线代英文课件\chapter1

IntroductiontoLinearAlgebraLeeW.JohnsonR.DeanRiessJimmyT.ArnoldEmail:mxb502@163.comOrganizationChapteroneMatricesandsystemsoflinearequationsChaptertwoVectorsin2-spaceand3-space(optional)ChapterthreeThevectorspaceRnChapterfourTheeigenvalueproblemChapterfiveVectorspacesandlineartransformationsChaptersixdeterminantsChaptersevenEigenvaluesandapplicationsChapter1MatricesandSystemsofLinearEquationsOverviewInthischapterwediscusssystemsoflinearequationsandmethods(suchasGauss-Jordanelimination)forsolvingthesesystems.WeintroducematricesasaconvenientlanguagefordescribingsystemsandtheGauss-Jordansolutionmethod.Wenextintroducetheoperationsofadditionandmultiplicationformatricesandshowhowtheseoperationsenableustoexpressalinearsysteminmatrix-vectortermsasAx=b.CoresectionsIntroductiontomatricesandsystemsoflinearequationsEchelonformandGauss-JordaneliminationConsistentsystemsoflinearequationsMatrixoperationsAlgebraicpropertiesofmatrixoperationsLinearindependenceandnonsingularmatricesMatrixinversesandtheirproperties1.1introductiontomatricesandsystemsoflinearequationsAlinearequationinnunknownsisanequationthatcanbeputintheformThecoefficientsa1,a2,…anandtheconstantbareknown,andx1,,x2,…,xndenotetheunknowns.Aequationiscalledlinearbecauseeachtermhasdegreeoneinthevariablesx1,,x2,…,xn.Otherwisetheequationiscallednonlinear.Example1:Whichofthefollowingequationsarelinear?An(mn)systemoflinearequationsisasetofequationsoftheform:Asolutiontosystem(*)isasequences1,s2,…,snofnumbersthatissimultaneouslyasolutionforeachequationinthesystem.Thedoublesubscriptnotationusedforthecoefficientsisnecessarytoprovidean“address”foreachcoefficient.Forexample,a32appearsinthethirdequationasthecoefficientofx2.1.Geometricinterpretationsofsolutionsets(1)(22)systemoflinearequations.Thetwolinesarecoincident(thesameline),sothereareinfinitelymanysolutions.Thetwolinesareparallel(nevermeet),sotherearenosolutions.Thetwolinesintersectatasinglepoint,sothereisauniquesolution.(2)(23)systemoflinearequations.Thetwoplanesmightbecoincident.Inthiscase,thesystemhasinfinitelymanysolutions.Thetwoplanesmightbeparallel.Inthiscase,thesystemhasnosolution.Thetwoplanesmightintersectinaline.Inthiscase,thesystemhasinfinitelymanysolutions.(3)(33)systemoflinearequations.Thethreeplanesmightbecoincident,orintersectinaline.Thenthesystemhasinfinitelymanysolutions.Thethreeplanesareparallel,therearetwoplanesbeparallel,orthethreeplanesintersectthreelineswhichforeverytwolinesareparallel.Thenthesystemhasnosolution.Thethreeplanesintersectatasinglepoint.Inthiscase,thesystemhasauniquesolution.Remark:An(mn)systemoflinearequationshaseitherinfinitelymanysolutions,nosolution,orauniquesolution.Ingeneral,asystemofequationsiscalledconsistentifithasatleastonesolution,andthesystemiscalledinconsistentifithasnosolution.2.MatricesWebeginourintroductiontomatrixtheorybyrelatingmatricestotheproblemofsolvingsystemsoflinearequations.Initiallyweshowthatmatrixtheoryprovidesaconvenientandnaturalsymboliclanguagetodescribelinearsystems.Laterweshowthatmatrixtheoryisalsoanappropriateandpowerfulframeworkwithinwhichtoanalyzeandsolvemoregenerallinearproblems,suchasleast-squaresapproximations,representationsoflinearoperations,andeigenvalueproblems.Moregenerally,an(mn)matrixisarectangulararrayofnumbersoftheformThusan(mn)matrixhasmrowsandncolumns.ThesubscriptsfortheentryaijindicatethatthenumberappearsintheithrowandjthcolumnofA.3.MatrixrepresentationofalinearsystemThecoefficientmatrixforthesystemisa(mn)matrixA:Theaugmentedmatrixforthesystemisa[m(n+1)]matrixBwhichisusuallydenotedas[A|b],whereAisthecoefficientmatrixandb=[b1b2…bm]T.4.ElementaryoperationsAsweshallsee,therearetwostepsinvolvedinsolvingan(mn)systemofequations.Reductionofthesystem(thatis,theeliminationofvariables).Descriptionofsetofsolutions.Definition1.1.1:twosystemsoflinearequationsinnunknownsareequivalentprovidedthattheyhavethesamesetofsolutions.ElementaryOperations:Interchangetwoequations.Multiplyanequationbyanonzeroscalar.Addaconstantmultipleofoneequationtoanother.Theorem1.1.1:Ifoneoftheelementaryoperationsisappliedtoasystemoflinearequationsthentheresultingsystemisequivalenttotheoriginalsystem.ktimesthejthequationisaddedtotheithequation.Ei+kEjTheithequationismultipliedbythenonzeroscalark.kEiTheithandjthequationsareinterchanged.EiEjElementaryoperationperformedNotationExample2:UseelementaryoperationstosolvethesystemSolution:TheelementaryoperationE2+E1producesthefollowingequivalentsystem:Theoperation1/3E2thenleadstoFinally,usingtheoperationE1-E2,weobtainThismethodiscalledGauss-Jordanelimination.5.RowOperations:ktimesthejthrowisaddedtotheithequation.Ri+kRjTheithrowismultipliedbythenonzeroscalark.kRiTheithandjthrowsareinterchanged.RiRjElementaryRowOperationNotationDefinition1.1.2:Thefollowingoperations,performedontherowsofamatrix,arecalledelementaryrowoperations:1.Interchangetworows.2.Multiplyarowbyanonzeroscalar.3.Addaconstantmultipleofonerowtoanother.Wesaythattwo(mn)matrices,BandC,arerowequivalentifonecanbeobtainedfromtheotherbyasequenceofelementaryrowoperations.NowifBistheaugmentedmatrixforasystemoflinearequationsandifCisrowequivalenttoB,thenCistheaugmentedmatrixforanequivalentsystem.Thus,wecansolvealinearsystemwiththefollowingsteps:1.FormtheaugmentedmatrixBforthesystem.2.UseelementaryrowoperationstotransformBtoarowequivalentmatrixCwhichrepresentsa“simpler”system.3.SolvethesimplersystemthatisrepresentedbyC.Example3:Solution:Corollary:Suppose[A|b]and[C|d]areaugmentedmatrices,eachrepresentingadifferent(mn)systemoflinearequations.If[A|b]and[C|d]arerowequivalentmatrices,thenthetwosystemsarealsoequivalent.1.2EchelonformandGauss-JordaneliminationGivensystemofequationsAugmentedmatrixReducedmatrixReducedsystemofequationSolutionProcedureforsolvingasystemoflinearequations1.EchelonFormDefinition1.2.1:An(mn)matrixBisinechelonformif:Allrowsthatconsistentirelyofzerosaregroupedtogetheratthebottomofthematrix.Ineverynonzerorow,thefirstnonzeroentry(countingfromlefttoright)isa1.Ifthe(i+1)-strowcontainsnonzeroentries,thenthefirstnonzeroentryisinacolumntotherightofthefirstnonzeroentryintheithrow.Definition1.2.2:Amatrixthatisinechelonformisinreducedechelonformprovidedthatthefirstnonzeroentryinanyrowistheonlynonzeroentryinitscolumn.Example1:Foreachmatrixshown,chooseoneofthefollowingphrasestodescribethematrix.Thematrixisnotinechelonform.Thematrixisinechelonform,butnotinreducedechelonform.Thematrixisinreducedechelonform.2.SolvingalinearsystemwhoseaugmentedmatrixisinreducedechelonformExample2:Eachofthefollowingmatricesisinreducedechelonformandistheaugmentedmatrixforasystemoflinearequations.Ineachcase,givethesystemofequationsanddescribethesolution.Solution:MatrixBistheaugmentedmatrixforthefollowingsystem:Therefore,thesystemhastheuniquesolutionx1=3,x2=-2,andx3=7.MatrixCistheaugmentedmatrixforthefollowingsystem:Becausenovaluesforx1,x2,orx3cansatisfythethirdequation,thesystemisinconsistent.3.RecognizinganinconsistentsystemTheorem1.2.1:Let[A|b]betheaugmentedmatrixforan(mn)linearsystemofequations,andlet[A|b]beinreducedechelonform.Ifthelastnonzerorowof[A|b]hasitsleading1inthelastcolumn,thenthesystemofequationshasnosolution.Thatis,thesystemrepresentedby[A|b]isinconsistent.4.SolvingasystemofequationsStep1.Createtheaugmentedmatrixforthesystem.Step2.TransformthematrixinStep1toreducedechelonform.Step3.DecodethereducedmatrixfoundinStep2toobtainitsassociatedsystemofequations.Step4.ByexaminingthereducedsysteminStep3,describethesolutionsetfortheoriginalsystem.5.ReductiontoechelonformTheorem1.2.2:LetBbean(mn)matrix.Thereisaunique(mn)matrixCsuchthat:CisinreducedechelonformCisrowequivalenttoB.Reductiontoreducedechelonformforan(mn)matrix:(1)Locatethefirst(left-most)columnthatcontainsanonzeroentry.(2)Ifnecessary,interchangethefirstrowwithanotherrowsothatthefirstnonzerocolumnhasanonzeroentryinthefirstrow.(3)Ifadenotestheleadingnonzeroentryinrowone,multiplyeachentryinrowoneby1/a.(4)Addappropriatemultiplesofrowonetoeachoftheremainingrowssothateveryentrybelowtheleading1inrowoneisa0.(5)Temporarilyignorethefirstrowofthismatrixandrepeat(1)—(4)onthesubmatrixthatremains.Stoptheprocesswhentheresultingmatrixisinechelonform.(6)Havingreachedechelonformin(5),continueontoreducedechelonformasfollows:Proceedingupward,addmultiplesofeachnonzerorowtotherowsaboveinordertozeroallentriesabovetheleading1.Example3:UseelementaryrowoperationstotransformthefollowingmatrixtoreducedechelonformExercise:Example4:Solvethefollowingsystemofequations:Solution:transformtheaugmentedmatrixtoreducedechelonform.R1+R2R2+R1R4-3R1ThenmatrixaboverepresentsthefollowingsystemofequationsR1+2R3R4+2R3R1-2R2R3+3R2R4-4R2Remark:InEq.