TI-83/84 Plus Calculator, Third Edition provides extensive instruction for using the TI-83 and TI-84 Plus (and Silver Edition) calculators for statistics, with information on calculator functions, images of screen displays, and projects designed exclusively for the graphing calculator. Drawn from Triola's Elementary Statistics, Eleventh Edition, this text provides the same student-friendly approach with material presented in a real-world context.
The Third Edition contains more than 2,000 exercises, 87% are new, and 82% use real data. It also contains hundreds of examples; 86% are new and 94% use real data. By analyzing real data, readers are able to connect abstract concepts to the world at large, learning to think statistically and apply conceptual understanding using the same methods that professional statisticians employ.
Datasets and other resources (where applicable) for this book are available here.
Book Description:Pearson, 2010. Hardcover. Book Condition: New. Please read before purchase>> annotated teacher edition New no writing or marks includes all Students content and all answers. Includes sealed Cd and formulas charts no access code or other supplements. ship immediately - Expedited shipping available. Bookseller Inventory # 0026
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