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.
No part of this work may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying and recording, or by any information storage or retrieval system without the prior written permission of Houghton Mifflin Harcourt Publishing Company unless such copying is expressly permitted by federal copyright law. Address inquiries to College Permissions, Houghton Mifflin Harcourt Publishing Company, 222 Berkeley Street, Boston, MA 02116-3764.
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Library of Congress Control Number: 2007940572
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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