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首页 linear algebra demystified(David McMahon)

linear algebra demystified(David McMahon).pdf

linear algebra demystified(David…

tigerxI1S 2014-04-10 评分 0 浏览量 0 0 0 0 暂无简介 简介 举报

简介:本文档为《linear algebra demystified(David McMahon)pdf》,可适用于高等教育领域,主题内容包含LinearAlgebraDemystifiediDemystifiedSeriesAdvancedStatisticsDemystifiedMathPr符等。

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eofaMatrixSpaceofaMatrixQuizCONTENTSviiCHAPTERInnerProductSpacesTheVectorSpaceRnInnerProductsonFunctionSpacesPropertiesoftheNormAnInnerProductforMatrixSpacesTheGramSchmidtProcedureQuizCHAPTERLinearTransformationsMatrixRepresentationsLinearTransformationsintheSameVectorSpaceMorePropertiesofLinearTransformationsQuizCHAPTERTheEigenvalueProblemTheCharacteristicPolynomialTheCayleyHamiltonTheoremFindingEigenvectorsNormalizationTheEigenspaceofanOperatorASimilarMatricesDiagonalRepresentationsofanOperatorTheTraceandDeterminantandEigenvaluesQuizCHAPTERSpecialMatricesSymmetricandSkewSymmetricMatricesHermitianMatricesOrthogonalMatricesUnitaryMatricesQuizCHAPTERMatrixDecompositionLUDecompositionviiiCONTENTSSolvingaLinearSystemwithanLUFactorizationSVDDecompositionQRDecompositionQuizFinalExamHintsandSolutionsReferencesIndexPREFACEThisbookisforpeoplewhowanttogetaheadstartandlearnthebasicconceptsoflinearalgebraSuitableforselfstudyorasareferencethatputssolvingproblemswithineasyreach,thisbookcanbeusedbystudentsorprofessionalslookingforaquickrefresherIfyou’relookingforasimplifiedpresentationwithexplicitlysolvedproblemsforselfstudy,thisbookwillhelpyouIfyou’reastudenttakinglinearalgebraandneedaninformativeaidtokeepyouaheadofthegame,thisbookistheperfectsupplementtotheclassroomThetopicscoveredfitthoseusuallytaughtinaonesemesterundergraduatecourse,butthebookisalsousefultograduatestudentsasaquickrefresherThebookcanserveasagoodjumpingoffpointforstudentstoreadbeforetakingacourseThepresentationisinformalandtheemphasisisonshowingstudentshowtosolveproblemsthataresimilartothosetheyarelikelytoencounterinhomeworkandexaminationsEnhanceddetailisusedtouncovertechniquesusedtosolveproblemsratherthanleavingthehowandwhyofhomeworksolutionsasecretWhilelinearalgebrabeginswiththesolutionofsystemsoflinearequations,itquicklyjumpsoffintoabstracttopicslikevectorspaces,lineartransformations,determinants,andsolvingeigenvectorproblemsManystudentshaveahardtimestrugglingthroughthesetopicsIfyouarehavingahardtimegettingthroughyourcoursesbecauseyoudon’tknowhowtosolveproblems,thisbookshouldhelpyoumakeprogressAspartofaselfstudycourse,thisbookisagoodplacetogetafirstexposuretothesubjectoritisagoodrefresherifyou’vebeenoutofschoolforalongtimeAfterreadinganddoingtheexercisesinthisbookitwillbemucheasierforyoutotacklestandardlinearalgebratextbooksortomoveontoamoreadvancedtreatmentTheorganizationofthebookisasfollowsWebeginwithadiscussionofsolutiontechniquesforsolvinglinearsystemsofequationsAfterintroducingtheixCopyrightbyTheMcGrawHillCompanies,IncClickherefortermsofusexPREFACEnotionofmatrices,weillustratebasicmatrixalgebraoperationsandtechniquessuchasfindingthetransposeofamatrixorcomputingthetraceNextwestudydeterminants,vectors,andvectorspacesThisisfollowedbythestudyoflineartransformationsWethendevotesometimeshowinghowtofindtheeigenvaluesandeigenvectorsofamatrixThisisfollowedbyachapterthatdiscussesseveralspecialtypesofmatricesthatareimportantThisincludessymmetric,Hermitian,orthogonal,andunitarymatricesWefinishthebookwithareviewofmatrixdecompositions,specificallyLU,SVD,andQRdecompositionsEachchapterhasseveralexamplesthataresolvedindetailTheideaistoremovethemysteryandshowthestudenthowtosolveproblemsExercisesattheendofeachchapterhavebeendesignedtocorrespondtothesolvedproblemsinthetextsothatthestudentcanreinforceideaslearnedwhilereadingthechapterAfinalexam,withsimilarquestions,attheendofthebookgivesthestudentachancetoreinforcethesenotionsaftercompletingthetextDavidMcMahonCHAPTERSystemsofLinearEquationsAlinearequationwithnunknownsisanequationofthetypeaxaxanxn=bInmanysituations,wearepresentedwithmlinearequationsinnunknownsSuchasetisknownasasystemoflinearequationsandtakestheformaxaxanxn=baxaxanxn=bamxamxamnxn=bmCopyrightbyTheMcGrawHillCompanies,IncClickherefortermsofuseCHAPTERSystemsofLinearEquationsThetermsx,x,,xnaretheunknownsorvariablesofthesystem,whiletheaijarecalledcoefficientsThebiontherighthandsidearefixednumbersorscalarsThegoalistofindthevaluesofthex,x,,xnsuchthattheequationsaresatisfiedEXAMPLEConsiderthesystemxyz=xy=xyz=Does(x,y,z)=(,,)solvethesystemWhatabout(,,)SOLUTIONWesubstitutethevaluesof(x,y,z)intoeachequationTrying(x,y,z)=(,,)inthefirstequation,weobtain()()==andsothefirstequationissatisfiedUsingthesubstitutioninthesecondequation,wefind()()===Thesecondequationisnotsatisfiedtherefore,(x,y,z)=(,,)cannotbeasolutiontothissystemofequationsNowwetrythesecondsetofnumbers(,,)Substitutioninthefirstequationgives()()====Again,thefirstequationissatisfiedTryingthesecondequationgives()()==CHAPTERSystemsofLinearEquationsConsistentSystemAuniquesolutionoraninfinitenumberofsolutionsInconsistentSystemSystemhasnosolutionFigDescriptionofsolutionpossibilitiesThistimethesecondequationisalsosatisfiedFinally,thethirdequationworksouttobe()()==()===ThisshowsthatthethirdequationissatisfiedaswellThereforeweconcludethat(x,y,z)=(,,)isasolutiontothesystemConsistentandInconsistentSystemsWhenatleastonesolutionexistsforagivensystemoflinearequations,wecallthatsystemconsistentIfnosolutionexists,thesystemiscalledinconsistentThesolutiontoasystemisnotnecessarilyuniqueAconsistentsystemeitherhasauniquesolutionoritcanhaveaninfinitenumberofsolutionsWesummarizetheseideasinFigIfaconsistentsystemhasaninfinitenumberofsolutions,ifwecandefineasolutionintermsofsomeextraparametert,wecallthisaparametricsolutionMatrixRepresentationofaSystemofEquationsItisconvenienttowritedownthecoefficientsandscalarsinalinearsystemofequationsasarectangulararrayofnumberscalledamatrixEachrowinCHAPTERSystemsofLinearEquationsthearraycorrespondstooneequationForasystemwithmequationsinnunknowns,therewillbemrowsinthematrixThearraywillhavencolumnsEachofthefirstncolumnsisusedtowritethecoefficientsthatmultiplyeachoftheunknownvariablesThelastcolumnisusedtowritethenumbersfoundontherighthandsideoftheequationsConsiderthesetofequationsusedinthelastexample:xyz=xy=xyz=ThematrixusedtorepresentthissystemisWerepresentthissetofequationsxy=xy=bythematrixOnewaywecancharacterizeamatrixisbythenumberofrowsandcolumnsithasAmatrixwithmrowsandncolumnsisreferredtoasanmnmatrixSometimesmatricesaresquare,meaningthatthenumberofrowsequalsthenumberofcolumnsWerefertoagivenelementfoundinamatrixbyidentifyingitsrowandcolumnpositionThiscanbedoneusingthenotation(i,j)torefertotheelementlocatedatrowiandcolumnjRowsarenumberedstartingwithatthetopofthematrix,increasingaswemovedownthematrixColumnsarenumberedstartingwithonthelefthandsideAnalternativemethodofidentifyingelementsinamatrixistouseasubscriptnotationMatricesareoftenidentifiedwithitalicizedorboldcapitallettersSoA,B,CorA,B,CcanbeusedaslabelstoidentifymatricesThecorrespondingCHAPTERSystemsofLinearEquationssmallletteristhenusedtoidentifyindividualelementsofthematrix,withsubscriptsindicatingtherowandcolumnwherethetermislocatedForamatrixA,wecanuseaijtoidentifytheelementlocatedattherowandcolumnposition(i,j)Asanexample,considerthematrixB=Theelementlocatedatrowandcolumnofthismatrixcanbeindicatedbywriting(,)orbThisnumberisb=Theelementlocatedatrowandcolumnisb=ThesubscriptnotationisshowninFigAmatrixthatincludestheentirelinearsystemiscalledanaugmentedmatrixWecanalsomakeamatrixthatismadeuponlyofthecoefficientsthatmultiplytheunknownvariablesThisisknownasthecoefficientmatrixForthesystemxyz=xyz=xyz=thecoefficientmatrixisaijElementatrowiColumnjFigTheindexingofanelementfoundatrowiandcolumnjofamatrixCHAPTERSystemsofLinearEquationsA=WecanfindasolutiontoalinearsystemofequationsbyapplyingasetofelementaryoperationstotheaugmentedmatrixSolvingaSystemUsingElementaryOperationsThereexistthreeelementaryoperationsthatcanbeappliedtoasystemoflinearequationswithoutfundamentallychangingthatsystemTheseare•Exchangetworowsofthematrix•Replacearowbyascalarmultipleofitself,aslongasthescalarisnonzero•ReplaceonerowbyaddingthescalarmultipleofanotherrowLet’sintroducesomeshorthandnotationtodescribetheseoperationsanddemonstrateusingthematrixM=Toindicatetheexchangeofrowsand,wewriteRRThistransformsthematrixasfollows:Nowlet’sconsidertheoperationwherewereplacearowbyascalarmultipleofitselfLet’ssaywewantedtoreplacethefirstrowinthefollowingway:RRCHAPTERSystemsofLinearEquationsThematrixwouldbetransformedasInthethirdtypeofoperation,wereplaceaselectedrowbyaddingascalarmultipleofadifferentrowConsiderRRRThematrixbecomesThesolutiontothesystemisobtainedwhenthissetofoperationsbringsthematrixintotriangularformThistypeofeliminationissometimesknownasGaussianeliminationTriangularMatricesGenerally,thegoalofperformingtheelementaryoperationsonasystemistogetitinatriangularformAsystemthatisinanuppertriangularformisB=Thisaugmentedmatrixrepresentstheequationsxyz=yz=z=AsolutionforthelastvariablecanbefoundbyinspectionInthisexample,weseethatz=Tofindthevaluesoftheothervariables,weusebacksubstitutionWesubstitutethevaluewehavefoundintotheequationimmediatelyaboveitInthisCHAPTERSystemsofLinearEquationscase,insertthevaluefoundforzintothesecondequationThisallowsustosolvefory:yz=,z=y=y=(Notethatthesymbolisshorthandfortherefore)Eachtimeyouapplybacksubstitution,youobtainanequationthathasonlyoneunknownvariableNowwecansubstitutethevaluesy=andz=intothefirstequationtosolveforthefinalunknown,whichisx:x=x=x=AsystemthatistriangularissaidtobeinechelonformLet’sillustratethecompletesolutionofasystemoflinearequationsusingtheelementaryrowoperations(seeFig)PIVOTSOnceasystemhasbeenreduced,wecallthecoefficientofthefirstunknownineachrowapivotForexample,inthereducedsystemxyz=yz=z=sbelowdiagonalNonzeroitemscanbehereUppertriangularmatrixsabovediagonalNonzeroentriescanbehereLowertriangularmatrixFigAnillustrationofanuppertriangularmatrix,whichhassbelowthediagonal,andalowertriangularmatrix,whichhassabovethediagonalCHAPTERSystemsofLinearEquationsthepivotsareforthefirstrow,forthesecondrow,andforthelastrowThisisalsotruewhenrepresentingthesystemwithamatrixForinstance,ifthematrixA=isacoefficientmatrixforsomesystemoflinearequations,thenthepivotsare,,,andMOREONROWECHELONFORMAnechelonsystemhastwocharacteristics:•Anyrowsthatcontainallzerosarefoundatthebottomofthematrix•ThefirstnonzeroentryoneachrowisfoundtotherightofthefirstnonzeroentryintheprecedingrowAnechelonsystemgenerallyhastheformaxaxaxanxn=bajxj

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