Elementary Linear Algebra, 6th edition

INDEX OF APPLICATIONS

BIOLOGY AND LIFE SCIENCES Calories burned, 117 Population

of deer, 43 of rabbits, 459 Population growth, 458? 461, 472, 476, 477 Reproduction rates of deer, 115 Spread of a virus, 112

BUSINESS AND ECONOMICS Average monthly cable television rates, 119 Basic cable and satellite television, 173 Cable television service, 99, 101 Consumer preference model, 99, 101, 174 Consumer Price Index, 119 Demand

for a certain grade of gasoline, 115 for a rechargeable power drill, 115 Economic system, 107 Industries, 114, 119 Market research, 112 Net profit Microsoft, 38 Polo Ralph Lauren, 335 Number of stores Target Corporation, 354 Production levels guitars, 59 vehicles, 59 Profit from crops, 59 Retail sales of running shoes, 354 Revenue eBay, Inc., 354 Google, Inc., 354 Sales, 43 Advanced Auto Parts, 334 Auto Zone, 334 Circuit City Stores, 355 Dell, Inc., 335 Gateway, Inc., 334 Wal-Mart, 39 Subscribers of a cellular communications company, 170 Total cost of manufacturing, 59

COMPUTERS AND COMPUTER SCIENCE Computer graphics, 410 ? 413, 415, 418 Computer operator, 142

ELECTRICAL ENGINEERING Current flow in networks, 33, 36, 37, 40, 44 Kirchhoff's Laws, 35, 36

MATHEMATICS Area of a triangle, 164, 169, 173 Collinear points, 165, 169 Conic sections and rotation, 265?270, 271?272, 275 Coplanar points, 167, 170 Equation

of a line, 165?166, 170, 174 of a plane, 167?168, 170, 174 Fourier approximations, 346?350, 351?352, 355 Linear differential equations in calculus, 262?265,

270 ?271, 274 ?275 Quadratic forms, 463? 471, 473, 476 Systems of linear differential equations, 461? 463,

472? 473, 476 Volume of a tetrahedron, 166, 170

MISCELLANEOUS Carbon dioxide emissions, 334 Cellular phone subscribers, 120 College textbooks, 170 Doctorate degrees, 334 Fertilizer, 119 Final grades, 118 Flow

of traffic, 39, 40 of water, 39 Gasoline, 117 Milk, 117 Motor vehicle registrations, 115 Network of pipes, 39 of streets, 39, 40

Population, 118, 472, 476, 480 of consumers, 112 of smokers and nonsmokers, 112 of the United States, 38

Projected population of the United States, 173 Regional populations, 60 Television viewing, 112 Voting population, 60 World population, 330

NUMERICAL LINEAR ALGEBRA

Adjoint of a matrix, 158?160, 168?169, 173 Cramer's Rule, 161?163, 169?170, 173 Cross product of two vectors in space, 336?341, 350 ?351,

355 Cryptography, 102, 113?114, 118?119 Geometry of linear transformations in the plane, 407? 410,

413?414, 418 Idempotent matrix, 98 Leontief input-output models, 105, 114, 119 LU-factorization, 93?98, 116?117 QR-factorization, 356?357 Stochastic matrices, 98, 118

PHYSICAL SCIENCES Astronomy, 332 Average monthly temperature, 43 Periods of planets, 31 World energy consumption, 354

SOCIAL AND BEHAVIORAL SCIENCES Sports

average salaries of Major League Baseball players, 120 average salary for a National Football League player,

354 basketball, 43 Fiesta Bowl Championship Series, 41 Super Bowl I, 43 Super Bowl XLI, 41 Test scores, 120 ?121

STATISTICS Least squares approximations, 341?346, 351, 355 Least squares regression analysis, 108, 114 ?115, 119?120

Elementary Linear Algebra

SIXTH EDITION

RON LARSON The Pennsylvania State University

The Behrend College DAVI D C. FALVO The Pennsylvania State University The Behrend College

HOUGHTON MIFFLIN HARCOURT PUBLISHING COMPANY Boston New York

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Copyright ? 2009 by Houghton Mifflin Harcourt Publishing Company. All rights reserved.

