LinearAlgebraAndItsApplications3rdEditionStudyGuideLay

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L

INEAR

A

LGEBRA

AND

I

TS

A

PPLICATIONS

T

HIRD

E

DITION

U

PDATE

David C. Lay

University of Maryland – College Park

S

TUDY

G

UIDE

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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

1 2 3 4 5 6 BB 08 07 06 05

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Brief Contents

INTRODUCTION

HOW TO STUDY LINEAR ALGEBRA

C H A P T E R 1

LINEAR EQUATIONS IN LINEAR ALGEBRA

C H A P T E R 2

MATRIX ALGEBRA

C H A P T E R 3

DETERMINANTS

C H A P T E R 4

VECTOR SPACES

C H A P T E R 5

EIGENVALUES AND EIGENVECTORS

C H A P T E R 6

ORTHOGONALITY AND LEAST SQUARES

C H A P T E R 7

SYMMETRIC MATRICES

A P P E N D I C E S

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

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INTRODUCTION vii

Technology Support vii

Review Materials on the Web viii

HOW TO STUDY LINEAR ALGEBRA ix

Strategies for Success in Linear Algebra ix

C H A P T E R

1 LINEAR EQUATIONS IN LINEAR ALGEBRA

1.1 Systems of Linear Systems 1-1

1.2 Row Reduction and Echelon Forms 1-7

1.3 Vector Equations 1-10

1.4 The Matrix Equation Ax = b

1-15

Mastering Linear Algebra Concepts: Span

1.5 Solution Sets of Linear Systems 1-20

1.6 Applications of Linear Systems 1-25

1.7 Linear Independence 1-29

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 1-42

Supplementary Exercises 1-46

Glossary Checklist 1-47

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C H A P T E R

2 MATRIX ALGEBRA

2.1 Matrix Operations 2-1

2.2 The Inverse of a Matrix 2-4

2.3 Characterizations of Invertible Matrices 2-9

Expanded Table for the IMT 2-10

Mastering Linear Algebra Concepts: Reviewing and Reflecting

2.4 Partitioned Matrices 2-14

The Principles of Induction 2-20

2.5 Matrix Factorizations 2-20

Permuted LU Factorizations 2-24

2.6 The Leontief Input-Output Model 2-28

2.7 Applications to Computer Graphics 2-30

2.8 Subspaces of Rn

2-33

Mastering Linear Algebra Concepts: Subspace, Column Space, Null Space, Basis

2.9 Dimension and Rank 2-38

Expanded Table for the IMT

2-39

Mastering Linear Algebra Concepts: Dimension and Rank

Supplementary Exercises

2-42

Glossary Checklist 2-42

C H A P T E R

3 DETERMINANTS

3.1 Introduction to Determinants 3-1

3.2 Properties of Determinants 3-5

3.3 Cramer’s Rule, Volume and Linear Transformations 3-8

A Geometric Proof 3-12

Glossary Checklist 3-13

C H A P T E R

4 VECTOR SPACES

4.1 Vector Spaces and Subspaces 4-1

4.2 Null Spaces, Column Spaces, and Linear Transformations 4-4

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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

Isomorphic Vector Spaces

4-12

4.5 The Dimension of a Vector Space 4-16

4.6 Rank 4-19

Expanded Table for the IMT

4-21

Mastering Linear Algebra Concepts: Major Review of Key Concepts

4.7 Change of Basis 4-24

4.8 Applications to Difference Equations 4-27

The Casorati Test

4-31

4.9 Applications to Markov Chains 4-32

Glossary Checklist 4-35

C H A P T E R

5 EIGENVALUES AND EIGENVECTORS

5.1 Eigenvectors and Eigenvalues 5-1

5.2 The Characteristic Equation 5-5

Factoring a Polynomial

5-8

5.3 Diagonalization 5-9

Mastering Linear Algebra Concepts: Eigenvalue, Eigenvector, Eigenspace

5.4 Eigen

values and Linear Transformations 5-16

5.5 Complex Eigenvalues 5-19

5.6 Discrete Dynamical Systems 5-22

5.7 Applications to Differential Equations 5-26

5.8 Iterative Estimates for Eigenvalues 5-29

Glossary Checklist 5-34

C H A P T E R

6 ORTHOGONALITY AND LEAST SQUARES

6.1 Inner Product, Length, and Orthogonality 6-1

6.2 Orthogonal Sets 6-2

Mastering Linear Algebra Concepts: Orthogonal Basis

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6.3 Orthogonal Projections 6-5

6.4 The Gram-Schmidt Process 6-9

6.5 Least-Squares Problems 6-12

6.6 Applications to Linear Models 6-16

The Geometrey of a Linear Model

6-19

6.7 Inner Product Spaces 6-20

6.8 Applications of Inner Product Spaces 6-22

The Linearity of an Orthoganal Projection

6-25

Glossary Checklist 6-26

C H A P T E R

7 SYMMETRIC MATRICES

7.1 Diagonalization of Symmetric Matrices 7-1

7.2 Quadratic Forms 7-6

Mastering Linear Algebra Concepts: Diagonalization and Quadratic Forms

7.3 Constrained Optimization 7-8

7.4 The Singular Value Decomposition 7-10

Computing an SVD

7-11

7.5 Applications to Image Processing and Statistics 7-16

Supplementary Exercises 7-19

Glossary Checklist 7-19

A P P E N D I C E S

TECHNOLOGY INDEX OF PROCEDURES AND TERMS

Technology Index of Procedures and Terms TECH-1

INTRODUCTION TO MATLAB

Getting Started with Matlab ML-1

Script M-Files ML-3

Index of Matlab Commands ML-5

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NOTES FOR THE MAPLE COMPUTER

ALGEBRA SYSTEM

Getting Started with Maple MP-1

Study Guide Notes MP-4

Index of Maple Commands MP-22

NOTES FOR THE MATHEMATICA COMPUTER

ALGEBRA SYSTEM

Getting Started with Mathematica MM-1

Study Guide Notes MM-5

Index of Mathematica Commands MM-24

NOTES FOR THE TI-83+/86/89

GRAPHIC CALCULATORS

Getting Started with a TI-83+ Calculator TI-1

Getting Started with a TI-86 Calculator TI-2

Getting Started with a TI-89 Calculator TI-3

Study Guide Notes TI-4

Index of TI Calculator Commands TI-29

NOTES FOR THE HP-48G GRAPHIC CALCULATOR

Getting Started with an HP-48G Calculator HP-1

Study Guide Notes HP-2

Index of HP Calculator Commands HP-18

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