Elementary Linear Algebra, 6th edition

1 BIOLOGY AND LIFE SCIENCESC alories burned, 117 Populationof deer, 43of rabbits, 459 Population growth, 458 461, 472, 476, 477 Reproduction rates of deer, 115 Spread of a virus, 112 business AND ECONOMICSA verage 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 Demandfor a certain grade of gasoline, 115for a rechargeable power drill, 115 Economic system, 107 Industries, 114, 119 Market research, 112 Net profitMicrosoft, 38 Polo Ralph Lauren, 335 Number of storesTarget Corporation, 354 Production levelsguitars, 59vehicles, 59 Profit from crops, 59 Retail sales of running shoes, 354 RevenueeBay, Inc.

2 , 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 SCIENCEC omputer graphics, 410 413, 415, 418 Computer operator, 142 ELECTRICAL ENGINEERINGC urrent flow in networks, 33, 36, 37, 40, 44 Kirchhoff s Laws, 35, 36 MATHEMATICSArea of a triangle, 164, 169, 173 Collinear points, 165, 169 Conic sections and rotation, 265 270, 271 272, 275 Coplanar points, 167, 170 Equationof a line, 165 166, 170, 174of 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.

3 476 Volume of a tetrahedron, 166, 170 MISCELLANEOUSC arbon dioxide emissions, 334 Cellular phone subscribers, 120 College textbooks, 170 Doctorate degrees, 334 Fertilizer, 119 Final grades, 118 Flowof traffic, 39, 40of water, 39 Gasoline, 117 Milk, 117 Motor vehicle registrations, 115 Networkof pipes, 39of streets, 39, 40 INDEX OF APPLICATIONSP opulation, 118, 472, 476, 480of consumers, 112of smokers and nonsmokers, 112of 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 ALGEBRAA djoint 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, 119LU-factorization, 93 98, 116 117QR-factorization, 356 357 Stochastic matrices, 98, 118 PHYSICAL SCIENCESA stronomy, 332 Average monthly temperature, 43 Periods of planets, 31 World energy consumption, 354 SOCIAL AND BEHAVIORAL SCIENCESS portsaverage salaries of Major League Baseball players, 120average salary for a National Football League player,354basketball, 43 Fiesta Bowl Championship Series, 41 Super Bowl I, 43 Super Bowl XLI, 41 Test scores.

4 120 121 STATISTICSL east squares approximations, 341 346, 351, 355 Least squares regression analysis, 108, 114 115, 119 120 Elementary Linear AlgebraRON LARSONThe Pennsylvania State UniversityThe Behrend CollegeDAVID C. FALVOThe Pennsylvania State UniversityThe Behrend CollegeSIXTH EDITIONHOUGHTON MIFFLIN HARCOURT PUBLISHING COMPANYB ostonNew YorkPublisher:Richard Stratton Senior Sponsoring Editor:Cathy CantinSenior Marketing Manager:Jennifer JonesDiscipline Product Manager:Gretchen Rice KingAssociate Editor:Janine Tangney Associate Editor:Jeannine LawlessSenior Project Editor:Kerry FalveyProgram Manager:Touraj ZadehSenior Media Producer:Douglas WinickiSenior Content Manager:Maren KunertArt and Design Manager: Jill HaberCover Design Manager:Anne S.

5 KatzeffSenior Photo Editor:Jennifer Meyer DareSenior Composition Buyer:Chuck DuttonNew Title Project Manager:Susan PeltierManager of New Title Project Management:Pat O Neill Editorial Assistant:Amy HainesMarketing Assistant:Michael Moore Editorial Assistant:Laura CollinsCover image: Carl Reader/age fotostockCopyright 2009 by Houghton Mifflin Harcourt Publishing Company. All rights 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 informationstorage or retrieval system without the prior written permission of Houghton MifflinHarcourt Publishing Company unless such copying is expressly permitted by federal copyright law.

