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首页 线性代数

线性代数

线性代数

光芒四季
2009-03-03 0人阅读 举报 0 0 暂无简介

简介:本文档为《线性代数pdf》,可适用于工程科技领域

LINEARALGEBRASecondEditionKENNETHHOFFMANProfessorofMathematicsMassachusettsInstituteofTechnologyRAYKUNZEProfessorofMathematicsUniversityofCalifornia,IrvinePRENTICEHALL,INC,EnglewoodCliffs,NewJersey,byPrenticeHall,IncEnglewoodCliffs,NewJerseyAllrightsreservedNopartofthisbookmaybereproducedinanyformorbyanymeanswithoutpermissioninwritingfromthepublisherPRENTICEHALL~NTERNATXONAL,INC,LondonPRENTICEHALLOFAUSTRALIA,PTYLTD,SydneyPRENTICEHALLOFCANADA,LTD,TorontoPRENTICEHALLOFINDIAPRIVATELIMITED,NewDelhiPRENTICEHALLOFJAPAN,INC,TokyoCurrentprinting(lastdigit):LibraryofCongressCatalogCardNoPrintedintheUnitedStatesofAmericaPfreaceOuroriginalpurposeinwritingthisbookwastoprovideatextfortheundergraduatelinearalgebracourseattheMassachusettsInstituteofTechnologyThiscoursewasdesignedformathematicsmajorsatthejuniorlevel,althoughthreefourthsofthestudentsweredrawnfromotherscientificandtechnologicaldisciplinesandrangedfromfreshmenthroughgraduatestudentsThisdescriptionoftheMITaudienceforthetextremainsgenerallyaccuratetodayThetenyearssincethefirsteditionhaveseentheproliferationoflinearalgebracoursesthroughoutthecountryandhaveaffordedoneoftheauthorstheopportunitytoteachthebasicmaterialtoavarietyofgroupsatBrandeisUniversity,WashingtonUniversity(StLouis),andtheUniversityofCalifornia(Irvine)OurprincipalaiminrevisingLinearAlgebrahasbeentoincreasethevarietyofcourseswhichcaneasilybetaughtfromitOnonehand,wehavestructuredthechapters,especiallythemoredifficultones,sothatthereareseveralnaturalstoppingpointsalongtheway,allowingtheinstructorinaonequarteroronesemestercoursetoexerciseaconsiderableamountofchoiceinthesubjectmatterOntheotherhand,wehaveincreasedtheamountofmaterialinthetext,sothatitcanbeusedforarathercomprehensiveoneyearcourseinlinearalgebraandevenasareferencebookformathematiciansThemajorchangeshavebeeninourtreatmentsofcanonicalformsandinnerproductspacesInChapterwenolongerbeginwiththegeneralspatialtheorywhichunderliesthetheoryofcanonicalformsWefirsthandlecharacteristicvaluesinrelationtotriangulationanddiagonalizationtheoremsandthenbuildourwayuptothegeneraltheoryWehavesplitChaptersothatthebasicmaterialoninnerproductspacesandunitarydiagonalizationisfollowedbyaChapterwhichtreatssesquilinearformsandthemoresophisticatedpropertiesofnormaloperators,includingnormaloperatorsonrealinnerproductspacesWehavealsomadeanumberofsmallchangesandimprovementsfromthefirsteditionButthebasicphilosophybehindthetextisunchangedWehavemadenoparticularconcessiontothefactthatthemajorityofthestudentsmaynotbeprimarilyinterestedinmathematicsForwebelieveamathematicscourseshouldnotgivescience,engineering,orsocialsciencestudentsahodgepodgeoftechniques,butshouldprovidethemwithanunderstandingofbasicmathematicalconceptsamPrefaceOntheotherhand,wehavebeenkeenlyawareofthewiderangeofbackgroundswhichthestudentsmaypossessand,inparticular,ofthefactthatthestudentshavehadverylittleexperiencewithabstractmathematicalreasoningForthisreason,wehaveavoidedtheintroductionoftoomanyabstractideasattheverybeginningofthebookInaddition,wehaveincludedanAppendixwhichpresentssuchbasicideasasset,function,andequivalencerelationWehavefounditmostprofitablenottodwellontheseideasindependently,buttoadvisethestudentstoreadtheAppendixwhentheseideasariseThroughoutthebookwehaveincludedagreatvarietyofexamplesoftheimportantconceptswhichoccurThestudyofsuchexamplesisoffundamentalimportanceandtendstominimizethenumberofstudentswhocanrepeatdefinition,theorem,proofinlogicalorderwithoutgraspingthemeaningoftheabstractconceptsThebookalsocontainsawidevarietyofgradedexercises(aboutsixhundred),rangingfromroutineapplicationstooneswhichwillextendtheverybeststudentsTheseexercisesareintendedtobeanimportantpartofthetextChapterdealswithsystemsoflinearequationsandtheirsolutionbymeansofelementaryrowoperationsonmatricesIthasbeenourpracticetospendaboutsixlecturesonthismaterialItprovidesthestudentwithsomepictureoftheoriginsoflinearalgebraandwiththecomputationaltechniquenecessarytounderstandexamplesofthemoreabstractideasoccurringinthelaterchaptersChapterdealswithvectorspaces,subspaces,bases,anddimensionChaptertreatslineartransformations,theiralgebra,theirrepresentationbymatrices,aswellasisomorphism,linearfunctionals,anddualspacesChapterdefinesthealgebraofpolynomialsoverafield,theidealsinthatalgebra,andtheprimefactorizationofapolynomialItalsodealswithroots,Taylor’sformula,andtheLagrangeinterpolationformulaChapterdevelopsdeterminantsofsquarematrices,thedeterminantbeingviewedasanalternatingnlinearfunctionoftherowsofamatrix,andthenproceedstomultilinearfunctionsonmodulesaswellastheGrassmanringThematerialonmodulesplacestheconceptofdeterminantinawiderandmorecomprehensivesettingthanisusuallyfoundinelementarytextbooksChaptersandcontainadiscussionoftheconceptswhicharebasictotheanalysisofasinglelineartransformationonafinitedimensionalvectorspacetheanalysisofcharacteristic(eigen)values,triangulableanddiagonalizabletransformationstheconceptsofthediagonalizableandnilpotentpartsofamoregeneraltransformation,andtherationalandJordancanonicalformsTheprimaryandcyclicdecompositiontheoremsplayacentralrole,thelatterbeingarrivedatthroughthestudyofadmissiblesubspacesChapterincludesadiscussionofmatricesoverapolynomialdomain,thecomputationofinvariantfactorsandelementarydivisorsofamatrix,andthedevelopmentoftheSmithcanonicalformThechapterendswithadiscussionofsemisimpleoperators,toroundouttheanalysisofasingleoperatorChaptertreatsfinitedimensionalinnerproductspacesinsomedetailItcoversthebasicgeometry,relatingorthogonalizationtotheideaof‘bestapproximationtoavector’andleadingtotheconceptsoftheorthogonalprojectionofavectorontoasubspaceandtheorthogonalcomplementofasubspaceThechaptertreatsunitaryoperatorsandculminatesinthediagonalizationofselfadjointandnormaloperatorsChapterintroducessesquilinearforms,relatesthemtopositiveandselfadjointoperatorsonaninnerproductspace,movesontothespectraltheoryofnormaloperatorsandthentomoresophisticatedresultsconcerningnormaloperatorsonrealorcomplexinnerproductspacesChapterdiscussesbilinearforms,emphasizingcanonicalformsforsymmetricandskewsymmetricforms,