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Linear Algebra and Its Applications SIXTH EDITION
David C. Lay • Steven R. Lay • Judi J. McDonald
LinearAlgebra andItsApplications GLOBALEDITION
DavidC.Lay
UniversityofMaryland–CollegePark
StevenR.Lay
LeeUniversity
JudiJ.McDonald
WashingtonStateUniversity
Pearson Education Limited
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© Pearson Education Limited, 2022
The rights of David C. Lay, Steven R. Lay, and Judi J. McDonald to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.
AuthorizedadaptationfromtheUnitedStatesedition,entitledLinear Algebra and Its Applications,6thEdition,ISBN 978-0-13-585125-8byDavidC.Lay,StevenR.Lay,andJudiJ.McDonald , publishedbyPearsonEducation©2021.
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A catalogue record for this book is available from the British Library
ISBN 10: 1-292-35121-7
ISBN 13: 978-1-292-35121-6
eBook ISBN 13: 978-1-292-35122-3
Tomywife,Lillian,andourchildren, Christina,Deborah,andMelissa,whose support,encouragement,andfaithful prayersmadethisbookpossible.
AbouttheAuthors DavidC.Lay AsafoundingmemberoftheNSF-sponsoredLinearAlgebraCurriculumStudyGroup (LACSG),DavidLaywasaleaderinthemovementtomodernizethelinearalgebra curriculumandsharedthoseideaswithstudentsandfacultythroughhisauthorship ofthefirstfoureditionsofthistextbook.DavidC.LayearnedaB.A.fromAurora University(Illinois),andanM.A.andPh.D.fromtheUniversityofCaliforniaatLos Angeles.DavidLaywasaneducatorandresearchmathematicianformorethan40 years,mostlyattheUniversityofMaryland,CollegePark.Healsoservedasavisiting professorattheUniversityofAmsterdam,theFreeUniversityinAmsterdam,andthe UniversityofKaiserslautern,Germany.Hepublishedmorethan30researcharticleson functionalanalysisandlinearalgebra.Laywasalsoacoauthorofseveralmathematics texts,including IntroductiontoFunctionalAnalysis withAngusE.Taylor, Calculus andItsApplications,withL.J.GoldsteinandD.I.Schneider,and LinearAlgebra Gems—AssetsforUndergraduateMathematics,withD.Carlson,C.R.Johnson,and A.D.Porter.
DavidLayreceivedfouruniversityawardsforteachingexcellence,including,in 1996,thetitleofDistinguishedScholar-TeacheroftheUniversityofMaryland.In1994, hewasgivenoneoftheMathematicalAssociationofAmerica’sAwardsforDistinguishedCollegeorUniversityTeachingofMathematics.HewaselectedbytheuniversitystudentstomembershipinAlphaLambdaDeltaNationalScholasticHonorSociety andGoldenKeyNationalHonorSociety.In1989,AuroraUniversityconferredonhim theOutstandingAlumnusaward.DavidLaywasamemberoftheAmericanMathematicalSociety,theCanadianMathematicalSociety,theInternationalLinearAlgebra Society,theMathematicalAssociationofAmerica,SigmaXi,andtheSocietyforIndustrialandAppliedMathematics.Healsoservedseveraltermsonthenationalboardof theAssociationofChristiansintheMathematicalSciences.
InOctober2018,DavidLaypassedaway,buthislegacycontinuestobenefitstudents oflinearalgebraastheystudythesubjectinthiswidelyacclaimedtext.
StevenR.Lay StevenR.LaybeganhisteachingcareeratAuroraUniversity(Illinois)in1971,after earninganM.A.andaPh.D.inmathematicsfromtheUniversityofCaliforniaatLos Angeles.Hiscareerinmathematicswasinterruptedforeightyearswhileservingasa missionaryinJapan.UponhisreturntotheStatesin1998,hejoinedthemathematics facultyatLeeUniversity(Tennessee)andhasbeenthereeversince.Sincethenhehas supportedhisbrotherDavidinrefiningandexpandingthescopeofthispopularlinear algebratext,includingwritingmostofChapters8and9.Stevenisalsotheauthorofthree college-levelmathematicstexts: ConvexSetsandTheirApplications,Analysiswithan IntroductiontoProof,andPrinciplesofAlgebra.
In1985,StevenreceivedtheExcellenceinTeachingAwardatAuroraUniversity. HeandDavid,andtheirfather,Dr.L.ClarkLay,arealldistinguishedmathematicians, andin1989,theyjointlyreceivedtheOutstandingAlumnusawardfromtheiralma mater,AuroraUniversity.In2006,StevenwashonoredtoreceivetheExcellencein ScholarshipAwardatLeeUniversity.HeisamemberoftheAmericanMathematical Society,theMathematicsAssociationofAmerica,andtheAssociationofChristiansin theMathematicalSciences.
JudiJ.McDonald JudiJ.McDonaldbecameaco-authoronthefifthedition,havingworkedcloselywith Davidonthefourthedition.SheholdsaB.Sc.inMathematicsfromtheUniversity ofAlberta,andanM.A.andPh.D.fromtheUniversityofWisconsin.Asaprofessor ofMathematics,shehasmorethan40publicationsinlinearalgebraresearchjournals andmorethan20studentshavecompletedgraduatedegreesinlinearalgebraunderher supervision.SheisanassociatedeanoftheGraduateSchoolatWashingtonStateUniversityandaformerchairoftheFacultySenate.Shehasworkedwiththemathematics outreachprojectMathCentral(http://mathcentral.uregina.ca/)andisamemberofthe secondLinearAlgebraCurriculumStudyGroup(LACSG2.0).
Judihasreceivedthreeteachingawards:twoInspiringTeachingawardsattheUniversityofRegina,andtheThomasLutzCollegeofArtsandSciencesTeachingAwardat WashingtonStateUniversity.ShealsoreceivedtheCollegeofArtsandSciencesInstitutionalServiceAwardatWashingtonStateUniversity.Throughouthercareer,shehas beenanactivememberoftheInternationalLinearAlgebraSocietyandtheAssociation forWomeninMathematics.ShehasalsobeenamemberoftheCanadianMathematical Society,theAmericanMathematicalSociety,theMathematicalAssociationofAmerica, andtheSocietyforIndustrialandAppliedMathematics.