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Instructor's examination copy ISBN-13: 978-0-547-00481-5 ISBN-10: 0-547-00481-8

For orders, use student text ISBNs ISBN-13: 978-0-618-78376-2 ISBN-10: 0-618-78376-8

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Contents

A WORD FROM THE AUTHORS

vii

WHAT IS LINEAR ALGEBRA?

xv

CHAPTER 1 SYSTEMS OF LINEAR EQUATIONS

1

1.1 Introduction to Systems of Linear Equations

1

1.2 Gaussian Elimination and Gauss-Jordan Elimination

14

1.3 Applications of Systems of Linear Equations

29

Review Exercises

41

Project 1 Graphing Linear Equations

44

Project 2 Underdetermined and Overdetermined Systems of Equations

45

CHAPTER 2 MATRICES

46

2.1 Operations with Matrices

46

2.2 Properties of Matrix Operations

61

2.3 The Inverse of a Matrix

73

2.4 Elementary Matrices

87

2.5 Applications of Matrix Operations

98

Review Exercises

115

Project 1 Exploring Matrix Multiplication

120

Project 2 Nilpotent Matrices

121

iii

iv

Contents

CHAPTER 3 DETERMINANTS

122

3.1 The Determinant of a Matrix

122

3.2 Evaluation of a Determinant Using Elementary Operations

132

3.3 Properties of Determinants

142

3.4 Introduction to Eigenvalues

152

3.5 Applications of Determinants

158

Review Exercises

171

Project 1 Eigenvalues and Stochastic Matrices

174

Project 2 The Cayley-Hamilton Theorem

175

Cumulative Test for Chapters 1?3

177

CHAPTER 4 VECTOR SPACES

179

4.1 Vectors in Rn

179

4.2 Vector Spaces

191

4.3 Subspaces of Vector Spaces

198

4.4 Spanning Sets and Linear Independence

207

4.5 Basis and Dimension

221

4.6 Rank of a Matrix and Systems of Linear Equations

232

4.7 Coordinates and Change of Basis

249

4.8 Applications of Vector Spaces

262

Review Exercises

272

Project 1 Solutions of Linear Systems

275

Project 2 Direct Sum

276

CHAPTER 5 INNER PRODUCT SPACES

277

5.1 Length and Dot Product in Rn

277

5.2 Inner Product Spaces

292

5.3 Orthonormal Bases: Gram-Schmidt Process

306

5.4 Mathematical Models and Least Squares Analysis

320

5.5 Applications of Inner Product Spaces

336

Review Exercises

352

Project 1 The QR-Factorization

356

Project 2 Orthogonal Matrices and Change of Basis

357

Cumulative Test for Chapters 4 and 5

359

CHAPTER 6

6.1 6.2 6.3 6.4 6.5

LINEAR TRANSFORMATIONS

Introduction to Linear Transformations The Kernel and Range of a Linear Transformation Matrices for Linear Transformations Transition Matrices and Similarity Applications of Linear Transformations

Review Exercises Project 1 Reflections in the Plane (I) Project 2 Reflections in the Plane (II)

CHAPTER 7

7.1 7.2 7.3 7.4

EIGENVALUES AND EIGENVECTORS

Eigenvalues and Eigenvectors Diagonalization Symmetric Matrices and Orthogonal Diagonalization Applications of Eigenvalues and Eigenvectors

Review Exercises Project 1 Population Growth and Dynamical Systems (I) Project 2 The Fibonacci Sequence Cumulative Test for Chapters 6 and 7

CHAPTER 8

8.1 8.2 8.3 8.4 8.5

COMPLEX VECTOR SPACES (online)*

Complex Numbers Conjugates and Division of Complex Numbers Polar Form and DeMoivre's Theorem Complex Vector Spaces and Inner Products Unitary and Hermitian Matrices

Review Exercises Project Population Growth and Dynamical Systems (II)

Contents

v

361

361 374 387 399 407

416 419 420

421

421 435 446 458

474 477 478 479

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