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JinHo Kwak

Sungpyo

Hong

Department of Matbematics

Pohang University of Science

Department of Mathematics

Pohang University of Science

and Technology and Technology

Pohang, Kyungbuk 790-784

South Korea

Pohang, Kyungbuk 790-784

SouthKorea

Library of Cougress Cataloging-in-PubHeation Data

Kwak,

lin

Ho, 1948-

Linear algebra

lin

Ho Kwak, Sungpyo

Hong.-2nd

ed.

p.cm.

Includes

bibliographical

references and

index.

ISBN 978-0-8176-4294-5 ISBN 978-0-8176-8194-4 (eBook)

DOI 10.1007/978-0-8176-8194-4

1. Algebras, Linear. I. Hong, Sungpyo, 1948-

Title.

QAI84.2.K932004

512

.5-dc22

AMS Subject Classifications: 15-01

ISBN 978-0-8176-4294-5

Printed on acid-free paper.

@2004 Springer Science Business

Media New York

Originally

published by

Birkhlluser Boston in 2004

2004043751

CIP

All rights reserved. This work

may

not be translated or copied in

whole

or in part without the written

permission

of

the

publisher Springer Science Business

Media, LLC,

except

for

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whether

or not they

are

subject

to property

rights.

987654321

SPIN

10979327

www.birkhasuer science.com

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vi

Preface to the Second Edition

The major changes from the first edition are the following.

(1) In Chapter 2, Section 2.5.1 Miscellaneous examples for determinants is

added as an application.

(2) In Chapter 4, A homogeneous coordinate system is introduced for an appli

cation in computer graphics.

(3) InChapter5, Section 5.7 Relations of fundamental subspaces and Section 5.8

Orthogonal matrices and isometries are interchanged. Least squares solutions,

Polynomial approximations and Orthogonal projection matrices are collected

together in Section 5.9-Applications.

(4) Chapter 6 is entitled Diagonalization instead of Eigenvectors and Eigen

values. In Chapters 6 and 8, Recurrence relations, Linear difference equations

and Linear differential equations are described in more detail as applications

of

diagonalizations and the Jordan canonical forms

of

matrices.

(5) In Chapter 8, Section 8.5 The minimal polynomial of a matrix has been

added to introducemore easily accessible computational methods for n nd

e

A

, with

complete solutions of linear difference equations and linear differential equations .

(6) Chapter 8 Jordan Canonical Forms and Chapter 9 Quadratic Forms are

interchanged for a smooth continuation of the diagonalization problem of matrices.

Chapter 9 Quadratic Forms is extended to a complex case and includes many new

figures.

(7) The errors and typos found to date in the first edition have been corrected .

(8) Problems are refined to supplement the worked-out illustrative examples and

to enable the reader to check his or her understanding of new definitions or theorems.

Additional problems are added in the last exercise section of each chapter. More

answers, sometimes with brief hints, are added, including some corrections.

(9) In most examples, we begin with a brief explanatory phrase to enhance the

reader s understanding.

This textbook can

be used for a one- or two-semester course in linear algebra. A

theory oriented one-semester course may cover Chapter 1,Sections 1.1-1.4, 1.6-1.7;

Chapter 2 Sections 21 2 .3; Chapter 3 Sections 3.1-3.6; Chapter 4 Sections 41 4 .6;

Chapter 5 Sections 5.1-5.4; Chapter 6 Sections 6

1 6

.2; Chapter 7 Sections 7.1-7.4

with possible addition from Sections 1.8, 2.4 or 9.1-9.4. Selected applications are

included in each chapter as appropriate. For a beginning applied algebra course, an

instructormight include some ofthem in the syllabus athis orher discretion depending

on which area is to be emphasized or considered more interesting to the students.

Indefinitions, we use bold face for the word being defined, and sometimes an italic

or shadowbox to emphasize a sentence or undefined or post-defined terminology.

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Preface to the Second Edition vii

Acknowledgement: The authors would like to express our sincere appreciation

for the many opinions and suggestions from the readers of the first edition including

many of our colleagues at POSTECH. The authors are also indebted to Ki Hang Kim

and Fred Roush at Alabama State University and Christoph Dalitz at Hochschule

Niederrhein for improving the manuscript and selecting the newly added subjects in

this edition . Our thanks again go toMrs . Kathleen Roush for grammatical corrections

in the final manuscript, and also to the editing staff of Birkhauser for gladly accepting

the second edition for publication.

JinHo

Kwak

Sungpyo Hong

E-mail: jinkwak@postech.ac.kr

sungpyo@postech.ac.kr

January 2004

Pohang

South

Korea

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Preface to the First Edition

Linear algebra is one of the most important subjects in the study of science and engi

neering because of its widespread applications in social or natural science, computer

science, physics, or economics . As one of the most useful courses in undergradu

ate mathematics , it has provided essential tools for industrial scientists. The basic

concepts of linear algebra are vector spaces, linear transformations, matrices and

determinants, and they serve as an abstract language for stating ideas and solving

problems.

