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1、Chapter3VectorSpacesTheoperationsofadditionandscalarmultiplicationareusedinmanycontextsinmathematics.Regardlessofthecontext,however,theseoperationsusuallyobeythesamesetofalgebrarules.Thusageneraltheoryofmathematicalsystemsinvolvingadditionandscalarmultip
2、licationwillhaveapplicationtomanyareasinmathematics.Mathematicalsystemsofthisformarecalledvectorspacesorlinearspaces.1DefinitionandExamplesEuclideanVectorSpacesRnIngeneral,scalarmultiplicationandadditioninRnaredefinedbyandforanyandanyscalar。TheVectorSpac
3、eRm×nRm×ndenotethesetofallm×nmatriceswithrealentries.DefinitionLetVbeasetonwhichtheoperationsofadditionandscalarmultiplicationaredefined.Bythiswemeanthat,witheachpairofelementsxandyinV,wecanassociateauniqueelementsx+ythatisalsoinV,andwitheachelementxinVa
4、ndeachscalar,wecanassociateauniqueelementxinV.ThesetVtogetherwiththeoperationsofadditionandscalarmultiplicationissaidtoformavectorspaceifthefollowingaxiomsaresatisfied.VectorSpaceAxiomsA1.x+y=y+xforanyxandyinV.A2.(x+y)+z=x+(y+z)foranyx,y,zinV.A3.Thereexi
5、stsanelement0inVsuchthatx+0=xforeachx∈V.A4.Foreachx∈V,thereexistsanelement–xinVsuchthatx+(-x)=0.A5.α(x+y)=αx+αyforeachscalarαandanyxandyinV.A6.(α+β)x=αx+βxforanyscalarsαandβandanyx∈V.A7.(αβ)x=α(βx)foranyscalarsαandβandanyx∈V.A8.1·x=xforallx∈V.Theclosurep
6、ropertiesofthetwooperations:C1.Ifx∈Vandαisascalar,thenαx∈V.C2.Ifx,y∈V,thenx+y∈V.ExampleLetW={(a,1)areal}withadditionandscalarMultiplicationdefinedintheusualway.ExampleLetSbethesetofallorderedpairsofrealnumbers.DefinescalarmultiplicationandadditiononSbyWe
7、usethesymboltodenotetheadditionoperationforthissystemavoidconfusionwiththeusualadditionx+yofrowvectors.ShowthatS,withtheordinaryscalarmultiplicationandadditionoperation,isnotavectorspace.Whichoftheeightaxiomsfailtohold?TheVectorSpaceC[a,b]C[a,b]denotethe
8、setofallreal-valuedfunctionsthataredefinedandcontinuousontheclosedinterval[a,b].TheVectorSpacePnPndenotethesetofallpolynomialsofdegreelessthann.Theorem3.1.1IfVisavectorspaceanfxisanyElementofV,then0x=0.(2)x+y=0impliesthaty