AbouttheAuthors 3
Preface 12
ANotetoStudents 22
Chapter1 LinearEquationsinLinearAlgebra 25
INTRODUCTORYEXAMPLE: LinearModelsinEconomics andEngineering 25
1.1SystemsofLinearEquations 26
1.2RowReductionandEchelonForms 37
1.3VectorEquations 50
1.4TheMatrixEquation Ax D b 61
1.5SolutionSetsofLinearSystems 69
1.6ApplicationsofLinearSystems 77
1.7LinearIndependence 84
1.8IntroductiontoLinearTransformations 91
1.9TheMatrixofaLinearTransformation 99
1.10LinearModelsinBusiness,Science,andEngineering 109 Projects 117
SupplementaryExercises 117
Chapter2 MatrixAlgebra 121
INTRODUCTORYEXAMPLE: ComputerModelsinAircraftDesign 121
2.1MatrixOperations 122
2.2TheInverseofaMatrix 135
2.3CharacterizationsofInvertibleMatrices 145
2.4PartitionedMatrices 150
2.5MatrixFactorizations 156
2.6TheLeontiefInput–OutputModel 165
2.7ApplicationstoComputerGraphics 171
2.8Subspacesof Rn 179
2.9DimensionandRank 186 Projects 193
SupplementaryExercises 193
Chapter3 Determinants 195 INTRODUCTORYEXAMPLE: WeighingDiamonds 195
3.1IntroductiontoDeterminants 196
3.2PropertiesofDeterminants 203
3.3Cramer’sRule,Volume,andLinearTransformations 212 Projects 221
SupplementaryExercises 221
Chapter4 VectorSpaces 225 INTRODUCTORYEXAMPLE: Discrete-TimeSignalsandDigital SignalProcessing 225
4.1VectorSpacesandSubspaces 226
4.2NullSpaces,ColumnSpaces,RowSpaces,andLinear Transformations 235
4.3LinearlyIndependentSets;Bases 246
4.4CoordinateSystems 255
4.5TheDimensionofaVectorSpace 265
4.6ChangeofBasis 273
4.7DigitalSignalProcessing 279
4.8ApplicationstoDifferenceEquations 286 Projects 295
SupplementaryExercises 295
Chapter5 EigenvaluesandEigenvectors 297 INTRODUCTORYEXAMPLE: DynamicalSystemsandSpottedOwls 297
5.1EigenvectorsandEigenvalues 298
5.2TheCharacteristicEquation 306
5.3Diagonalization 314
5.4EigenvectorsandLinearTransformations 321
5.5ComplexEigenvalues 328
5.6DiscreteDynamicalSystems 335
5.7ApplicationstoDifferentialEquations 345
5.8IterativeEstimatesforEigenvalues 353
5.9ApplicationstoMarkovChains 359 Projects 369
SupplementaryExercises 369
Chapter6 OrthogonalityandLeastSquares 373
INTRODUCTORYEXAMPLE: ArtificialIntelligenceandMachine Learning 373
6.1InnerProduct,Length,andOrthogonality 374
6.2OrthogonalSets 382
6.3OrthogonalProjections 391
6.4TheGram–SchmidtProcess 400
6.5Least-SquaresProblems 406
6.6MachineLearningandLinearModels 414
6.7InnerProductSpaces 423
6.8ApplicationsofInnerProductSpaces 431 Projects 437
SupplementaryExercises 438
Chapter7 SymmetricMatricesandQuadraticForms 441
INTRODUCTORYEXAMPLE: MultichannelImageProcessing 441
7.1DiagonalizationofSymmetricMatrices 443
7.2QuadraticForms 449
7.3ConstrainedOptimization 456
7.4TheSingularValueDecomposition 463
7.5ApplicationstoImageProcessingandStatistics 473 Projects 481
SupplementaryExercises 481
Chapter8 TheGeometryofVectorSpaces 483
INTRODUCTORYEXAMPLE: ThePlatonicSolids 483
8.1AffineCombinations 484
8.2AffineIndependence 493
8.3ConvexCombinations 503
8.4Hyperplanes 510
8.5Polytopes 519
8.6CurvesandSurfaces 531 Project 542
SupplementaryExercises 543
Chapter9 Optimization 545
INTRODUCTORYEXAMPLE: TheBerlinAirlift 545
9.1MatrixGames 546
9.2LinearProgramming GeometricMethod 560
9.3LinearProgramming SimplexMethod 570
9.4Duality 585 Project 594
SupplementaryExercises 594
Chapter10 Finite-StateMarkovChains C-1 (AvailableOnline)
INTRODUCTORYEXAMPLE: GooglingMarkovChains C-1
10.1IntroductionandExamples C-2
10.2TheSteady-StateVectorandGoogle’sPageRank C-13
10.3CommunicationClasses C-25
10.4ClassificationofStatesandPeriodicity C-33
10.5TheFundamentalMatrix C-42
10.6MarkovChainsandBaseballStatistics C-54
Appendixes
AUniquenessoftheReducedEchelonForm 597 BComplexNumbers 599
Credits 604
Glossary 605
AnswerstoOdd-NumberedExercises A-1
Index I-1
ApplicationsIndex BiologyandEcology
Estimatingsystolicbloodpressure,422
Laboratoryanimaltrials,367
Molecularmodeling,173–174
Netprimaryproductionofnutrients, 418–419
Nutritionproblems,109–111,115
Predator-preysystem,336–337,343
Spottedowlsandstage-matrixmodels, 297–298,341–343
BusinessandEconomics
Accelerator-multipliermodel,293
Averagecostcurve,418–419
Carrentalfleet,116,368
Costvectors,57
Equilibriumprices,77–79,82
Exchangetable,82
Feasibleset,460,562
Grossdomesticproduct,170
Indifferencecurves,460–461
Intermediatedemand,165
Investment,294
Leontiefexchangemodel,25,77–79
Leontiefinput–outputmodel,25, 165–171
Linearprogramming,26,111–112,153, 484,519,522,560–566
Loanamortizationschedule,293
Manufacturingoperations,57,96
Marginalpropensitytoconsume,293
Markovchains,311,359–368,C-1–C-63
Maximizingutilitysubjecttoabudget constraint,460–461
Populationmovement,113,115–116, 311,361
Priceequation,170
Totalcostcurve,419
Valueaddedvector,170
Variablecostmodel,421
ComputersandComputerScience
Béziercurvesandsurfaces,509,531–532
CAD,537,541
Colormonitors,178
Computergraphics,122,171–177, 498–500
Craysupercomputer,153
Datastorage,66,163