This book is based on lectures delivered over several years in a sophomore-level

linear algebra course designed for science and engineering students. The primary

purpose of this book is to give a careful presentation of the basic concepts of linear

algebra as a coherent part ofmathematics, and to illustrate its power and utility through

applications to other disciplines . We have tried to emphasize computational skills

along with mathematical abstractions , which have an integrity and beauty of their

own. The book includes a variety of interesting applications with many examples not

only to help students understand new concepts but also to practice wide applications

ofthe subject to such areas as differential equations, statistics, geometry, and physics.

Some of those applications may not be central to the mathematical development and

may be omitted or selected in a syllabus at the discretion of the instructor. Most

basic concepts and introductory motivations begin with examples in Euclidean space

or solving a system of linear equations, and are gradually examined from different

points of view to derive general principles .

For students who have finished a year of calculus, linear algebra may be the first

course inwhich the subject isdeveloped inanabstract way, and weoften findthat many

students struggle with the abstractions and miss the applications . Our experience is

that, to understand the material, students should practice with many problems, which

are sometimes omitted. To encourage repeated practice, we placed in the middle of

the text not only many examples but also some carefully selected problems, with

answers or helpful hints. We have tried to make this book as easily accessible and

clear as possible , but certainly there may be some awkward expressions in several

ways. Any criticism or comment from the readers will be appreciated .

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x Preface to the First Edition

We are very grateful to many colleagues in Korea, especially to the faculty mem

bers in the mathematics department at Pohang University of Science and Technology

(POSTECH), who helped us over the years with various aspects of this book. For

their valuable suggestions and comments, we would like to thank the students at

POSTECH, who have used photocopied versions of the text over the past several

years. Wewould also like to acknowledge the invaluable assistance we have received

from the teaching assistants who have checked and added some answers or hints

for the problems and exercises in this book. Our thanks also go to Mrs. Kathleen

Roush who made this book much more readable with grammatical corrections in the

final manuscript. Our thanks finally go to the editing staff of Birkhauser for gladly

accepting our book for publication.

Jin Ho Kwak

Sungpyo Hong

April 997 Pohang South Korea

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Contents

Preface to the Second Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Preface to the First Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

1 Linear Equations and Matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.1 Systems of linear equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Gaussian elimination. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Sums and scalar multiplications of matrices. . . . . . . . . . . . . . . . . . . . . 11

1.4 Products of matrices

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.5 Block matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.6 Inverse matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.7 Elementary matrices and finding

I

23

1 8 LDU factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

1.9 Applications .

. . .

. .

.

. . .

. . . . . . 34

1.9.1 Cryptography.. . .

. . . . . . .

.

.

.

.

. .

34

1.9.2 Electrical network 36

1.9.3 Leontief model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

1.10 Exercises 40

2 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.1 Basic properties of the determinant. . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.2 Existence and uniqueness of the determinant. . . . . . . . . . . . . . . . . . 50

2.3 Cofactor expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56

2.4 Cramer s rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

2.5 Applications . . . .

.

64

2.5.1 Miscellaneous examples for determinants. . . . . . . . . . . . . . 64

2.5.2 Area and volume. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

2.6 Exercises

. .

. .

.

.

. . . 72

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Contents xiii

6 Diagonalization 201

6.1 Eigenvalues and eigenvectors 201

6.2 Diagonalization of matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

6.3 Applications 212

6.3.1 Linear recurrence relations . . . . . . . . .

212

6.3.2 Linear difference equations 221

6.3.3 Linear differential equations I . . . . . . . . . . . . . . . . . . . . . . . . . . 226

6.4 Exponential matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

6.5 Applications continued . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

6.5.1 Linear differential equations II 235

6.6 Diagonalization of linear transformations . . . . . . . . . . . . . . . . . . . . . . . 240

6.7 Exercises . . .

. . . 242

7 Complex Vector Spaces 247

7.1 The n-space

and complex vector spaces 247

7.2 Hermitian and unitary matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

254

7.3 Unitarily diagonalizable matrices 258

7.4 Normal matrices 262

7.5 Application .

.

. 265

7.5.1 The spectral theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

7.6 Exercises . .

. .

. . 269

8 Jordan Canonical

Forms

273

8.1 Basic properties of Jordan canonical forms . . . . . . . . . . . . . .

273

8.2 Generalized eigenvectors 281

8.3 The power A

k

and the exponential e

A

289

8.4 Cayley-Hamilton theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294

8.5 The minimal polynomial of a matrix 299

8.6 Applications. 302

8.6.1 The power matrix

A

k

again 302

8.6.2 The exponential matrix e

A

again 306

8.6.3 Linear difference equations again . . . . . . . . . . . . . . . . . . . . .

309

8.6.4 Linear differential equations again. . . . . . . . . . . . . . . . . . . . . . 310

8.7 Exercises 315

9

Quadratic Forms

319

9.1 Basic properties of quadratic forms 319

9.2 Diagonalization of quadratic forms 324

9.3 A classification of level surfaces 327

9.4 Characterizations of definite forms 332

9.5 Congruence relation 335

9.6 Bilinear and Hermitian forms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339

9.7 Diagonalization of bilinear or Hermitian forms 342

9.8 Applications 348

9.8.1 Extrema of real-valued functions on jRn . . 348

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xiv Contents

9.8.2 Constrainedquadratic optimization 353

9.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356

SelectedAnswersand

mots

361

Bibliography 383

Index

385

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Linear Algebra