Linear Algebra

196 Pages·1995·22.671 MB·English

by  Harold M. Edwards (auth.)

Harold M. Edwards
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1995
Springer Science+Business Media, LLC
Harold M. Edwards
Courant Institute
New York University
New York, NY 10012
Library of Congress Cataloging In-Publication Data
Edwards, Harold M.
Linear algebra I Harold M. Edwards.
p. cm.
ISBN 978-0-8176-4370-6 ISBN 978-0-8176-4446-8 (eBook)
DOI 10.1007/978-0-8176-4446-8
1. Algebra, Linear. I. Title.
QA184.E355 1994 94-35356
512.9'43--dc20 CIP
Printed on acid-free paper
© 1995 Harold M. Edwards
Originally published by Birkhiiuser Boston in 1995
All rights reserved. No part of this publication may be reproduced, stored in a retrieval
system, or transmitted, in any form or by any means, electronic, mechanical, photocopy
ing, recording, or otherwise, without prior permission of the copyright owner.
ISBN 978-0-8176-4370-6
Layout, design, and macros by Martin Stock, Cambridge, MA
Typeset by TEXniques, Inc., Brighton, MA
9 8 765 432 1
Contents
Preface ix
Matrix Multiplication
I linear Substitutions
2 Composition of linear Substitutions 2
3 Matrices 3
4 Matrix Multiplication 4
5 The Computation of Matrix Products 5
6 Associativity 6
Examples 7
Exercises 8
2 Equivalence of Matrices.
Reduction to Diagonal Form
I Tilts 10
2 Composition with Tilts 12
3 Equivalence of Matrices 13
4 Unimodular Matrices 14
5 On Algorithms 14
6 The I x 2 Case IS
7 The General Case 16
8 Finding Explicit Equivalences 19
Examples 20
Exercises 23
3 Matrix Division
Division 25
2 Division on the left by a Diagonal Matrix 26
3 Division on the Left in the General Case 27
4 Matrix Addition 28
5 Zero Divisors 29
Examples 30
Exercises 32
vi Contents
4 Determinants
I Introduction 34
2 The 2 x 2 Case 34
3 The 3 x 3 Case 35
4 The 4 x 4 Case 37
5 The General Case 38
6 The Determinant of a Product 40
7 The Evaluation of Determinants 41
Examples 43
Exercises 45
Supplementary Unit Exterior Algebra 47
5 Testing for Equivalence
I Introduction 50
2 Strongly Diagonal Matrices 50
3 Equivalence of Strongly Diagonal Matrices 54
4 Conclusion 55
5 The Rank of a Matrix 55
Examples 56
Exercises 57
Supplementary Unit: Finitely Generated Abelian Groups 58
6 Matrices with Rational Number Entries
I Introduction 60
2 Matrix Division 61
3 Inverses 62
Examples 63
Exercises 65
Supplementary Unit Sets and Vector Spaces 66
7 The Method of Least Squares
I Introduction 68
2 The Transpose of a Matrix 68
3 Mates 69
'" Mates as Generalized Inverses 71
Examples 73
Exercises 77
Contents vii
8 Matrices with Polynomial Entries
I Polynomials 78
2 Equivalence of Matrices of Polynomials 79
3 Unimodular Matrices 80
4 The I x 2 Case 80
5 The General Case 81
6 Determinants 82
7 Strongly Diagonal Form 82
8 Wider Equivalence 85
Examples 86
Exercises 90
9 Similarity of Matrices
I Introduction 91
2 A Necessary Condition for Similarity 92
3 The Necessary Condition Is Sufficient 94
4 Matrices Similar to Given Matrices 95
5 Rational Canonical Form 97
6 The Minimum Polynomial of a Matrix 99
7 Diagonalizable Matrices 102
8 Other Coefficient Fields 102
Examples 103
Exercises 109
10 The Spectral Theorem
I Introduction III
2 Orthogonal Partitions of Unity I II
3 Spectral Representations 113
4 Symmetric Matrices with Spectral Representations 114
5 Sign Changes in Polynomials 116
6 The Algebraic Theorem 119
7 Real Numbers 122
8 The Spectral Theorem 123
9 Matrix Inversion 123
10 Diagonalizing Symmetric Matrices 124
Examples 125
Exercises 130
viii Contents
Appendix Linear Programming
I The Problem 132
2 Standard Form for the Constraints 133
3 Vertices 134
4 Feasible Vertices 135
5 Solution Vertices 136
6 The Simplex Method 137
7 Implementation of the Simplex Algorithm 138
8 Termination of the Simplex Method 139
9 Finding a Feasible Vertex 141
10 Summary 143
Examples 144
Answers to Exercises 151
Index 183
Preface
Although the title "Linear Algebra" describes the place of this book in the math
ematics curriculum, a better description of its actual content would be "The
Arithmetic of Matrices." The ability to compute with matrices-even better,
the ability to imagine computations with matrices-is useful in any pursuit that
involves mathematics, from the purest of number theory to the most applied of
economics or engineering. The goal of this book is to help students acquire that
ability.
Accordingly, the emphasis throughout the book is on algorithms. A by