Error-detectinganderror-correcting codes,447,471
Gametheory,519
High-endcomputergraphicsboards,176 Homogeneouscoordinates,172–173,174
Parallelprocessing,25,132
Perspectiveprojections,175–176
Vectorpipelinearchitecture,153 Virtualreality,174
VLSImicrochips,150
Wire-framemodels,121,171
ControlTheory
Controllablesystem,296
Controlsystemsengineering,155
Decoupledsystem,340,346,349
Deepspaceprobe,155
State-spacemodel,296,335
Steady-stateresponse,335
Transferfunction(matrix),155
ElectricalEngineering
Branchandloopcurrents,111–112 Circuitdesign,26,160
Currentflowinnetworks,111–112, 115–116
Discrete-timesignals,228,279–280
Inductance-capacitancecircuit,242
Kirchhoff’slaws,161
Laddernetwork,161,163–164 Laplacetransforms,155,213 Linearfilters,287–288
Low-passfilter,289,413
Minimalrealization,162 Ohm’slaw,111–113,161 RCcircuit,346–347
RLCcircuit,254
Seriesandshuntcircuits,161
Transfermatrix,161–162,163
Engineering
Aircraftperformance,422,437
BoeingBlendedWingBody,122
Cantileveredbeam,293
CFDandaircraftdesign,121–122
Deflectionofanelasticbeam,137,144
Deformationofamaterial,482
Equilibriumtemperatures,36,116–117, 193
Feedbackcontrols,519
Flexibilityandstiffnessmatrices,137, 144
Heatconduction,164
Imageprocessing,441–442,473–474, 479
LUfactorizationandairflow,122
Movingaveragefilter,293
Superpositionprinciple,95,98,112
Mathematics
Areaandvolume,195–196,215–217
Attractors/repellersinadynamical system,338,341,343,347,351
Bessel’sinequality,438
Bestapproximationinfunctionspaces, 426–427
Cauchy-Schwarzinequality,427
Conicsectionsandquadraticsurfaces, 481
Differentialequations,242,345–347 Fourierseries,434–436
Hermitepolynomials,272 Hypercube,527–529
Interpolatingpolynomials,49,194
Isomorphism,188,260–261
Jacobianmatrix,338
Laguerrepolynomials,272
Laplacetransforms,155,213
Legendrepolynomials,430
Lineartransformationsincalculus,241, 324–325
Simplex,525–527
Splines,531–534,540–541
Triangleinequality,427
Trigonometricpolynomials,434
NumericalLinearAlgebra
Bandmatrix,164
Blockdiagonalmatrix,153,334
Choleskyfactorization,454–455,481
Companionmatrix,371
Conditionnumber,147,149,211,439, 469
Effectiverank,190,271,465
Floatingpointarithmetic,33,45,221
Fundamentalsubspaces,379,439, 469–470
Givensrotation,119 Grammatrix,482 Gram–Schmidtprocess,405 Hilbertmatrix,149
Householderreflection,194 Ill-conditionedmatrix(problem),147 Inversepowermethod,356–357 Iterativemethods,353–359
Jacobi’smethodforeigenvalues,312 LAPACK,132,153
Large-scaleproblems,119,153 LUfactorization,157–158,162–163,164
Operationcounts,142,158,160,206
Outerproducts,133,152
Parallelprocessing,25 Partialpivoting,42,163
Polardecomposition,482 Powermethod,353–356 Powersofamatrix,129 Pseudoinverse,470,482
QRalgorithm,312–313,357
QRfactorization,403–404,405,413,438
Rank-revealingfactorization,163,296, 481
Rayleighquotient,358,439
Relativeerror,439
Schurcomplement,154 Schurfactorization,439
Singularvaluedecomposition,163, 463–473
Sparsematrix,121,168,206
Spectraldecomposition,446–447
Spectralfactorization,163
Tridiagonalmatrix,164 Vandermondematrix,194,371
Vectorpipelinearchitecture,153
PhysicalSciences
Cantileveredbeam,293 Centerofgravity,60 Chemicalreactions,79,83
Crystallattice,257,263
Decomposingaforce,386
Gaussianelimination,37 Hooke’slaw,137
Interpolatingpolynomial,49,194
Kepler’sfirstlaw,422
Landsatimage,441–442
Linearmodelsingeologyandgeography, 419–420
Massestimatesforradioactive substances,421
Mass-springsystem,233,254
Modelforglacialcirques,419
ModelforsoilpH,419
Paulispinmatrices,194
Periodicmotion,328
Quadraticformsinphysics,449–454
Radardata,155
Seismicdata,25
Spaceprobe,155
Steady-stateheatflow,36,164
Superpositionprinciple,95,98,112
Three-momentequation,293
Trafficflow,80
Trendsurface,419
Weather,367
Windtunnelexperiment,49
Statistics
Analysisofvariance,408,422
Covariance,474–476,477,478,479
Fullrank,465
Least-squareserror,409
Least-squaresline,413,414–416
Linearmodelinstatistics,414–420
Markovchains,359–360
Mean-deviationformfordata,417,475
Moore-Penroseinverse,471
Multichannelimageprocessing, 441–442,473–479
Multipleregression,419–420
Orthogonalpolynomials,427
Orthogonalregression,480–481
Powersofamatrix,129
Principalcomponentanalysis,441–442, 476–477
Quadraticformsinstatistics,449
Regressioncoefficients,415
Sumsofsquares(inregression),422, 431–432
Trendanalysis,433–434
Variance,422,475–476
Weightedleast-squares,424,431–432
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Preface Theresponseofstudentsandteacherstothefirstfiveeditionsof LinearAlgebraandIts Applications hasbeenmostgratifying.This SixthEdition providessubstantialsupport bothforteachingandforusingtechnologyinthecourse.Asbefore,thetextprovides amodernelementaryintroductiontolinearalgebraandabroadselectionofinteresting classicalandleading-edgeapplications.Thematerialisaccessibletostudentswiththe maturitythatshouldcomefromsuccessfulcompletionoftwosemestersofcollege-level mathematics,usuallycalculus.