aswellasgroupspreservingnondegenerateforms,especiallytheorthogonal,unitary,pseudoorthogonalandLorentzgroupsWefeelthatanycoursewhichusesthistextshouldcoverChapters,,andPrefaceVthoroughly,possiblyexcludingSectionsandwhichdealwiththedoubledualandthetransposeofalineartransformationChaptersand,onpolynomialsanddeterminants,maybetreatedwithvaryingdegreesofthoroughnessInfact,polynomialidealsandbasicpropertiesofdeterminantsmaybecoveredquitesketchilywithoutseriousdamagetotheflowofthelogicinthetexthowever,ourinclinationistodealwiththesechapterscarefully(excepttheresultsonmodules),becausethematerialillustratessowellthebasicideasoflinearalgebraAnelementarycoursemaynowbeconcludednicelywiththefirstfoursectionsofChapter,togetherwith(thenew)ChapterIftherationalandJordanformsaretobeincluded,amoreextensivecoverageofChapterisnecessaryOurindebtednessremainstothosewhocontributedtothefirstedition,especiallytoProfessorsHarryFurstenberg,LouisHoward,DanielKan,EdwardThorp,toMrsJudithBowers,MrsBettyAnn(Sargent)RoseandMissPhyllisRubyInaddition,wewouldliketothankthemanystudentsandcolleagueswhoseperceptivecommentsledtothisrevision,andthestaffofPrenticeHallfortheirpatienceindealingwithtwoauthorscaughtinthethroesofacademicadministrationLastly,specialthanksareduetoMrsSophiaKoulourasforbothherskillandhertirelesseffortsintypingtherevisedmanuscriptKMHRAKContentsChapterLinearEquationsFieldsSystemsofLinearEquationsMatricesandElementaryRowOperationsRowReducedEchelonMatricesMatrixMultiplicationInvertibleMatricesChapterVectorSpacesVectorSpacesSubspacesBasesandDimensionCoordinatesSummaryofRowEquivalenceComputationsConcerningSubspacesChapterLinearTransformationsLinearTransformationsTheAlgebraofLinearTransformationsIsomorphismRepresentationofTransformationsbyMatricesLinearFunctionalsTheDoubleDualTheTransposeofaLinearTransformationViContentsviiChapterPolynomialsAlgebrasTheAlgebraofPolynomialsLagrangeInterpolationPolynomialIdealsThePrimeFactorizationofaPolynomialChapterDeterminantsCommutativeRingsDeterminantFunctionsPermutationsandtheUniquenessofDeterminantsAdditionalPropertiesofDeterminantsModulesMultilinearFunctionsTheGrassmanRingChapterElementaryCanonicalFormsIntroductionCharacteristicValuesAnnihilatingPolynomialsInvariantSubspacesSimultaneousTriangulationSimultaneousDiagonalizationDirectSumDecompositionsInvariantDirectSumsThePrimaryDecompositionTheoremChapterTheRationalandJordanFormsCyclicSubspacesandAnnihilatorsCyclicDecompositionsandtheRationalFormTheJordanFormComputationofInvariantFactorsSummarySemiSimpleOperatorsChapterInnerProductSpacesInnerProductsInnerProductSpacesLinearFunctionalsandAdjointsUnitaryOperatorsNormalOperatorsContentsChapterOperatorsonInnerProductSpacesIntroductionFormsonInnerProductSpacesPositiveFormsMoreonFormsSpectralTheoryFurtherPropertiesofNormalOperatorsChapterBilinearFormsIBilinearFormsSymmetricBilinearFormsSkewSymmetricBilinearFormsGroupsPreservingBilinearFormsAppendixASetsAFunctionsAEquivalenceRelationsAQuotientSpacesAEquivalenceRelationsinLinearAlgebraATheAxiomofChoiceBibliographyIndexLinearEquationslFieldsWeassumethatthereaderisfamiliarwiththeelementaryalgebraofrealandcomplexnumbersForalargeportionofthisbookthealgebraicpropertiesofnumberswhichweshalluseareeasilydeducedfromthefollowingbrieflistofpropertiesofadditionandmultiplicationWeletFdenoteeitherthesetofrealnumbersorthesetofcomplexnumbersAdditioniscommutative,xy=yxforallxandyinFAdditionisassociative,x(Yx>=(xY)forall,y,andzinFThereisauniqueelement(zero)inFsuchthat=x,foreveryxinFToeachxinFtherecorrespondsauniqueelement(x)inFsuchthatx(x)=Multiplicationiscommutative,xy=yxforallxandyinFMultiplicationisassociative,forallx,y,andxinFdYZ>=(XY>ZLinearEquationsChapThereisauniquenonzeroelement(one)inFsuchthat~=,foreveryxinFToeachnonzeroxinFtherecorrespondsauniqueelementxl(orlx)inFsuchthatxx’=Multiplicationdistributesoveradditionthatis,x(yZ)=xyxz,forallx,y,andzinFSupposeonehasasetFofobjectsx,y,x,andtwooperationsontheelementsofFasfollowsThefirstoperation,calledaddition,associateswitheachpairofelements,yinFanelement(xy)inFthesecondoperation,calledmultiplication,associateswitheachpairx,yanelementzyinFandthesetwooperationssatisfyconditions(l)()aboveThesetF,togetherwiththesetwooperations,isthencalledafieldRoughlyspeaking,afieldisasettogetherwithsomeoperationsontheobjectsinthatsetwhichbehavelikeordinaryaddition,subtraction,multiplication,anddivisionofnumbersinthesensethattheyobeytheninerulesofalgebralistedaboveWiththeusualoperationsofadditionandmultiplication,thesetCofcomplexnumbersisafield,asisthesetRofrealnumbersFormostofthisbookthe‘numbers’weusemayaswellbetheelementsfromanyfieldFToallowforthisgenerality,weshallusetheword‘scalar’ratherthan‘number’NotmuchwillbelosttothereaderifhealwaysassumesthatthefieldofscalarsisasubfieldofthefieldofcomplexnumbersAsubfieldofthefieldCisasetFofcomplexnumberswhichisitselfafieldundertheusualoperationsofadditionandmultiplicationofcomplexnumbersThismeansthatandareinthesetF,andthatifxandyareelementsofF,soare(xy),x,xy,andzl(ifx#)AnexampleofsuchasubfieldisthefieldRofrealnumbersfor,ifweidentifytherealnumberswiththecomplexnumbers(aib)forwhichb=,theandofthecomplexfieldarerealnumbers,andifxandyarereal,soare(xy),Z,zy,andxl(ifx#)WeshallgiveotherexamplesbelowThepointofourdiscussingsubfieldsisessentiallythis:IfweareworkingwithscalarsfromacertainsubfieldofC,thentheperformanceoftheoperationsofaddition,subtraction,multiplication,ordivisiononthesescalarsdoesnottakeusoutofthegivensubfieldEXAMPLEThesetofpositiveintegers:,,,,isnotasubfieldofC,foravarietyofreasonsForexample,isnotapositiveintegerfornopositiveintegernisnapositiveintegerfornopositiveintegernexceptislnapositiveintegerEXAMPLEThesetofintegers:,,,,,,,isnotasubfieldofC,becauseforanintegern,lnisnotanintegerunlessnisorSecSystemsofLinearEquationsWiththeusualoperationsofadditionandmultiplication,thesetofintegerssatisfiesalloftheconditions(l)()exceptcondition()EXAMPLEThesetofrationalnumbers,thatis,numbersoftheformpq,wherepandqareintegersandq#,isasubfieldofthefieldofcomplexnumbersThedivisionwhichisnotpossiblewithinthesetofintegersispossiblewithinthesetofrationalnumbersTheinterestedreadershouldverifythatanysubfieldofCmustcontaineveryrationalnumberEXAMPLEThesetofallcomplexnumbersoftheformyG,wherexandyarerational,isasubfieldofCWeleaveittothereadertoverifythisIntheexamplesandexercisesofthisbook,thereadershouldassumethatthefieldinvolvedisasubfieldofthecomplexnumbers,unlessitisexpresslystatedthatthefieldismoregeneralWedonotwanttodwellonthispointhowever,weshouldindicatewhyweadoptsuchaconventionIfFisafield,itmaybepossibletoaddtheunittoitselfafinitenumberoftimesandobtain(seeExercisefollowingSection):=Thatdoesnothappeninthecomplexnumberfield(orinanysubfieldthereof)IfitdoeshappeninF,thentheleastnsuchthatthesumofnl’sisiscalledthecharacteristicofthefieldFIfitdoesnothappeninF,then(forsomestrangereason)FiscalledafieldofcharacteristiczeroOften,whenweassumeFisasubfieldofC,whatwewanttoguaranteeisthatFisafieldofcharacteristiczerobut,inafirstexposuretolinearalgebra,itisusuallybetternottoworrytoomuchaboutcharacteristicsoffieldsSystemsofLinearEquationsSupposeFisafieldWeconsidertheproblemoffindingnscalars(elementsofF)x,,x,whichsatisfytheconditionsXlAxaAl=y(l)XIxAznxn=yA:,x:A,zxzAnxn=jwhereyl,,ymandAi,im,jn,aregivenelementsofFWecall(ll)asystemofmlinearequationsinnunknownsAnyntuple(xi,,x,)ofelementsofFwhichsatisfieseachoftheLinearEquationsChapequationsin(ll)iscalledasolutionofthesystemIfyl=yZ==ym=,wesaythatthesystemishomogeneous,orthateachoftheequationsishomogeneousPerhapsthemostfundamentaltechniqueforfindingthesolutionsofasystemoflinearequationsisthetechniqueofeliminationWecanillustratethistechniqueonthehomogeneoussystemxxx=xx=Ifweadd()timesthesecondequationtothefirstequation,weobtainX=or,x=xIfweaddtimesthefirstequationtothesecondequation,weobtainxx=or,x=xSoweconcludethatif(xl,x,x)isasolutionthenx=x=xConversely,onecanreadilyverifythatanysuchtripleisasolutionThusthesetofsolutionsconsistsofalltriples(a,a,a)Wefoundthesolutionstothissystemofequationsby‘eliminatingunknowns,’thatis,bymultiplyingequationsbyscalarsandthenaddingtoproduceequationsinwhichsomeofthexjwerenotpresentWewishtoformalizethisprocessslightlysothatwemayunderstandwhyitworks,andsothatwemaycarryoutthecomputationsnecessarytosolveasysteminanorganizedmannerForthegeneralsystem(ll),supposeweselectmscalarscl,,c,,multiplythejthequationbyciandthenaddWeobtaintheequation(ClCmAml)Xl*(Clac,A,n)xn=cyG‘Suchanequationweshallcallalinearcombinationoftheequationsin(ll)Evidently,anysolutionoftheentiresystemofequations(ll)willalsobeasolutionofthisnewequationThisisthefundamentalideaoftheeliminationprocessIfwehaveanothersystemoflinearequationsXBlnXn=XlU)lx*Bk’nxn=z,,inwhicheachofthekequationsisalinearcombinationoftheequationsin(ll),theneverysolutionof(ll)isasolutionofthisnewsystemOfcourseitmayhappenthatsomesolutionsof(l)arenotsolutionsof(ll)ThisclearlydoesnothappenifeachequationintheoriginalsystemisalinearcombinationoftheequationsinthenewsystemLetussaythat

用户评价(6)

  • 10.44.7.248 呵呵,英文的,看看再说,不过好像比较难啃奥1

    2012-09-28 00:43:36

  • terrencew 有没有习题答案啊

    2011-10-17 18:05:35

  • UC 日,全英文的,要把他看完,估计要几年

    2010-02-04 17:47:36

  • 灯客 英文就暂时不下了 等英语搞好了再来 谢谢

    2010-01-24 18:54:44

  • zwwang 英文的啊,有没有中文的

    2009-11-19 22:45:30

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