(1)wehaveanicedescriptionofalloftheinfinitelymanysolutionstotheoriginalsystem—itiscalledthegeneralsolutionforthesystem.Forthisexample,x2andx5areindependentvariablesandcanbeassignedvaluesarbitrarily.Thevariablesx1,x3,andx4aredependentvariables,andtheirvaluesaredeterminedbythevaluesassignedtox2andx5.Particularsolution.Solvingtheprecedingsystem,wefind:Exercises:P2728,30,49,53Threepeopleplayagameinwhichtherearealwaystwowinnersandoneloser.Theyhavetheunderstandingthatthelosergiveseachwinneranamountequaltowhatthewinneralreadyhas.Afterthreegames,eachhaslostjustonceandeachhas$24.Withhowmuchmoneydideachbegin?P24.example1.3ConsistentSystemsofLinearEquations1.SolutionpossibilitiesforaconsistentlinearsystemOurgoalistodeduceasmuchinformationaspossibleaboutthesolutionsetofsystem(1)withoutactuallysolvingthesystem.Theorem1.3.1:Letthematrix[C|d]isinreducedechelonform.Thesystemrepresentedbythematrix[C|d]isinconsistentifandonlyif[C|d]hasarowoftheform[0,0,…,0,1].Theorem1.3.2:Everyvariablecorrespondingtoaleading1in[C|d]isadependentvariable.Theorem1.3.3:Letrdenotethenumberofnonzerorowsin[C|d].Then,rn+1.Theorem1.3.4:Letrdenotethenumberofnonzerorowsin[C|d].Ifthesystemrepresentedby[C|d]isconsistent,thenrn.Theorem1.3.6:Consideran(mn)systemoflinearequations.Ifmm,thenthissetislinearlydependent.Definition1.7.2:An(nn)matrixAisnonsingulariftheonlysolutiontoAx=0isx=0.Furthermore,AissaidtobesingularifAisnotnonsingular.Theorem1.7.2:The(nn)matrixA=[A1,A2,,An]isnonsingularifandonlyis{A1,A2,,An}isalinearlyindependentset.Theorem1.7.3:LetAbean(nn)matrix.TheequationAx=bhasauniquesolutionforevery(n1)columnvectorbifandonlyifAisnonsingular.Exercises:P7849,501.9Matrixinversesandtheirproperties1.ThematrixinverseDefinition1.9.1:LetAbean(nn)matrix.WesaythatAisinvertibleifwecanfindan(nn)matrixA-1suchthatThematrixA-1iscalledaninverseforA.Example1:Letfinditsinversematrix.2.UsinginversestosolvesystemsoflinearequationsAx=bx=A-1bAX=BX=A-1BwhereAisan(nn)matrix,BandXare(nm)matrices.3.ExistenceofinversesLemma:LetP,Q,andRbe(nn)matricessuchthatPQ=R.IfeitherPorQissingular,thensoisR.Theorem1.9.1:LetAbean(nn)matrix.ThenAhasaninverseifandonlyifAisnonsingular.4.CalculatingtheinverseComputationofA-1Step1.Formthe(n2n)matrix[A|I].Step2.Useelementaryrowoperationstotransform[A|I]totheform[I|B].Step3.Readingformthisfinalform,A-1=B.Example1:solution：Example2:solution：ZerorownotinverseExample2:Findtheinverseofthe(nn)matrixExercise1.Findtheinverseofthe(nn)matrix2.FindthethematrixXsuchthatAXB=C,whereTheorem1.9.2:LetAbean(nn)matrix.ThenAisnonsingularifandonlyifAisrowequivalenttoI.5.PropertiesofmatrixinversesTheorem1.9.3:LetAandBbe(nn)matrices,eachofwhichhasaninverse.Then:1.A-1hasaninverse,and(A-1)-1=A.2.ABhasaninverse,and(AB)-1=B-1A-1.3.Ifkisanonzeroscalar,thenkAhasaninverse,and(kA)-1=(1/k)A-1.4.AThasaninverse,and(AT)-1=(A-1)T.Theorem1.9.4:LetAbean(nn)matrix.Thefollowingareequivalent:1.Aisnonsingular;thatis,theonlysolutionofAx=0.2.ThecolumnvectorsofAarelinearlyindependent.3.Ax=balwayshasauniquesolution.4.Ahasaninverse.5.AisrowequivalenttoI.6.Ill-conditionedmatrixInapplicationstheequationAx=boftenservesasamathematicalmodelforaphysicalproblem.InthesecasesitisimportanttoknowwhethersolutionstoAx=baresensitivetosmallchangesintheright-handsideb.ifsmallchangesinbcanleadtorelativelylargechangesinthesolutionx,thenthematrixAiscalledill-conditioned.Exercises:P10450,54,58,P1064,5Partitionedmatrix(orblockmatrix)LetAandBareinvertiblematrices,then