Themaingoalofthetextistohelpstudentsmasterthebasicconceptsandskillsthey willuselaterintheircareers.Thetopicsherefollowtherecommendationsoftheoriginal LinearAlgebraCurriculumStudyGroup(LACSG),whichwerebasedonacareful investigationoftherealneedsofthestudentsandaconsensusamongprofessionalsin manydisciplinesthatuselinearalgebra.IdeasbeingdiscussedbythesecondLinear AlgebraCurriculumStudyGroup(LACSG2.0)havealsobeenincluded.Wehopethis coursewillbeoneofthemostusefulandinterestingmathematicsclassestakenby undergraduates.
What’sNewinThisEdition The SixthEdition hasexcitingnewmaterial,examples,andonlineresources.Aftertalkingwithhigh-techindustryresearchersandcolleaguesinappliedareas,weaddednew topics,vignettes,andapplicationswiththeintentionofhighlightingforstudentsand facultythelinearalgebraicfoundationalmaterialformachinelearning,artificialintelligence,datascience,anddigitalsignalprocessing.
ContentChanges • Sincematrixmultiplicationisahighlyusefulskill,weaddednewexamplesinChapter2toshowhowmatrixmultiplicationisusedtoidentifypatternsandscrubdata. Correspondingexerciseshavebeencreatedtoallowstudentstoexploreusingmatrix multiplicationinvariousways.
• Inourconversationswithcolleaguesinindustryandelectricalengineering,weheard repeatedlyhowimportantunderstandingabstractvectorspacesistotheirwork.After readingthereviewers’commentsforChapter4,wereorganizedthechapter,condensingsomeofthematerialoncolumn,row,andnullspaces;movingMarkovchainsto theendofChapter5;andcreatinganewsectiononsignalprocessing.Weviewsignals
asaninfinitedimensionalvectorspaceandillustratetheusefulnessoflineartransformationstofilteroutunwanted“vectors”(a.k.a.noise),analyzedata,andenhance signals.
• BymovingMarkovchainstotheendofChapter5,wecannowdiscussthesteady statevectorasaneigenvector.Wealsoreorganizedsomeofthesummarymaterialon determinantsandchangeofbasistobemorespecifictothewaytheyareusedinthis chapter.
• InChapter6,wepresentpatternrecognitionasanapplicationoforthogonality,and thesectiononlinearmodelsnowillustrateshowmachinelearningrelatestocurve fitting.
• Chapter9onoptimizationwaspreviouslyavailableonlyasanonlinefile.Ithasnow beenmovedintotheregulartextbookwhereitismorereadilyavailabletofacultyand students.Afteranopeningsectiononfindingoptimalstrategiestotwo-personzerosumgames,therestofthechapterpresentsanintroductiontolinearprogramming— fromtwo-dimensionalproblemsthatcanbesolvedgeometricallytohigherdimensionalproblemsthataresolvedusingtheSimplexMethod.
OtherChanges • Inthehigh-techindustry,wheremostcomputationsaredoneoncomputers,judging thevalidityofinformationandcomputationsisanimportantstepinpreparingand analyzingdata.Inthisedition,studentsareencouragedtolearntoanalyzetheirown computationstoseeiftheyareconsistentwiththedataathandandthequestionsbeing asked.Forthisreason,wehaveadded“ReasonableAnswers”adviceandexercisesto guidestudents.
• Wehaveaddedalistofprojectstotheendofeachchapter(availableonlineandin MyLabMath).Someoftheseprojectswerepreviouslyavailableonlineandhavea widerangeofthemesfromusinglineartransformationstocreatearttoexploring additionalideasinmathematics.Theycanbeusedforgroupworkortoenhancethe learningofindividualstudents.
• PowerPointlectureslideshavebeenupdatedtocoverallsectionsofthetextandcover themmorethoroughly.
DistinctiveFeatures EarlyIntroductionofKeyConcepts Manyfundamentalideasoflinearalgebraareintroducedwithinthefirstsevenlectures, intheconcretesettingof Rn,andthengraduallyexaminedfromdifferentpointsofview. Latergeneralizationsoftheseconceptsappearasnaturalextensionsoffamiliarideas, visualizedthroughthegeometricintuitiondevelopedinChapter1.Amajorachievement ofthistextisthatthelevelofdifficultyisfairlyeventhroughoutthecourse.
AModernViewofMatrixMultiplication Goodnotationiscrucial,andthetextreflectsthewayscientistsandengineersactually uselinearalgebrainpractice.Thedefinitionsandproofsfocusonthecolumnsofamatrix ratherthanonthematrixentries.Acentralthemeistoviewamatrix–vectorproduct Ax asalinearcombinationofthecolumnsof A.Thismodernapproachsimplifiesmany arguments,andittiesvectorspaceideasintothestudyoflinearsystems.
LinearTransformations Lineartransformationsforma“thread”thatiswovenintothefabricofthetext.Their useenhancesthegeometricflavorofthetext.InChapter1,forinstance,lineartransformationsprovideadynamicandgraphicalviewofmatrix–vectormultiplication.
