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1、CHAPTER3LinearMappingsChapter3abstractstheconceptofamatrixasalinearmappingofonelinearspaceintoanother.AgainIpointoutthatnogreatergeneralityisachieved,sowhathasbeengained?Firstofall,simplicityofnotation;wecanrefertomappingsbysinglesymbols,insteadofrectangul
2、ararraysofnumbers.Theabstractviewleadstosimple,transparentproofs.Thisisstrikinglyillustratedbytheproofoftheassociativelawofmatrixmultiplicationandbytheproofofthebasicresultthatthecolumnrankofamatrixequalsitsrowrank.Manyimportantmappingsarenotpresentedinmat
3、rixform;see,forexample,thefirsttwoapplicationspresentedinthischapter.Lastbutnotleast,theabstractviewisindispensableforinfinite-dimensionalspaces.Theretheviewofmappingsasinfinitematriceshasheldupprogressuntilitwasreplacedbyanabstractconcept.Amappingfromones
4、etXintoanothersetUisafunctionwhoseargumentsarepointsofXandwhosevaluesarepointsofU:f(X)=U.Inthischapterwediscussaclassofveryspecialmappings:(i)BothX,calledthedomainspace,andU,calledthetargetspace,arelinearspacesoverthesamefield.(ii)AmappingT:XUiscalledlinea
5、rifitisadditive,thatis,satisfiesT(x+))=T(x)+T(y)forallx,yinX,andifitishomogeneous,thatis,satisfiesT(kx)=kT(x)LinearAlgebraandItsApplications.SecondEdition,byPeterD.LaxCopyright2007JohnWiley&Sons,Inc.1920LINEARALGEBRAANDITSAPPLICATIONSforallxinXandallkinK.T
6、hevalueofTatxiswrittenmultiplicativelyasTx;theadditivepropertybecomesthedistributivelaw:T(x+y)=Tx+Ty.Othernamesforlinearmappingarelineartransformationandlinearoperator.Example1.Anyisomorphism.Example2.X=Upolynomialsofdegreelessthannins;T=d/ds.Example3.X=U=
7、082,Trotationaroundtheoriginbyangle0.Example4.Xanylinearspace,UtheonedimensionalspaceK,TanylinearfunctiononX.ExampleS.X=U=Differentiablefunctions,Tlineardifferentialoperator.Example6.X=U=Co(08),(Tf)(x)=ff(y)(x-y)2dy.Example7.X=08",U=08',u=Txdefinedbyu+tljx
8、j,M.Hereu=(UI,...,u",),x=Theorem1.(a)TheimageofasubspaceofXunderalinearmapTisasubspaceofU.(b)TheinverseimageofasubspaceofU,thatisthesetofallvectorsinXmappedbyTintothesubspace,isasubspaceofX.EXERCISEI.ProveThe
