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1、CHAPTER4MatricesInExample7ofChapter3wedefinedaclassofmappingsT:R"->U8mwheretheithcomponentofu=Txisexpressedintermsofthecomponentsxlofxbytheformulau(tjlxl,t=l,...,to(1)andthetjlarearbitraryscalars.Thesemappingsarelinear;conversely,wehavethefollowingtheorem.Theo
2、rem1.EverylinearmapTx=ufromR"toR'canbewritteninform(1).Proof.Thevectorxcanbeexpressedasalinearcombinationoftheunitvectorse1,...,ewhereejhasjthcomponent1,allothers0:x=Exlel.(2)SinceTislinearu=Tx=>xjTel.(3)DenotetheithcomponentofTelbytjl:tij=(Tel);.(4)LinearAlge
3、braandItsApplications,SecondEdition,byPeterD.LaxCopyrightR)2007JohnWiley&Sons,Inc.32MATRICES33Itfollowsfrom(3)and(4)thattheithcomponentu,ofItisUi=xjtij,exactlyasinformula(1).Itisconvenientandtraditionaltoarrangethecoefficientstijappearingin(1)inarectangulararr
4、ay,t1It12...tin(5)121t,n!...tawSuchanarrayiscalledanmbyn(mxn)matrix,mbeingthenumberofrows,nthenumberofcolumns.Amatrixthathasthesamenumberofrowsandcolumnsiscalledasquarematrix.ThenumberstijarecalledtheentriesofthematrixT.AccordingtoTheorem1,thereisa1-to-Icorres
5、pondencebetweenmxnmatricesandlinearmappingsT:Ifs"->R"'.Weshalldenotethe(ij)thentrytijofthematrixidentifiedwithTbyTij=(T)ij.(5)'AmatrixTcanbethoughtofasarowofcolumnvectors,oracolumnofrowvectors:riT=(ci,...,cn)_cj=ri=(tii,...,tin)(6)r,,,Accordingto(4),theithcomp
6、onentofTejistij;accordingto(6),theithcomponentofcjistij.ThusTej=cj.(7)Thisformulashowsthat,asconsequenceofthedecisiontoputtijintheithrowandjthcolumn,theimageofejunderTappearsasacolumnvector.Tobeconsistent,weshallwriteallvectorsinU=68ascolumnvectors:uium34LINEA
7、RALGEBRAANDITSAPPLICATIONSWeshallalsowriteelementsofX=I8"ascolumnvectors:x=Thematrixrepresentation(6)ofalinearmaplfrom118"toIisasinglerowvectorofncomponents:(8)Wedefineby(8)theproductofarowvectorrwithacolumnvectorx,inthisorder.Itcanbeusedtogiveacompactdescript
8、ionofformula(1)givingtheactionofamatrixonacolumnvector:rixTx=(9)r,,,Xwherer1,...,r,,,aretherowsofthematrixT.InChapter3wehavedescribedthealgebraoflinearmappings.Sincematricesreprese