6 Address inquiries to College Permissions, Houghton Mifflin HarcourtPublishing Company, 222 Berkeley Street, Boston, MA in the of Congress Control Number: 2007940572 Instructor s examination copyISBN-13: 978-0-547-00481-5 ISBN-10: 0-547-00481-8 For orders, use student text ISBNsISBN-13: 978-0-618-78376-2 ISBN-10: 0-618-78376-8123456789-DOC-12 11 10 09 08A WORD FROM THE AUTHORSviiWHAT IS Linear ALGEBRA?xvSYSTEMS OF Linear EQUATIONS1 Introduction to Systems of Linear Equations1 Gaussian Elimination and Gauss-Jordan Elimination14 Applications of Systems of Linear Equations29 Review Exercises41 Project 1 Graphing Linear Equations44 Project 2 Underdetermined and Overdetermined Systems of Equations 45 MATRICES 46 Operations with Matrices 46 Properties of Matrix Operations 61 The Inverse of a Matrix 73 Elementary Matrices 87 Applications of Matrix Operations 98 Review Exercises 115 Project 1 Exploring Matrix Multiplication 120 Project 2 Nilpotent Matrices 121iiiContentsCHAPTER 122 The Determinant of a Matrix 122 Evaluation of a Determinant Using Elementary Operations 132 Properties of Determinants 142 Introduction to

7 Eigenvalues 152 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 VECTOR SPACES 179 Vectors in Rn179 Vector Spaces 191 Subspaces of Vector Spaces 198 Spanning Sets and Linear Independence 207 Basis and Dimension 221 Rank of a Matrix and Systems of Linear Equations 232 Coordinates and Change of Basis 249 Applications of Vector Spaces 262 Review Exercises 272 Project 1 Solutions of Linear Systems 275 Project 2 Direct Sum 276 INNER PRODUCT SPACES 277 Length and Dot Product in Rn277 Inner Product Spaces 292 Orthonormal Bases: Gram-Schmidt Process 306 Mathematical Models and Least Squares Analysis 320 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 TRANSFORMATIONS 361 Introduction to Linear Transformations 361 The Kernel and Range of a Linear Transformation 374 Matrices for Linear Transformations 387 Transition Matrices and Similarity 399 Applications of Linear Transformations 407 Review Exercises 416 Project 1 Reflections in the Plane (I) 419 Project 2 Reflections in the Plane (II)

8 420 EIGENVALUES AND EIGENVECTORS 421 Eigenvalues and Eigenvectors 421 Diagonalization 435 Symmetric Matrices and Orthogonal Diagonalization 446 Applications of Eigenvalues and Eigenvectors 458 Review Exercises 474 Project 1 Population Growth and Dynamical Systems (I)477 Project 2 The Fibonacci Sequence 478 Cumulative Test for Chapters 6 and 7 479 COMPLEX VECTOR SPACES (online)*Complex NumbersConjugates and Division of Complex NumbersPolar Form and DeMoivre's TheoremComplex Vector Spaces and Inner ProductsUnitary and Hermitian MatricesReview ExercisesProjectPopulation Growth and Dynamical Systems (II)CHAPTER PROGRAMMING (online)*Systems of Linear InequalitiesLinear Programming Involving Two VariablesThe Simplex Method: MaximizationThe Simplex Method: MinimizationThe Simplex Method: Mixed ConstraintsReview ExercisesProjectCholesterol LevelsNUMERICAL METHODS (online)*Gaussian Elimination with Partial PivotingIterative Methods for Solving Linear SystemsPower Method for Approximating EigenvaluesApplications of Numerical MethodsReview ExercisesProjectPopulation GrowthMATHEMATICAL INDUCTION AND OTHERA1 FORMS OF PROOFSONLINE TECHNOLOGY GUIDE (online)*ANSWER KEYA9 INDEXA59 CHAPTER *Available online at !

9 We have designed Elementary Linear Algebra, Sixth edition , for the introductorylinear algebra embarking on a Linear algebra course should have a thorough knowledge ofalgebra, and familiarity with analytic geometry and trigonometry. We do not assume thatcalculus is a prerequisite for this course, but we do include examples and exercises requir-ing calculus in the text. These exercises are clearly labeled and can be omitted if students will encounter mathematical formalism for the first time in this a result, our primary goal is to present the major concepts of Linear algebra clearly andconcisely. To this end, we have carefully selected the examples and exercises to balancetheory with applications and geometrical order and coverage of topics were chosen for maximum efficiency, effectiveness,and balance.

10 For example, in Chapter 4 we present the main ideas of vector spaces andbases, beginning with a brief look leading into the vector space concept as a natural exten-sion of these familiar examples. This material is often the most difficult for students, butour approach to Linear independence, span, basis, and dimension is carefully explained andillustrated by examples. The eigenvalue problem is developed in detail in Chapter 7, but welay an intuitive foundation for students earlier in Section , Section , and Chapter online Chapters 8, 9, and 10 cover complex vector spaces, Linear program-ming, and numerical methods. They can be found on the student website for this text at read on to learn more about the features of the Sixth hope you enjoy this new edition of Elementary Linear Word from the AuthorsAcknowledgmentsWe would like to thank the many people who have helped us during various stages of theproject.