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Linear Algebra with Applications This page intentionally left blank Linear Algebra with Applications Fifth Edition Otto Bretscher Colby College EditorinChief:ChristineHoag CoverDesigner:SuzanneDuda SeniorAcquisitionsEditor:WilliamHoffman CoverArt:ColorizationbyJorgensen ExecutiveMarketingManager:JeffWeidenaar Fernandez/NASA MarketingAssistant:CaitlinCrain Full-ServiceProjectManagement: SeniorProductionProjectManager: IntegraSoftwareServices,Ltd. BethHouston Composition:IntegraSoftwareServices,Ltd. Manager,CoverVisual Printer/Binder:EdwardsBrothersMalloy ResearchandPermissions:JayneConte CoverPrinter:Lehigh/Phoenix The cover shows the Mars rover Curiosity, casting a long shadow onto Gale crater, facing Aeolis Mons. Linear Algebraplaysacentralroleinmanyaspectsoftheplanning,design,andcontrolofaspacemission.Forexample, datacompressionisusedforinterplanetarycommunication(seePage411),anderror-correctioncodesincreasethe reliabilityofdatatransmission(seePage121). Creditsandacknowledgmentsborrowedfromothersourcesandreproduced,withpermission,inthistextbookappear ontheappropriatepagewithintext. Copyright©2013,2009,2005byPearsonEducation,Inc.Allrightsreserved.ManufacturedintheUnitedStatesof America.ThispublicationisprotectedbyCopyright,andpermissionshouldbeobtainedfromthepublisherpriorto anyprohibitedreproduction,storageinaretrievalsystem,ortransmissioninanyformorbyanymeans,electronic, mechanical, photocopying, recording, or likewise. To obtain permission(s) to use material from this work, please submitawrittenrequesttoPearsonEducation,Inc.,PermissionsDepartment,OneLakeStreet,UpperSaddleRiver, NewJersey07458,oryoumayfaxyourrequestto201-236-3290. Many of the designations by manufacturers and sellers to distinguish their products are claimed as trademarks. Wherethosedesignationsappearinthisbook,andthepublisherwasawareofatrademarkclaim,thedesignations havebeenprintedininitialcapsorallcaps. LibraryofCongressCataloging-in-PublicationData Bretscher,Otto. Linearalgebrawithapplications/OttoBretscher.–5thed. p.cm. Includesindex. ISBN-13:978-0-321-79697-4 ISBN-10:0-321-79697-7 1. Algebras,Linear–Textbooks.I.Title. QA184.2.B732013 512’.5–dc23 2012017551 10 9 8 7 6 5 4 3 2 1 EBM 16 15 14 13 12 ISBN-10: 0-321-79697-7 ISBN-13: 978-0-321-79697-4 To my parents Otto and Margrit Bretscher-Zwicky with love and gratitude This page intentionally left blank Contents Preface ix 1 Linear Equations 1 1.1 Introduction toLinearSystems 1 1.2 Matrices,Vectors,andGauss–JordanElimination 8 1.3 OntheSolutionsofLinearSystems;MatrixAlgebra 25 2 Linear Transformations 41 2.1 Introduction toLinearTransformationsandTheir Inverses 41 2.2 LinearTransformationsinGeometry 58 2.3 MatrixProducts 75 2.4 TheInverseofaLinearTransformation 88 3 Subspaces of Rn and Their Dimensions 110 3.1 ImageandKernelofaLinearTransformation 110 3.2 SubspacesofRn;BasesandLinearIndependence 122 3.3 TheDimensionofaSubspaceofRn 133 3.4 Coordinates 147 4 Linear Spaces 166 4.1 Introduction toLinearSpaces 166 4.2 LinearTransformationsandIsomorphisms 178 4.3 TheMatrixofaLinearTransformation 186 5 Orthogonality and Least Squares 202 5.1 Orthogonal ProjectionsandOrthonormal Bases 202 5.2 Gram–Schmidt ProcessandQRFactorization 218 5.3 Orthogonal TransformationsandOrthogonal Matrices 225 5.4 LeastSquaresandDataFitting 236 5.5 InnerProductSpaces 249 vii viii Contents 6 Determinants 265 6.1 Introduction toDeterminants 