LINEAR ALGEBRA AND ITS APPLICATIONS

THIRD EDITION UPDATE

David C. Lay

University of Maryland – College Park

Copyright © 2006 Pearson Addison-Wesley. All rights reserved.

Reproduced by Pearson Addison-Wesley from electronic files supplied by the author. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley, 75 Arlington Street, Boston, MA 02116. 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, photocopying, recording, or otherwise, without the prior written permission of the publisher. Printed in the United States of America. ISBN 0-321-28066-0

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Copyright © 2006 Pearson Addison-Wesley. All rights reserved.

Brief Contents

INTRODUCTION HOW TO STUDY LINEAR ALGEBRA CHAPTER 1 CHAPTER 2 CHAPTER 3 CHAPTER 4 CHAPTER 5 CHAPTER 6 CHAPTER 7 APPENDICES LINEAR EQUATIONS IN LINEAR ALGEBRA MATRIX ALGEBRA DETERMINANTS VECTOR SPACES EIGENVALUES AND EIGENVECTORS ORTHOGONALITY AND LEAST SQUARES SYMMETRIC MATRICES TECHNOLOGY INDEX OF PROCEDURES AND TERMS INTRODUCTION TO MATLAB NOTES FOR THE MAPLE COMPUTER ALGEBRA SYSTEM NOTES FOR THE MATHEMATICA COMPUTER ALGEBRA SYSTEM NOTES FOR THE TI-83+/86/89 GRAPHIC CALCULATORS

NOTES FOR THE HP-48G GRAPHIC CALCULATOR

Copyright © 2006 Pearson Addison-Wesley. All rights reserved.

Contents

INTRODUCTION

Technology Support vii viii Review Materials on the Web

vii

HOW TO STUDY LINEAR ALGEBRA

Strategies for Success in Linear Algebra ix

ix

CHAPTER 1

LINEAR EQUATIONS IN LINEAR ALGEBRA

1.1 1.2 1.3 1.4 Systems of Linear Systems 1-1 1-7 Row Reduction and Echelon Forms Vector Equations 1-10 1-15

The Matrix Equation Ax = b

Mastering Linear Algebra Concepts: Span 1.5 1.6 1.7 Solution Sets of Linear Systems Applications of Linear Systems Linear Independence 1-29 1-20 1-25

Mastering Linear Algebra Concepts: Linear Independence 1.8 Introduction to Linear Transformations 1-33

Mastering Linear Algebra Concepts: Linear Transformation 1.9 Matrix of a Linear Transformation 1-37

Mastering Linear Algebra Concepts: Existence and Uniqueness 1.10 Linear Models in Business, Science, and Engineering Supplementary Exercises Glossary Checklist 1-47 1-46 1-42

Copyright © 2006 Pearson Addison-Wesley. All rights reserved.

CHAPTER 2

MATRIX ALGEBRA

2.1 2.2 2.3 Matrix Operations 2-1 2-4 2-9 The Inverse of a Matrix

Characterizations of Invertible Matrices Expanded Table for the IMT 2-10

Mastering Linear Algebra Concepts: Reviewing and Reflecting 2.4 Partitioned Matrices 2-14 2-20

The Principles of Induction 2.5 Matrix Factorizations

2-20 2-24 2-28 2-30

Permuted LU Factorizations 2.6 2.7 2.8

The Leontief Input-Output Model Applications to Computer Graphics Subspaces of Rn 2-33

Mastering Linear Algebra Concepts: Subspace, Column Space, Null Space, Basis 2.9 Dimension and Rank 2-38 2-39

Expanded Table for the IMT

Mastering Linear Algebra Concepts: Dimension and Rank Supplementary Exercises Glossary Checklist 2-42 2-42

CHAPTER

3

DETERMINANTS

3.1 3.2 3.3 Introduction to Determinants Properties of Determinants 3-1 3-5 3-8

Cramer’s Rule, Volume and Linear Transformations A Geometric Proof Glossary Checklist 3-12 3-13

CHAPTER 4

VECTOR SPACES

4.1 4.2 Vector Spaces and Subspaces 4-1 4-4 Null Spaces, Column Spaces, and Linear Transformations

Copyright © 2006 Pearson Addison-Wesley. All rights reserved.

Mastering Linear Algebra Concepts: Vector Space, Subspace, Col A and Nul A 4.3 Linearly Independent Sets; Bases 4-7

Mastering Linear Algebra Concepts: Basis 4.4 Coordinate Systems 4-11 4-12 4-16

Isomorphic Vector Spaces 4.5 4.6

The Dimension of a Vector Space Rank 4-19 4-21

Expanded Table for the IMT

Mastering Linear Algebra Concepts: Major Review of Key Concepts 4.7 4.8 Change of Basis 4-24 4-27

Applications to Difference Equations...

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