EigenvaluesandDynamicalSystems Eigenvaluesappearfairlyearlyinthetext,inChapters5and7.Becausethismaterialis spreadoverseveralweeks,studentshavemoretimethanusualtoabsorbandreviewthese criticalconcepts.Eigenvaluesaremotivatedbyandappliedtodiscreteandcontinuous dynamicalsystems,whichappearinSections1.10,4.8,and5.9,andinfivesectionsof Chapter5.SomecoursesreachChapter5afteraboutfiveweeksbycoveringSections 2.8and2.9insteadofChapter4.Thesetwooptionalsectionspresentallthevectorspace conceptsfromChapter4neededforChapter5.
OrthogonalityandLeast-SquaresProblems Thesetopicsreceiveamorecomprehensivetreatmentthaniscommonlyfoundinbeginningtexts.TheoriginalLinearAlgebraCurriculumStudyGrouphasemphasized theneedforasubstantialunitonorthogonalityandleast-squaresproblems,because orthogonalityplayssuchanimportantroleincomputercalculationsandnumericallinear algebraandbecauseinconsistentlinearsystemsarisesoofteninpracticalwork.
PedagogicalFeatures Applications Abroadselectionofapplicationsillustratesthepoweroflinearalgebratoexplain fundamentalprinciplesandsimplifycalculationsinengineering,computerscience, mathematics,physics,biology,economics,andstatistics.Someapplicationsappear inseparatesections;othersaretreatedinexamplesandexercises.Inaddition,each chapteropenswithanintroductoryvignettethatsetsthestageforsomeapplication oflinearalgebraandprovidesamotivationfordevelopingthemathematicsthat follows.
AStrongGeometricEmphasis Everymajorconceptinthecourseisgivenageometricinterpretation,becausemanystudentslearnbetterwhentheycanvisualizeanidea.Therearesubstantiallymoredrawings herethanusual,andsomeofthefigureshaveneverbeforeappearedinalinearalgebra text.InteractiveversionsofmanyofthesefiguresappearinMyLabMath.
Examples Thistextdevotesalargerproportionofitsexpositorymaterialtoexamplesthandomost linearalgebratexts.Therearemoreexamplesthananinstructorwouldordinarilypresent inclass.Butbecausetheexamplesarewrittencarefully,withlotsofdetail,studentscan readthemontheirown.
TheoremsandProofs Importantresultsarestatedastheorems.Otherusefulfactsaredisplayedintintedboxes, foreasyreference.Mostofthetheoremshaveformalproofs,writtenwiththebeginner studentinmind.Inafewcases,theessentialcalculationsofaproofareexhibitedina carefullychosenexample.Someroutineverificationsaresavedforexercises,whenthey willbenefitstudents.
PracticeProblems AfewcarefullyselectedPracticeProblemsappearjustbeforeeachexerciseset.Completesolutionsfollowtheexerciseset.Theseproblemseitherfocusonpotentialtrouble spotsintheexercisesetorprovidea“warm-up”fortheexercises,andthesolutionsoften containhelpfulhintsorwarningsaboutthehomework.
Exercises Theabundantsupplyofexercisesrangesfromroutinecomputationstoconceptualquestionsthatrequiremorethought.Agoodnumberofinnovativequestionspinpointconceptualdifficultiesthatwehavefoundonstudentpapersovertheyears.Eachexercise setiscarefullyarrangedinthesamegeneralorderasthetext;homeworkassignments arereadilyavailablewhenonlypartofasectionisdiscussed.Anotablefeatureofthe exercisesistheirnumericalsimplicity.Problems“unfold”quickly,sostudentsspend littletimeonnumericalcalculations.Theexercisesconcentrateonteachingunderstandingratherthanmechanicalcalculations.Theexercisesinthe SixthEdition maintainthe integrityoftheexercisesfrompreviouseditions,whileprovidingfreshproblemsfor studentsandinstructors.
Exercisesmarkedwiththesymbol T aredesignedtobeworkedwiththeaidof a“matrixprogram”(acomputerprogram,suchasMATLAB,Maple,Mathematica, MathCad,orDerive,oraprogrammablecalculatorwithmatrixcapabilities,suchasthose manufacturedbyTexasInstruments).
True/FalseQuestions Toencouragestudentstoreadallofthetextandtothinkcritically,wehavedeveloped over300simpletrue/falsequestionsthatappearthroughoutthetext,justafterthecomputationalproblems.Theycanbeanswereddirectlyfromthetext,andtheyprepare studentsfortheconceptualproblemsthatfollow.Studentsappreciatethesequestionsaftertheygetusedtotheimportanceofreadingthetextcarefully.Basedonclasstesting anddiscussionswithstudents,wedecidednottoputtheanswersinthetext.(The Study Guide,inMyLabMath,tellsthestudentswheretofindtheanswerstotheodd-numbered questions.)Anadditional150true/falsequestions(mostlyattheendsofchapters)test understandingofthematerial.ThetextdoesprovidesimpleT/Fanswerstomostofthese supplementaryexercises,butitomitsthejustificationsfortheanswers(whichusually requiresomethought).
WritingExercises AnabilitytowritecoherentmathematicalstatementsinEnglishisessentialforallstudentsoflinearalgebra,notjustthosewhomaygotograduateschoolinmathematics.
Thetextincludesmanyexercisesforwhichawrittenjustificationispartoftheanswer. Conceptualexercisesthatrequireashortproofusuallycontainhintsthathelpastudent getstarted.Forallodd-numberedwritingexercises,eitherasolutionisincludedatthe backofthetextorahintisprovidedandthesolutionisgiveninthe StudyGuide.
Projects Alistofprojects(availableonline)havebeenidentifiedattheendofeachchapter.They canbeusedbyindividualstudentsoringroups.Theseprojectsprovidetheopportunity forstudentstoexplorefundamentalconceptsandapplicationsinmoredetail.Twoofthe projectsevenencouragestudentstoengagetheircreativesideanduselineartransformationstobuildartwork.
ReasonableAnswers Manyofourstudentswillenteraworkforcewhereimportantdecisionsarebeingmade basedonanswersprovidedbycomputersandothermachines.TheReasonableAnswers boxesandexerciseshelpstudentsdevelopanawarenessoftheneedtoanalyzetheir answersforcorrectnessandaccuracy.