265 6.2 PropertiesoftheDeterminant 277 6.3 GeometricalInterpretations oftheDeterminant; Cramer’sRule 294 7 Eigenvalues and Eigenvectors 310 7.1 Diagonalization 310 7.2 FindingtheEigenvaluesofaMatrix 327 7.3 FindingtheEigenvectorsofaMatrix 339 7.4 MoreonDynamicalSystems 347 7.5 ComplexEigenvalues 360 7.6 Stability 375 8 Symmetric Matrices and Quadratic Forms 385 8.1 SymmetricMatrices 385 8.2 QuadraticForms 394 8.3 SingularValues 403 9 Linear Differential Equations 415 9.1 AnIntroduction toContinuousDynamicalSystems 415 9.2 TheComplex Case:Euler’sFormula 429 9.3 LinearDifferentialOperatorsandLinearDifferential Equations 442 AppendixAVectors 457 AppendixBTechniquesofProof 467 AnswerstoOdd-NumberedExercises 471 SubjectIndex 499 NameIndex 507 Preface (with David Steinsaltz) A police officer on patrol at midnight, so runs an old joke, notices a man crawling about on his hands and knees under a streetlamp. He walks over toinvestigate,whereuponthemanexplainsinatiredandsomewhatslurred voicethathehaslosthishousekeys.Thepolicemanofferstohelp,andforthenext five minutes he too is searching on his hands and knees. At last he exclaims, “Are youabsolutelycertainthatthisiswhereyoudroppedthekeys?” “Here? Absolutely not. I dropped them a block down, in the middle of the street.” “Thenwhythedevilhaveyougotmehuntingaroundthislamppost?” “Becausethisiswherethelightis.” Itismathematics,andnotjust(asBismarckclaimed)politics,thatconsistsin“the artofthepossible.”Ratherthansearchinthedarknessforsolutionstoproblemsof pressing interest, we contrive a realm of problems whose interest lies above all in thefactthatsolutionscanconceivablybefound. Perhapsthelargestpatchoflightsurroundsthetechniquesofmatrixarithmetic and algebra, and in particular matrix multiplication and row reduction. Here we mightbeginwithDescartes,sinceitwashewhodiscoveredtheconceptualmeeting- pointofgeometryandalgebraintheidentificationofEuclideanspacewithR3;the techniquesandapplicationsproliferatedsincehisday.Toorganizeandclarifythose istheroleofamodernlinearalgebracourse. Computers and Computation Anessentialissuethatneedstobeaddressedinestablishingamathematicalmethod- ologyistheroleofcomputationandofcomputingtechnology.Arethepropersub- jects of mathematics algorithms and calculations, or are they grand theories and abstractionsthatevadetheneedforcomputation?Iftheformer,isitimportantthat thestudentslearntocarryoutthecomputationswithpencilandpaper,orshouldthe algorithm“pressthecalculator’s x−1 button”beallowedtosubstituteforthetradi- tionalmethodoffindinganinverse?Ifthelatter,shouldtheabstractions betaught through elaborate notational mechanisms or through computational examples and graphs? Weseektotakeaconsistentapproachtothesequestions:Algorithmsandcom- putations are primary, and precisely for this reason computers are not. Again and again we examine the nitty-gritty of row reduction or matrix multiplication in or- dertoderivenewinsights.Mostoftheproofs,whetherofrank-nullitytheorem,the volume-change formula for determinants, or the spectral theorem for symmetric matrices,areinthiswaytiedtohands-onprocedures. The aim is not just to know how to compute the solution to a problem, but to imaginethecomputations.Thestudentneedstoperformenoughrowreductionsby handtobeequipped tofollow alineofargument oftheform: “Ifwecalculate the reducedrow-echelonformofsuchamatrix...,”andtoappreciateinadvancethe possibleoutcomesofaparticularcomputation. ix