## PREFACE TO THE SECOND EDITION

The principal change from the first edition is the addition of

a new chapter on linear programming. While linear programming

is one of the most widely used and successful applications

of linear algebra, it rarely appears in a text such as this. In

the new Chapter Ten the theoretical basis of the simplex algorithm

is carefully explained and its geometrical interpretation

is stressed.

Some further applications of linear algebra have been

added, for example the use of Jordan normal form to solve

systems of linear differential equations and a discussion of extremal

values of quadratic forms.

On the theoretical side, the concepts of coset and quotient

space are thoroughly explained in Chapter 5. Cosets have

useful interpretations as solutions sets of systems of linear

equations. In addition the Isomorphisms Theorems for vector

spaces are developed in Chapter Six: these shed light on the

relationship between subspaces and quotient spaces.

The opportunity has also been taken to add further exercises,

revise the exposition in several places and correct a few

errors. Hopefully these improvements will increase the usefulness

of the book to anyone who needs to have a thorough

knowledge of linear algebra and its applications.

I am grateful to Ms. Tan Rok Ting of World Scientific

for assistance with the production of this new edition and for

patience in the face of missed deadlines. I thank my family

for their support during the preparation of the manuscript.

## PREFACE TO THE FIRST EDITION

A rough and ready definition of linear algebra might be: that

part of algebra which is concerned with quantities of the first

degree. Thus, at the very simplest level, it involves the solution

of systems of linear equations, and in a real sense this

elementary problem underlies the whole subject. Of all the

branches of algebra, linear algebra is the one which has found

the widest range of applications. Indeed there are few areas

of the mathematical, physical and social sciences which have

not benefitted from its power and precision. For anyone working

in these fields a thorough knowledge of linear algebra has

become an indispensable tool. A recent feature is the greater

mathematical sophistication of users of the subject, due in

part to the increasing use of algebra in the information sciences.

At any rate it is no longer enough simply to be able to

perform Gaussian elimination and deal with real vector spaces

of dimensions two and three.

The aim of this book is to give a comprehensive introduction

to the core areas of linear algebra, while at the same

time providing a selection of applications. We have taken the

point of view that it is better to consider a few quality applications

in depth, rather than attempt the almost impossible task

of covering all conceivable applications that potential readers

might have in mind.

The reader is not assumed to have any previous knowledge

of linear algebra – though in practice many will – but

is expected to have at least the mathematical maturity of a

student who has completed the calculus sequence. In North

America such a student will probably be in the second or third

year of study.

The book begins with a thorough discussion of matrix

operations. It is perhaps unfashionable to precede systems

of linear equations by matrices, but I feel that the central

position of matrices in the entire theory makes this a logical

and reasonable course. However the motivation for the introduction

of matrices, by means of linear equations, is still

provided informally. The second chapter forms a basis for

the whole subject with a full account of the theory of linear

equations. This is followed by a chapter on determinants, a

topic that has been unfairly neglected recently. In practice it

is hard to give a satisfactory definition of the general n x n

determinant without using permutations, so a brief account

of these is given.

Chapters Five and Six introduce the student to vector

spaces. The concept of an abstract vector space is probably

the most challenging one in the entire subject for the nonmathematician,

but it is a concept which is well worth the

effort of mastering. Our approach proceeds in gentle stages,

through a series of examples that exhibit the essential features

of a vector space; only then are the details of the definition

written down. However I feel that nothing is gained

by ducking the issue and omitting the definition entirely, as is

sometimes done.

Linear tranformations are the subject of Chapter Six.

After a brief introduction to functional notation, and numerous

examples of linear transformations, a thorough account

of the relation between linear transformations and matrices is

given. In addition both kernel and image are introduced and

are related to the null and column spaces of a matrix.

Orthogonality, perhaps the heart of the subject, receives

an extended treatment in Chapter Seven. After a gentle introduction

by way of scalar products in three dimensions —

which will be familiar to the student from calculus — inner

product spaces are denned and the Gram-Schmidt procedure

is described. The chapter concludes with a detailed account

of The Method of Least Squares, including the problem of

finding optimal solutions, which texts at this level often fail

to cover.