ComputationalTopics Thetextstressestheimpactofthecomputeronboththedevelopmentandpracticeof linearalgebrainscienceandengineering.FrequentNumericalNotesdrawattention toissuesincomputinganddistinguishbetweentheoreticalconcepts,suchasmatrix inversion,andcomputerimplementations,suchasLUfactorizations.
Acknowledgments DavidLaywasgratefultomanypeoplewhohelpedhimovertheyearswithvarious aspectsofthisbook.HewasparticularlygratefultoIsraelGohbergandRobertEllisfor morethanfifteenyearsofresearchcollaboration,whichgreatlyshapedhisviewoflinear algebra.AndhewasprivilegedtobeamemberoftheLinearAlgebraCurriculumStudy GroupalongwithDavidCarlson,CharlesJohnson,andDuanePorter.Theircreative ideasaboutteachinglinearalgebrahaveinfluencedthistextinsignificantways.Heoften spokefondlyofthreegoodfriendswhoguidedthedevelopmentofthebooknearlyfrom thebeginning—givingwisecounselandencouragement—GregTobin,publisher;Laurie Rosatone,formereditor;andWilliamHoffman,formereditor.
JudiandStevenhavebeenprivilegedtoworkonrecenteditionsofProfessorDavid Lay’slinearalgebrabook.Inmakingthisrevision,wehaveattemptedtomaintainthe basicapproachandtheclarityofstylethathasmadeearliereditionspopularwithstudents andfaculty.WethankEricSchulzforsharinghisconsiderabletechnologicalandpedagogicalexpertiseinthecreationoftheelectronictextbook.Hishelpandencouragement wereessentialinbringingthefiguresandexamplestolifeintheWolframCloudversion ofthistextbook.
MathewHudelsonhasbeenavaluablecolleagueinpreparingthe SixthEdition; he isalwayswillingtobrainstormaboutconceptsorideasandtestoutnewwritingand exercises.HecontributedtheideafornewvignetteforChapter3andtheaccompanying
project.Hehashelpedwithnewexercisesthroughoutthetext.HarleyWestonhasprovidedJudiwithmanyyearsofgoodconversationsabouthow,why,andwhoweappeal towhenwepresentmathematicalmaterialindifferentways.KaterinaTsatsomeros’ artisticsidehasbeenadefiniteassetwhenweneededartworktotransform(thefish andthesheep),improvedwritinginthenewintroductoryvignettes,orinformationfrom theperspectiveofcollege-agestudents.
WeappreciatetheencouragementandsharedexpertisefromNellaLudlow,Thomas Fischer,AmyJohnston,CassandraSeubert,andMikeManzano.Theyprovidedinformationaboutimportantapplicationsoflinearalgebraandideasfornewexamplesand exercises.Inparticular,thenewvignettesandmaterialinChapters4and6wereinspired byconversationswiththeseindividuals.
WeareenergizedbySepidehStewartandtheothernewLinearAlgebraCurriculumStudyGroup(LACSG2.0)members:SheldonAxler,RobBeezer,EugeneBoman, MinervaCatral,GuershonHarel,DavidStrong,andMeganWawro.Initialmeetingsof thisgrouphaveprovidedvaluableguidanceinrevisingthe SixthEdition.
Wesincerelythankthefollowingreviewersfortheircarefulanalysesandconstructivesuggestions:
MailaC.Brucal-Hallare, NorfolkStateUniversity
StevenBurrow, CentralTexasCollege KristenCampbell, ElginCommunityCollege J.S.Chahal, BrighamYoungUniversity CharlesConrad, VolunteerStateCommunityCollege KevinFarrell, LyndonStateCollege R.DarrellFinney, WilkesCommunityCollege ChrisFuller, CumberlandUniversity XiaofengGu, UniversityofWestGeorgia JeffreyJauregui, UnionCollege JeongMi-Yoon, UniversityofHouston–Downtown ChristopherMurphy, GuilfordTech.C.C. MichaelT.Muzheve, TexasA&MU.–Kingsville CharlesI.Odion, HoustonCommunityCollege IasonRusodimos, PerimeterC.atGeorgiaStateU.DesmondStephens, FloridaAg.andMech.U. RebeccaSwanson, ColoradoSchoolofMines JiyuanTao, LoyolaUniversity–Maryland CaseyWynn, KenyonCollege AmyYielding, EasternOregonUniversity TaoyeZhang, PennStateU.–WorthingtonScranton HoulongZhuang, ArizonaStateUniversity
WeappreciatetheproofreadingandsuggestionsprovidedbyJohnSamonsand JenniferBlue.Theircarefuleyehashelpedtominimizeerrorsinthisedition.
WethankKristinaEvans,PhilOslin,andJeanChoefortheirworkinsettingup andmaintainingtheonlinehomeworktoaccompanythetextinMyLabMath,andfor continuingtoworkwithustoimproveit.Thereviewsoftheonlinehomeworkdone byJoanSaniuk,RobertPierce,DoronLubinskyandAdrianaCorinaldesiweregreatly appreciated.WealsothankthefacultyatUniversityofCaliforniaSantaBarbara,UniversityofAlberta,WashingtonStateUniversityandtheGeorgiaInstituteofTechnology fortheirfeedbackontheMyLabMathcourse.JoeVeterehasprovidedmuchappreciated technicalhelpwiththe StudyGuide and Instructor’sSolutionsManual.
WethankJeffWeidenaar,ourcontentmanager,forhiscontinuedcareful,wellthought-outadvice.ProjectManagerRonHamptonhasbeenatremendoushelpguiding usthroughtheproductionprocess.WearealsogratefultoStaceySveumandRosemary Morton,ourmarketers,andJonKrebs,oureditorialassociate,whohavealsocontributed tothesuccessofthisedition.
StevenR.LayandJudiJ.McDonald
AcknowledgmentsfortheGlobalEdition PearsonwouldliketoacknowledgeandthankthefollowingfortheirworkontheGlobal Edition.
