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1、CHAPTERIFundamentalsThisfirstchapteraimstointroducethenotionofanabstractlinearspacetothosewhothinkofvectorsasarraysofcomponents.Iwanttopointoutthattheclassofabstractlinearspacesisnolargerthantheclassofspaceswhoseelementsarearrays.Sowhatisgainedbythisabstraction?Firstofall,thefreedomtouseasinglesymbo
2、lforanarray;thiswaywecanthinkofvectorsasbasicbuildingblocks,unencumberedbycomponents.Theabstractviewleadstosimple,transparentproofsofresults.Moretothepoint,theelementsofmanyinterestingvectorspacesarenotpresentedintermsofcomponents.Forinstance,takealinearordinarydifferentialequationofdegreen;thesetof
3、itssolutionsformavectorspaceofdimensionn,yettheyarenotpresentedasarrays.Eveniftheelementsofavectorspacearepresentedasarraysofnumbers,theelementsofasubspaceofitmaynothaveanaturaldescriptionasarrays.Take,forinstance,thesubspaceofallvectorswhosecomponentsadduptozero.Lastbutnotleast,theabstractviewofvec
4、torspacesisindispensableforinfinite-dimensionalspaces;eventhoughthistextisstrictlyaboutfinite-dimensionalspaces,itisagoodpreparationforfunctionalanalysis.Linearalgebraabstractsthetwobasicoperationswithvectors:theadditionofvectors,andtheirmultiplicationbynumbers(scalars).Itisastonishingthatonsuchslen
5、derfoundationsanelaboratestructurecanbebuilt,withromanesque,gothic,andbaroqueaspects.Itisevenmoreastoundingthatlinearalgebrahasnotonlytherighttheoremsbutalsotherightlanguageformanymathematicaltopics,includingapplicationsofmathematics.AlinearspaceXoverafieldKisamathematicalobjectinwhichtwooperationsa
6、redefined:Addition,denotedby+,asin(1)LinearAlgebraandItsApplications.SecondEdition,byPeterD.LaxCopyright`'!2007JohnWiley&Sons,Inc.12LINEARALGEBRAANDITSAPPLICATIONSandassumedtobecommutative:x+y=y+x,(2)andassociative:x+(y+z)=(x+y)+z,(3)andtoformagroup,withtheneutralelementdenotedas0:x+0=x.(4)Theinvers
7、eofadditionisdenotedby-:x+(-x)=-x-x=0.(5)EXERCISEI.Showthatthezeroofvectoradditionisunique.ThesecondoperationismultiplicationofelementsofXbyelementskofthefieldK:kx.Theresultofthismultiplicationisavect
