Linear Algebra

453 Pages·1998·138.369 MB·English

by  C Y Hsiung, G Y Mao

IlliTihli
IinlnllnM I
C Y Hsiunq
GYMao
World Scientifi
Linear
Algebra
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Linear
Jgebra
C Y Hsiung
Wuhan University
GYMao
Wuhan University of Technology
World Scientific
Singapore'New Jersey'London 'Hong Kong
Published by
World Scientific Publishing Co. Pie. Ltd.
P O Box 128, Farrer Road, Singapore 912805
USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
LINEAR ALGEBRA
Copyright © 1998 by World Scientific Publishing Co. Pie. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
electronic or mechanical, including photocopying, recording or any information storage and retrieval
system now known or to be invented, without written permission from the Publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright
Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to
photocopy is not required from the publisher.
ISBN 981-02-3092-3
This book is printed on acid-free paper.
Printed in Singapore by Uto Print
PREFACE
This book introduces the basic properties and operations of linear algebra as
well as its basic theory and concepts. It is based on the first author's lecture
notes on Linear Algebra during his teaching in the Mathematics Department
of Wuhan University. Some amendments were made and many new ideas were
added to the notes before the book was finalized.
The book consists of eight chapters. The first six chapters introduce linear
equations and matrices, and the last two chapters introduce linear spaces and
linear transformations. They are arranged progressively from easy to difficult,
simple to complex, and specific to general, which makes it easier for the readers
to study on his own. A summary at the beginning of each chapter and each
section gives the reader some idea of the topics and aims to be dealt with in the
text. Both basic concepts and manipulation skills are equally emphasized, and
enough examples are given for their illustration. There are also many exercises
at the end of each section, with the answers atta.ched to the end of the book
for reference.
Basic concepts are specially stressed and great pains have been taken
to explain the underlying thoughts and the approach. Furthermore, between
chapters and between sections, a brief leader is given to preserve the coherence
and continuity of the text.
Chapter 1 was written by Professor Jian-Ke Lu, who gave the definition
of determinants in a very special way, quite different from those in other text­
books. This definition is easier for the reader to understand and master, and
facilitates the proofs of some of their properties. The author would like to
thank Bang-Teng Xu for his very helpful and meticulous work on the proofs
of the other seven chapters.
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CONTENTS
Preface v
1 Determinants 1
1.1. Concept of Determinants . 1
1.2. Basic Properties of Determinants 12
1.3. Development of a Determinant 20
1.4. Cramer's Theorem 35
2 Systems of Linear Equations 41
2.1. Linear Relations between Vectors 42
2.2. Systems of Homogeneous Linear Equations . . . 56
2.3. Systems of Fundamental Solutions 62
2.4. Systems of Nonhomogeneous Linear Equations . 70
2.5. Elementary Operations 80
3 Matrix Operations 92
3.1. Matrix Addition and Matrix Multiplication . . 92
3.2. Diagonal, Symmetric, and Orthogonal Matrices . 112
3.3. Invertible Matrices 124
4 Quadratic Forms 141
4.1. Standard Forms of General Quadratic Forms . . 142
4.2. Classification of Real Quadratic Forms . . .. 155
*4.3. Bilinear Forms 170
5 Matrices Similar to Diagonal Matrices 173
5.1. Eigenvalues and Eigenvectors 174
5.2. Diagonalization of Matrices 184
vii
viii
5.3. Diagonalization of Real Symmetric Matrices 198
*5.4. Canonical Form of Orthogonal Matrices 212
*5.5. Cayley-Hamilton Theorem and Minimum Polynomials . . .. 217
6 Jordan Canonical Form of Matrices 230
6.1. Necessary and Sufficient Condition for Two Matrices
to be Similar 230
6.2. Canonical Form of A-Matrices 236
6.3. Necessary and Sufficient Condition for Two A-Matrices
to be Equivalent 242
6.4. Jordan Canonical Forms 252
7 Linear Spaces and Linear Transformations 273
7.1. Concept of Linear Spaces 273
7.2. Bases and Coordinates 286
7.3. Linear Transformations 304
7.4. Matrix Representation of Linear Transformations 321
*7.5. Linear Transformations from One Linear Space into Another . . 337
*7.6. Dual Spaces and Dualistic Transformations 341
8 Inner Product Spaces 349
8.1. Concept of Inner Product Spaces 349
8.2. Orthonormal Bases 362
8.3. Orthogonal Linear Transformations 373
8.4. Linear Spaces over Complex Numbers with Inner Products . . . 381
*8.5. Normal Operators 392
Answers to Selected Exercises 395
Index 440
CHAPTER 1
DETERMINANTS
In many practical problems, relations among variables may be simply
expressed directly or approximately in terms of linear functions so that it is
necessary to investigate such functions. Linear algebra is a branch of mathe­
matics dealing mainly with linear functions, in which systems of linear
equations constitute its basic and also important part. In linear algebra, the
notion of determinants is fundamental. Theory of determinants is established
to satisfy the need for solving systems of linear equations. It has wide appli­
cations in mathematics, as well as in other scientific branches (for instance,
physics, dynamics, etc.). The present chapter will mainly deal with the follow­
ing three problems:
1. Formulation of the concept of determinants,
2. Derivation of their basic properties and study of the relating calculations,
3. Solving systems of linear equations by using them as a tool.
1.1. Concept of Determinants
In high school, we already learned how to solve a system of linear equations
in 2 or 3 unknowns by using determinants of order 2 or 3 respectively. Is it
possible, in general, to solve a system of linear equations in n unknowns in an
analogous way?
Determinants of an arbitrary order were introduced for such needs. The
aim of the present section is to establish the concept of determinants of order
n so as to answer the above-mentioned problem 1. We shall first recall results
familiar to us in high school.
1