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首页 A Concise Introduction to Linear Algebra(Geza S…

A Concise Introduction to Linear Algebra(Geza Schay).pdf

A Concise Introduction to Linea…

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简介:本文档为《A Concise Introduction to Linear Algebra(Geza Schay)pdf》,可适用于高等教育领域,主题内容包含GézaSchayAConciseIntroductiontoLinearAlgebraDepartmentofMathematicsGézaSch符等。

GézaSchayAConciseIntroductiontoLinearAlgebraDepartmentofMathematicsGézaSchayUniversityofMassachusettsBoston,MA,USAMathematicsSubjectClassification():AxxLibraryofCongressControlNumber:PrintedonacidfreepaperSpringerScienceBusinessMedia,LLCSpringerNewYorkDordrechtHeidelbergLondonAllrightsreservedThisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewrittenpermissionofthepublisher(SpringerScienceBusinessMedia,LLC,SpringStreet,NewYork,NY,USA),exceptforbriefexcerptsinconnectionwithreviewsorscholarlyanalysisUseinconnectionwithanyformofinformationstorageandretrieval,electronicadaptation,computerware,orbysimilarordissimilarmethodologynowknownorhereafterdevelopedisforbiddenTheuseinthispublicationoftradenames,trademarks,servicemarks,andsimilarterms,eveniftheyarenotidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyaresubjecttoproprietaryrightssoftThisbookisbasedontheauthor’sIntroductiontoLinearAlgebra,publishedbyJonesBartlettinISBNDOI(eBook)SpringerispartofSpringerScienceBusinessMedia(wwwbirkhsersciencecom)äuForMATLABandSimulinkproductinformation,pleasecontact:TheMathWorks,IncAppleHillDrive,Natick,MA,USATel:Fax:Email:infomathworkscomWeb:mathworkscomMATLABisaregisteredtrademarkofTheMathWorks,IncContentsPrefaceviiAnalyticGeometryofEuclideanSpacesVectorsLengthandDotProductofVectorsinRnLinesandPlanesSystemsofLinearEquations,MatricesGaussianEliminationTheTheoryofGaussianEliminationHomogeneousandInhomogeneousSystems,Gauss–JordanEliminationTheAlgebraofMatricesTheInverseandtheTransposeofaMatrixVectorSpacesandSubspacesGeneralVectorSpacesSubspacesSpanandIndependenceofVectorsBasesDimension,OrthogonalComplementsChangeofBasisLinearTransformationsRepresentationofLinearTransformationsbyMatricesPropertiesofLinearTransformationsApplicationsofLinearTransformationsinComputerGraphicsOrthogonalProjectionsandBasesOrthogonalProjectionsandLeastSquaresApproximationsOrthogonalBasesvviContentsDeterminantsDeterminants:DefinitionandBasicPropertiesFurtherPropertiesofDeterminantsTheCrossProductofVectorsinREigenvaluesandEigenvectorsEigenvaluesandEigenvectors,BasicPropertiesDiagonalizationofMatricesPrincipalAxesComplexMatricesNumericalMethodsLUFactorizationScaledPartialPivotingTheComputationofEigenvaluesandEigenvectorsAAppendicesAImplicationandEquivalenceAComplexNumbersFurtherReadingIndexPrefaceThisbookisdesignedforaonesemester,postcalculuslinearalgebracourse,primarilyintendedformathematics,physics,andcomputersciencemajorsWhilebasiccalculusisaprerequisiteforsuchacourse,verylittleofitisusedinthebookCertainly,multivariablecalculusisnotrequiredVectorsaretreatedfullyinChapter,butforclassesfamiliarwiththem,thischaptermaybeskippedorjustreviewedbrieflyComplexnumbers,series,andexponentialsarepresentedbrieflyinanappendix,buttheyareneededonlyinSection,whichmaynotbecoveredinsomecoursesTheselectionoftopicsconformstoalargeextenttotherecommendationsoftheLinearAlgebraCurriculumStudyGroupThemaindifferencesarethatthebookbeginswithachapteronEuclideanvectorgeometry,mostlyinthreedimensionsdeterminantsaretreatedmorefullyandareplacedjustbeforeeigenvalues,whichiswheretheyareneededtheLUfactorizationisrelegatedtoChapteronnumericalmethodsandthefactsaboutlineartransformationsarecollectedinonechapterandaretreatedinmoredetailThisbookisconsiderablyshorterthanthetopagesofmostintroductorylinearalgebrabooks,whicharemoresuitablefortwoorthreesemestercoursesWhilemanyapplicationsarepresented,theyaremostlytakenfromphysics,andseveralnewoneshavebeenaddedinthesecondeditionHowever,theseexamplesgiveonlyaglimpseofhowthesubjectisusedinotherfields,andfurtherdetailsarelefttotextsinthosefieldsThereis,though,asectiononcomputergraphicsandachapteronnumericalmethodsAlso,mostsectionscontainMATLABexercisesOntheotherhand,wehopethatthestudent’sinterestwillbearousednotonlybythepossibleapplications,butalsobythegeometricalbackgroundandthebeautifulstructureoflinearalgebraNevertheless,forreadersespeciallyinterestedinapplications,alistoftheonesdiscussedfollowsthisprefaceThemoredifficultexercisesandtheoremsaremarkedbyanasteriskSomeexercisesareusedtodevelopnewtopics,whoseinclusioninthemaintextwouldhavedisruptedtheflowofideasThesymbolsandareusedtoindicatetheendofproofsandexamples,respectivelyDavidCarlson,CharlesRJohnson,DavidCLay,ADuanePorterTheLinearAlgebraCurriculumStudyGroupRecommendationsfortheFirstCourseinLinearAlgebraCollegeMathematicsJournal,:()–viiviiiPrefaceInthissecondedition,inresponsetotheconcernsofsomeusersofthefirstedition,manyoftheearlierproofsandexplanationshavebeenexpandedandafewnewonesaddedAlso,exercisesinvolvinglaboriouscomputationshavebeenreplacedbysimplerones,andsomenewoneshavebeenaddedForewordtoInstructors•ThebrevitymentionedabovemakesthebookeasiertouseImportantpointsarenotdrownedinaseaofdetail,andinstructorsandstudentsdonothavetosearchforwhattokeepandwhattoomitInaminimalcourse,however,thefollowingsectionsmaybeomittedentirely:Sectiononcomputergraphics,Sectiononorthogonalprojectionsandleastsquares,Sectiononcofactorexpansionsofdeterminants,Cramer’srule,etc,Sectiononthecrossproduct,Sectionsandonprincipalaxesandcomplexmatrices,andChapteronnumericalmethodsTheorem(TheExchangeTheorem)mayalsobeomitted,sinceanalternativedirectproofofthedimensiontheoremisprovidedinthenewedition•ThegeometriccontentisheavilyemphasizedInfact,asmentionedabove,thebookbeginswithachapteronEuclideanvectorgeometry,mostlyinthreedimensionsMostothersimilartextbooksstartwiththesolutionoflinearsystemsWebelievethatthisearlyintroductionofthegeometricalbackgroundhelpsstudentstovisualizetheconceptsoflinearalgebraandprovideseasyconcreteexamplesAdditionally,manystudentsinthiscourse,eg,computersciencemajors,arenotrequiredtotakemultivariablecalculus,anddonotseethisimportantmaterialanywhereelse•Inthefirstchapter,theequationsofplanesaregiveninbothparametricandnonparametricform,incontrasttomostcalculusbooks,whichpresentonlythenonparametricformManyexamplesandexercisesillustratethetransitionfromoneformtotheotherHowever,weavoidusingthecrossproductatthisstage,becauseitisonlyavailableinRWeusethemethodofsolvingsimultaneousequationstoobtainanormalvectortoaplane,andthistopicisrevisitedasanexampletoGaussianeliminationOntheotherhand,Sectionisdevotedtothecrossproductasanillustrationoftheuseofdeterminants,anditisonlyatthatpointthatitisusedtoobtainanormalvectortoaplane•The“backandforth”processbetweenparametricandnonparametricequationsforlinesandplaneslaysthegroundworkforthesametransitionbetweendescribingasubspaceofRnasasetoflinearcombinationsorasthesolutionsetofahomogeneoussystemoflinearequations,thatis,asthecolumnspaceofamatrixorthespaceofanothermatrixAnothergeneralizationofthisissueisfindingorthogonalcomplementsofsubspacesofRngivenineitherform•Manybooksusethenotation‖p‖forthelengthofavectorpinRn,butweprefer|p|,becauseinRlengthistheabsolutevalue,andthereisnoPrefaceixreasontochangenotationforhigherdimensions,justastherewasnoneinusingforadditionofbothscalarsandofvectorsThenotation‖p‖isleftforothernorms•ImportantconceptsarepresentedasdefinitionsandtheoremsStudentsareadvisedtomemorizethemItisnotenoughjusttounderstandthematerialthemainconceptsmustberememberedwelltobeabletobuildonthem•ExceptfortheSpectralTheoreminthecomplexcaseandtheoremsfromotherfieldsofmathematics,alltheoremsareprovedItisthuslefttotheinstructortoadjustthelevelofthecoursefromthecomputationaltothefairlytheoreticalbyomittingasmanyorasfewproofsasdesired•Greatcarehasbeentakentomotivateeverynewconcept,eventhosethatmanybooksdonot,suchasdotproduct,matrixoperations,linearindependence(notjustintwoorthreedimensions),determinants,eigenvalues,andeigenvectors•Thelettersymbolsareselectedtoreflecttheconnectionsbetweenrelatedquantities,aprincipleoftenignoredinotherlinearalgebrabooksVectorsandtheircomponents,matricesandtheircolumnandrowvectorsandentriesaredenotedbythesameletterswithdifferentfonts,likev,viandA,ai,aj,aijThemainexceptionistheunitmatrix,whichis,bowingtotradition,denotedbyI,itscolumnsbyei,anditsentriesbyδij•Onlystandardnotationisused,sothatstudentswhostudyfurther,willhavenodifficultyinreadingappliedormoreadvancedtextsNonstandardnotation,suchastheuseofalistinparenthesesforcolumnvectorsandinbracketsforrowvectors,oraiorAiforarowvectorofamatrix,foundinsomeotherintroductorylinearalgebrabooks,isavoidedWeuseaiforthecolumnvectorsofamatrixAandaiforitsrowvectorsThisisstandardnotationinmoreadvancedbooks(See,eg,IntroductiontoLinearandNonlinearProgrammingbyDavidGLuenberger,AddisonWesley,)WealsousexA=(xA,xA,,xAn)TforthecoordinatevectorofavectorxrelativetoanorderedbasisorbasismatrixA(Comparethis,eg,withthenotationxB=(c,c,cn)TofLinearAlgebraandItsApplicationsbyDavidLay,AddisonWesley,,wherethebracketsontheleftaresuperfluous,thecoordinatesofxaredenotedbytheunrelatedletterc,andthebasisBisnotindicatedontheright,nottomentionthatweneedanorderedbasisorbasismatrixhere)Ournotationmakesthenotoriouslymessytopicofchangeofbasismuchsimpler•Similarityofmatricesisintroducedinthecontextofchangingbases•MostintroductorylinearalgebrabooksintroducedeterminantsbyunmotivatedformulasThisbookintroducesthembythreesimpleproperties,expandingontheapproachinStrangGilbertStrang,LinearAlgebraanditsApplications,rdedHarcourtBrace,SanDiego,xPreface•MATLABexercisesattheendofmostsectionsreinforceandexpandthelinearalgebramaterialTheyalsoprovidesomeintroductiontoMATLAB,butshouldbeusedinconjunctionwithaMATLABmanual•Theappendixonimplicationandequivalenceintroducesthestudentinaninformalwaytocertaincrucialelementsofproofs,andishighlyrecommendedreadingformost•Alldisplayedequationsarenumbered,andinthenewedition,mnemonicheadingsareappendedtoalldefinitions,theorems,figures,andexamplesThesenumbersandheadingsshouldmakereferencestotheseitemseasierandmaketheirconnectionsmoretransparentForewordtoStudentsLinearalgebraisprobablyyourfirstmathematicscourseinwhichthetheoryisjustasimportantasthecomputationsTostudyfromthisbookyouhavetocarefullyreadthetextwithpaperandpencilinhandThebookstartsoutgently,withanalyticgeometry,butsoonthealgebratakesoverandthesubjectbecomesmoreabstract,whichmaycausesomedifficultyforsomeofyouStudyingthiskindofmathematicsinvolvesthreeinterwovensteps:YoumustunderstandthematerialYoumustlearntheconceptsthoroughlysothatyourememberthemandcanapplythemknowledgeablyYoumustpracticeit,doingexercisesEachofthesestepsisnecessaryandsupportstheothersInmanyothersubjects,understandingisnotaproblem,andsomanystudentsbelievethatoncetheypassthathurdle,theyhavedoneenoughNottrue:Ifyouunderstandsomethinginclass,thatdoesnotmeanyouwillknowitthenextdayYoumuststudyaftereveryclassandmakesurethatyouareabletoexplainthematerialinyourownwordssothatyoudonotforgetitIfyoudon’t,thenyouhavetostartoveragainonyourown,withtheclassattendancewastedYouwillneedtostudyseveralhoursaftereveryclassThisisespeciallyimportant,becausemostconceptsarebuiltuponeachotherForinstance,vectors,introducedinSection,areusedthroughoutthebookmatricesintroducedinSectionareusedthroughouttherestofthebook,andsoonOntheotherhand,youcannotdomathematicsbyrotememorizationwithoutunderstanding,becausethesubjectisgenerallytoocomplicatedforthatAlso,doingthatwoulddefeatthewholepurposeofstudyingmathematics,whichisthecomprehensionofitslogicandtheabilitytouseitinapplicationsnotjustinthosethatwerepresented,butinothersimilar(orevensomewhatdifferent)applicationsPrefacexiWorkingoutsolutionstotheexercisesreinforcesboththelearningandtheunderstandingofthematerialandisoftenalsousefulinitsownright,becausemanyexercisesinvolveimportantapplicationsofthetheoryInstudyinglinearalgebra,youhavetothoroughlyunderstandandrememberthedefinitionsfirst,sinceeverythingelseisbuiltonthemIfyoudon’trememberadefinition,youcannotpossiblyunderstandthetheorythatdependsonitandtheexercisesthatmakeuseofitNextinimportancecomethetheorems,lemmas(minororauxiliarytheorems),andcorollariesTheseareusuallyprecededbyintroductoryexamplesandfollowedbyfurtherexamplesthatilluminatevariousaspectsandapplicationsofthetheoremsYoumuststudytheseexamplestogetherwiththetheoremsandtheirproofsItispermissibletoreadeverythingjustsuperficiallyatfirst,togetabasicunderstanding,butafterthat,youmuststudyitagainindetailWhenstudyingatheorem,isolatetheconditionsorhypotheseswhichmakeittickTrytoseewheretheseconditionsareusedintheproof,andwhatwouldhappenifaconditionwerechangedoromittedAfterpinpointingtheconditions,dothesamefortheconclusions,andlast,trytofollowthestepsoftheproofThisiswherethepaperandpencilcomein:WritethesestepsdownClosethebookandwritedowntheconditions,theconclusions,orthewholestatementthatyouarestudyingTrytofillinstepsthatarejustbrieflyindicatedintheproofsIftheproofhasareferencetosomeearliermaterial,besuretolookitupandexplaintoyourselfhowitisusedThesameadviceappliestothefollowupexamplesaswell:makesureyouseewheretheconditionsofthetheoremareusedandwhytheyarenecessary,andfollowthecomputationsonpaperThereisanappendixonimplicationandequivalence,whichintroducesinaninformalwaycertaincrucialelementsofproofsItishighlyrecommendedreadingforallthosewhohavenotseenthismaterialbeforeFinally,afteryouhavegonethroughthestepslistedabove,youwillbereadytotackleexercisesTheoddnumberedoneshavesolutionsavailableinaStudents’SolutionManualonthebook’swebpageDothoseexercisesfirsttheyareusuallysimilartoexamplesinthetextDon’tlookatthesolutionbeforemakingareallyseriousattempttosolveaproblemonyourownIfaproblemlookstoodifficultatfirst,thenlookatasimilarexampleinthetextorgobackandreviewthedefinitionortheoremthattheproblemisintendedtoillustrateAproblemthatyouhavesolvedstaysmuchbetterinyourmindthanonethatyouhavemerelyread,anditsstructurebecomesmuchclearerBut,ofcourse,onceyouhavesolvedaproblem,thereisnoharminlookingupthesolutionYoumayevenlearnadifferentwayofsolvingit,orfindanerrorinyoursolution(orperhapsinthesolutionmanual)Ifyoufollowtheadviceabove,youwillprobablyfindlinearalgebratobeaveryinterestingandenjoyablesubject,butifyoudon’t,t

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