product of this emphasis is the complete disappearance of set theory, a disap
pearance that will greatly disturb teachers accustomed to the standard linear
algebra course but will not, and should not, disturb students in the least. The
material in the book will be helpful, in addition to its other uses, in learning the
language and the peculiar habits of mind that are set theory (see the Supplemen
tary Unit of Chapter 6). The standard linear algebra course attempts to reverse
the order and to use set theory to teach linear algebra-an approach that is as
silly as it is unsuccessful.
Each chapter has an "Examples" section giving sample applications of the
algorithms contained in the chapter. The student should refer to the examples
while reading the chapter. The chapter necessarily deals with the most general
case, but no mathematician ever understands the most general case first. One
famous mathematician, David Hilbert, gave his students the excellent advice,
"Always begin with the simplest example."
I mentioned imaginary calculation in the first paragraph. The emphasis on
algorithms in this course lends itself very well to the use of computers, and I
hope that teachers will be able to take advantage of local computer facilities and
the particular levels of computer competence their students have to go further
with the computational aspect of the subject. However, I do believe that the main
thing is to be able to imagine the computation. The best theorems of mathematics
have the form, "If you do calculation X, you will find result Y." To understand
the meaning of such a theorem you must be able to imagine the calculation, but
the actual calculation is rarely needed; if it is needed, you need to be able to
imagine alternative ways of doing it in order to select a good one. The main
reason for doing computations in a linear algebra course is to develop this form
x Unear Algebra
of imagination. It is not altogether clear that doing large examples on a computer
is better for this purpose than doing small examples with pencil and paper. It is
best, no doubt, to do both.
Some of the proofs, particularly in the last few chapters, involve lengthy
arguments. My answer to the perennial question 'Are we expected to know the
proofs for the exam?' is 'no,' not because proofs are unimportant, but because
a proof cannot be known, it must be understood, and understanding is very hard
to test on an exam. The most natural exam question is a particular example
that uses the theorem. Preparing for such a question is the best way to study
the material of the course, because there is no way to work through an example
without referring to the proof of the most general case, just as there is no way
to grasp the general case covered by the proof without applying it to specific
examples. After enough work with specific examples, the meaning of the proof
should begin to become clear.
Synopsis
The first algorithm of matrix arithmetic is matrix multiplication. As Chapter 1
explains, this operation arises naturally as the operation of composition of linear
substitutions. The naturalness of the idea is confirmed by its appearance in a
wide variety of apparently disparate contexts in mathematics.
Matrix multiplication is such a simple operation that it scarcely deserves to
be called an algorithm. The same can certainly not be said of matrix division. If
a matrix product AX = Y is known, and if one of the factors A is known, to what
extent is the other factor X determined, and how can all possible X s be found?
The algorithmic answer to this question in Chapter 3 depends on the notion of
matrix equivalence developed in Chapter 2.
In the definition of Chapter 2, a matrix is equivalent to another if it can
be transformed into the other by a sequence of operations of four elementary
types: Adding a column to an adjacent column, adding a row to an adjacent
row, subtracting a column from an adjacent column, or subtracting a row from
an adjacent row. As is proved by an algorithm in Chapter 2, every matrix is
equivalent to a diagonal one. The solution of the division problem consists of
two simple observations; first, that the division problem AX = Y is easy to solve
when A is diagonal; second, that solution of the division problem for one matrix
A leads easily to its solution for any equivalent matrix.
The algorithm of Chapter 2 gives a method of finding equivalences betw~n
matrices, but it does not give a method for proving that matrices are inequivalent.