Contributors JoséLuisZuletaEstrugo, ÉcolePolytechniqueFédéraledeLausanne MohamadRafiSegiRahmat, UniversityofNottinghamMalaysia
Reviewers SibelDoğruAkgöl, AtilimUniversity HossamM.Hassan, CairoUniversity KwaKiamHeong, UniversityofMalaya YanghongHuang, UniversityofManchester NatanaelKarjanto, SungkyunkwanUniversity SomitraSanadhya, IndraprasthaInstituteofInformationTechnology VeroniqueVanLierde, AlAkhawaynUniversityinIfrane
Get the most out of MyLab Math MyLab Math for Linear Algebra and Its Applications
Lay, Lay, McDonald
MyLab Math features hundreds of assignable algorithmic exercises that mirror range of author-created resources, so your students have a consistent experience.
eText with Interactive Figures The eText includes Interactive Figures that bring the geometry of linear algebra to life. Students can manipulate with matrices to provide a deeper geometric understanding of key concepts and examples.
Teaching with Interactive Figures as a teaching tool for classroom demonstrations. Instructors can illustrate concep for students to visualize, leading to greater conceptual understanding.
Supporting Instruction MyLab Math provides resources to help you assess and improve student results and unparalleled flexibility to create a course tailored to you and your students.
PowerPoint® Lecture Slides Fully editable PowerPoint slides are available for all sections of the text. The slides include definitions, theorems, examples and solutions. When used in the classroom, these slides allow the instructor to focus on teaching, rather than writing on the board. PowerPoint slides are available to students (within the Video and Resource Library in MyLab Math) so that they can follow along.
Sample Assignments Sample Assignments are crafted to maximize student performance in the course. They make course set-up easier by giving instructors a starting point for each section.
Comprehensive Gradebook The gradebook includes enhanced reporting functionality, such as item analysis and a reporting dashboard to course. Student performance data are presented at the class, section, and program levels in an accessible, visual manner so you’ll have the information you need to keep your students on track.
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Resources for Success Instructor Resources Online resources can be downloaded from MyLab Math or from www.pearsonglobaleditions.com.
Instructor’s Solution Manual Includes fully worked solutions to all exercises in the text and teaching notes for many sections.
PowerPoint® Lecture Slides These fully editable lecture slides are available for all sections of the text.
Instructor’s Technology Manuals Each manual provides detailed guidance for integrating technology throughout the course, written by faculty who teach with the software and this text. Available For MATLAB, Maple, Mathematica, and Texas Instruments graphing calculators.
TestGen® TestGen (www pearsoned.com/testgen) enables instructors to build, edit, print, and administer tests using a computerized bank of questions developed to cover all the objectives of the text.
Student Resources Additional resources to enhance student success. All resources can be downloaded from MyLab Math.
Study Guide Pr ovid es d eta
e v er y t hi r d odd-nu
c o m pl ete e xpl a n
a n o dd - nu m b ere d w r i t ing e x erc i se h as a Hin t in t h e a n s w ers. Sp ec i a l s ub sect ion s o f t h e S tud y G u i d e s ugg est h o w to master k e y c on ce p ts of t h e co u rse. Fre qu e n t “ W ar ning s ” id e n t ify c o mm on errors a nd s how how t o p re v e n t t h em. MA TL AB box es in tr odu c e c o mma nd s as t h e y are n ee d e d . App e ndix e s in t h e S tud y G u i d e p ro vid e c o m p ara bl e info rmat i o n a bou t M a pl e, M at h emat i ca, a nd T I g ra phing ca l c ul at o rs. Av a il a bl e wi t hin My La b math.
Getting Started with Technology A quick-start guide for students to the technology they may use in this course. Available for MATLAB, Maple, Mathematica, or Texas Instrument graphing calculators. Downloadable from MyLab Math.
Projects Exploratory projects, written by experienced faculty members, invite students to discover applications of linear algebra
pearson.com/myla b/math
ANotetoStudents Thiscourseispotentiallythemostinterestingandworthwhileundergraduatemathematicscourseyouwillcomplete.Infact,somestudentshavewrittenorspokentous aftergraduationandsaidthattheystillusethistextoccasionallyasareferenceintheir careersatmajorcorporationsandengineeringgraduateschools.Thefollowingremarks offersomepracticaladviceandinformationtohelpyoumasterthematerialandenjoy thecourse.
Inlinearalgebra,the concepts areasimportantasthe computations. Thesimple numericalexercisesthatbegineachexercisesetonlyhelpyoucheckyourunderstanding ofbasicprocedures.Laterinyourcareer,computerswilldothecalculations,butyou willhavetochoosethecalculations,knowhowtointerprettheresults,analyzewhether theresultsarereasonable,thenexplaintheresultstootherpeople.Forthisreason,many exercisesinthetextaskyoutoexplainorjustifyyourcalculations.Awrittenexplanation isoftenrequiredaspartoftheanswer.IfyouareworkingonquestionsinMyLabMath, keepanotebookforcalculationsandnotesonwhatyouarelearning.Forodd-numbered exercisesinthetextbook,youwillfindeitherthedesiredexplanationoratleastagood hint.Youmustavoidthetemptationtolookatsuchanswersbeforeyouhavetriedtowrite outthesolutionyourself.Otherwise,youarelikelytothinkyouunderstandsomething wheninfactyoudonot.
Tomastertheconceptsoflinearalgebra,youwillhavetoreadandrereadthetext carefully.Newtermsareinboldfacetype,sometimesenclosedinadefinitionbox. Aglossaryoftermsisincludedattheendofthetext.Importantfactsarestatedas theoremsorareenclosedintintedboxes,foreasyreference.Weencourageyoutoread thePrefacetolearnmoreaboutthestructureofthistext.Thiswillgiveyouaframework forunderstandinghowthecoursemayproceed.
Inapracticalsense,linearalgebraisalanguage.Youmustlearnthislanguagethe samewayyouwouldaforeignlanguage—withdailywork.Materialpresentedinone sectionisnoteasilyunderstoodunlessyouhavethoroughlystudiedthetextandworked theexercisesfortheprecedingsections.Keepingupwiththecoursewillsaveyoulots oftimeanddistress!