Chapter Eight introduces the reader to the theory of

eigenvectors and eigenvalues, still one of the most powerful

tools in linear algebra. Included is a detailed account of applications

to systems of linear differential equations and linear

recurrences, and also to Markov processes. Here we have not

shied away from the more difficult case where the eigenvalues

of the coefficient matrix are not all different.

The final chapter contains a selection of more advanced

topics in linear algebra, including the crucial Spectral Theorem

on the diagonalizability of real symmetric matrices. The

usual applications of this result to quadratic forms, conies

and quadrics, and maxima and minima of functions of several

variables follow.

Also included in Chapter Nine are treatments of bilinear

forms and Jordan Normal Form, topics that are often not considered

in texts at this level, but which should be more widely

known. In particular, canonical forms for both symmetric and

skew-symmetric bilinear forms are obtained. Finally, Jordan

Normal Form is presented by an accessible approach that requires

only an elementary knowledge of vector spaces.

Chapters One to Eight, together with Sections 9.1 and

9.2, correspond approximately to a one semester course taught

by the author over a period of many years. As time allows,

other topics from Chapter Nine may be included. In practice

some of the contents of Chapters One and Two will already be

familiar to many readers and can be treated as review. Full

proofs are almost always included: no doubt some instructors

may not wish to cover all of them, but it is stressed that for

maximum understanding of the material as many proofs as

possible should be read. A good supply of problems appears

at the end of each section. As always in mathematics, it is an

indispensible part of learning the subject to attempt as many

problems as possible.

This book was originally begun at the suggestion of

Harriet McQuarrie. I thank Ms. Ho Hwei Moon of World

Scientific Publishing Company for her advice and for help with

editorial work. I am grateful to my family for their patience,

and to my wife Judith for her encouragement, and for assistance

with the proof-reading.

## CONTENTS

Preface to the Second Edition vii

Preface to the First Edition ix

Chapter One Matrix Algebra

1.1 Matrices 1

1.2 Operations with Matrices 6

1.3 Matrices over Rings and Fields 24

Chapter Two Systems of Linear Equations

2.1 Gaussian Elimination 30

2.2 Elementary Row Operations 41

2.3 Elementary Matrices 47

Chapter Three Determinants

3.1 Permutations and the Definition of a

Determinant 57

3.2 Basic Properties of Determinants 70

3.3 Determinants and Inverses of Matrices 78

Chapter Four Introduction to Vector Spaces

4.1 Examples of Vector Spaces 87

4.2 Vector Spaces and Subspaces 95

4.3 Linear Independence in Vector Spaces 104

Chapter Five Basis and Dimension

5.1 The Existence of a Basis 112

5.2 The Row and Column Spaces of a Matrix 126

5.3 Operations with Subspaces 133

Chapter Six Linear Transformations

6.1 Functions Defined on Sets 152

6.2 Linear Transformations and Matrices 158

6.3 Kernel, Image and Isomorphism 178

Chapter Seven Orthogonality in Vector Spaces

7.1 Scalar Products in Euclidean Space 193

7.2 Inner Product Spaces 209

7.3 Orthonormal Sets and the Gram-Schmidt

Process 226

7.4 The Method of Least Squares 241

Chapter Eight Eigenvectors and Eigenvalues

8.1 Basic Theory of Eigenvectors and Eigenvalues 257

8.2 Applications to Systems of Linear Recurrences 276

8.3 Applications to Systems of Linear Differential

Equations 288

Chapter Nine More Advanced Topics

9.1 Eigenvalues and Eigenvectors of Symmetric and

Hermitian Matrices 303

9.2 Quadratic Forms 313

9.3 Bilinear Forms 332

9.4 Minimum Polynomials and Jordan Normal

Form 347

Chapter Ten Linear Programming

10.1 Introduction to Linear Programming 370

10.2 The Geometry of Linear Programming 380

10.3 Basic Solutions and Extreme Points 391

10.4 The Simplex Algorithm 399

Appendix Mathematical Induction 415

Answers to the Exercises 418

Bibliography 430

Index 432