NumericalNotes WehopeyoureadtheNumericalNotesinthetext,evenifyouarenotusingacomputeror graphingcalculatorwiththetext.Inreallife,mostapplicationsoflinearalgebrainvolve numericalcomputationsthataresubjecttosomenumericalerror,eventhoughthaterror maybeextremelysmall.TheNumericalNoteswillwarnyouofpotentialdifficultiesin
usinglinearalgebralaterinyourcareer,andifyoustudythenotesnow,youaremore likelytorememberthemlater.
IfyouenjoyreadingtheNumericalNotes,youmaywanttotakeacourselaterin numericallinearalgebra.Becauseofthehighdemandforincreasedcomputingpower, computerscientistsandmathematiciansworkinnumericallinearalgebratodevelop fasterandmorereliablealgorithmsforcomputations,andelectricalengineersdesign fasterandsmallercomputerstorunthealgorithms.Thisisanexcitingfield,andyour firstcourseinlinearalgebrawillhelpyouprepareforit.
StudyGuide Tohelpyousucceedinthiscourse,wesuggestthatyouusethe StudyGuide availablein MyLabMath.Notonlywillithelpyoulearnlinearalgebra,italsowillshowyouhow tostudymathematics.Atstrategicpointsinyourtextbook,marginalnoteswillremind youtocheckthatsectionofthe StudyGuide forspecialsubsectionsentitled“Mastering LinearAlgebraConcepts.”Thereyouwillfindsuggestionsforconstructingeffective reviewsheetsofkeyconcepts.Theactofpreparingthesheetsisoneofthesecretsto successinthecourse,becauseyouwillconstruct linksbetweenideas. Theselinksare the“glue”thatenablesyoutobuildasolidfoundationforlearningand remembering the mainconceptsinthecourse.
The StudyGuide containsadetailedsolutiontomorethanathirdoftheoddnumberedexercises,plussolutionstoallodd-numberedwritingexercisesforwhich onlyahintisgivenintheAnswerssectionofthisbook.The Guide isseparatefrom thetextbecauseyoumustlearntowritesolutionsbyyourself,withoutmuchhelp.(We knowfromyearsofexperiencethateasyaccesstosolutionsinthebackofthetextslows themathematicaldevelopmentofmoststudents.)The Guide alsoprovideswarningsof commonerrorsandhelpfulhintsthatcallattentiontokeyexercisesandpotentialexam questions.
Ifyouhaveaccesstotechnology—MATLAB,Octave,Maple,Mathematica,oraTI graphingcalculator—youcansavemanyhoursofhomeworktime.The StudyGuide is your“labmanual”thatexplainshowtouseeachofthesematrixutilities.Itintroduces newcommandswhentheyareneeded.Youwillalsofindthatmostsoftwarecommands youmightuseareeasilyfoundusinganonlinesearchengine.Specialmatrixcommands willperformthecomputationsforyou!
Whatyoudoinyourfirstfewweeksofstudyingthiscoursewillsetyourpattern forthetermanddeterminehowwellyoufinishthecourse.Pleaseread“HowtoStudy LinearAlgebra”inthe StudyGuide assoonaspossible.Manystudentshavefoundthe strategiesthereveryhelpful,andwehopeyouwill,too.
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1 LinearEquationsin LinearAlgebra IntroductoryExample
LINEARMODELSINECONOMICS ANDENGINEERING Itwaslatesummerin1949.HarvardProfessorWassily Leontiefwascarefullyfeedingthelastofhispunchedcards intotheuniversity’sMarkIIcomputer.Thecardscontained informationabouttheU.S.economyandrepresenteda summaryofmorethan250,000piecesofinformation producedbytheU.S.BureauofLaborStatisticsaftertwo yearsofintensivework.LeontiefhaddividedtheU.S. economyinto500“sectors,”suchasthecoalindustry, theautomotiveindustry,communications,andsoon. Foreachsector,hehadwrittenalinearequationthat describedhowthesectordistributeditsoutputtotheother sectorsoftheeconomy.BecausetheMarkII,oneofthe largestcomputersofitsday,couldnothandletheresulting systemof500equationsin500unknowns,Leontiefhad distilledtheproblemintoasystemof42equationsin 42unknowns.
ProgrammingtheMarkIIcomputerforLeontief’s 42equationshadrequiredseveralmonthsofeffort,andhe wasanxioustoseehowlongthecomputerwouldtaketo solvetheproblem.TheMarkIIhummedandblinkedfor 56hoursbeforefinallyproducingasolution.Wewill discussthenatureofthissolutioninSections1.6and2.6.
Leontief,whowasawardedthe1973NobelPrize inEconomicScience,openedthedoortoanewera inmathematicalmodelingineconomics.Hiseffortsat Harvardin1949markedoneofthefirstsignificantuses ofcomputerstoanalyzewhatwasthenalarge-scale
mathematicalmodel.Sincethattime,researchersin manyotherfieldshaveemployedcomputerstoanalyze mathematicalmodels.Becauseofthemassiveamountsof datainvolved,themodelsareusually linear;thatis,they aredescribedby systemsoflinearequations
Theimportanceoflinearalgebraforapplicationshas risenindirectproportiontotheincreaseincomputing power,witheachnewgenerationofhardwareandsoftware triggeringademandforevengreatercapabilities.Computer scienceisthusintricatelylinkedwithlinearalgebrathrough theexplosivegrowthofparallelprocessingandlarge-scale computations.
Scientistsandengineersnowworkonproblemsfar morecomplexthanevendreamedpossibleafewdecades ago.Today,linearalgebrahasmorepotentialvaluefor studentsinmanyscientificandbusinessfieldsthanany otherundergraduatemathematicssubject!Thematerialin thistextprovidesthefoundationforfurtherworkinmany interestingareas.Hereareafewpossibilities;otherswill bedescribedlater.
Oilexploration. Whenashipsearchesforoffshore oildeposits,itscomputerssolvethousandsof separatesystemsoflinearequations everyday. Theseismicdatafortheequationsareobtained fromunderwatershockwavescreatedbyexplosions fromairguns.Thewavesbounceoffsubsurface
