如果您无法下载资料，请参考说明：
1、部分资料下载需要金币，请确保您的账户上有足够的金币
2、已购买过的文档，再次下载不重复扣费
3、资料包下载后请先用软件解压，在使用对应软件打开
Theoperationsofadditionandscalarmultiplicationareusedinmanycontextsinmathematics.Regardlessofthecontext,however,theseoperationsusuallyobeythesamesetofalgebrarules.Thusageneraltheoryofmathematicalsystemsinvolvingadditionandscalarmultiplicationwillhaveapplicationtomanyareasinmathematics.Mathematicalsystemsofthisformarecalledvectorspacesorlinearspaces.1DefinitionandExamplesTheVectorSpaceRm×nDefinitionLetVbeasetonwhichtheoperationsofadditionandscalarmultiplicationaredefined.Bythiswemeanthat,witheachpairofelementsxandyinV,wecanassociateauniqueelementsx+ythatisalsoinV,andwitheachelementxinVandeachscalar,wecanassociateauniqueelementxinV.ThesetVtogetherwiththeoperationsofadditionandscalarmultiplicationissaidtoformavectorspaceifthefollowingaxiomsaresatisfied.A1.x+y=y+xforanyxandyinV.A2.(x+y)+z=x+(y+z)foranyx,y,zinV.A3.Thereexistsanelement0inVsuchthatx+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.Theclosurepropertiesofthetwooperations:C1.Ifx∈Vandαisascalar,thenαx∈V.C2.Ifx,y∈V,thenx+y∈V.ExampleLetSbethesetofallorderedpairsofrealnumbers.DefinescalarmultiplicationandadditiononSbyTheVectorSpaceC[a,b]C[a,b]denotethesetofallreal-valuedfunctionsthataredefinedandcontinuousontheclosedinterval[a,b].TheVectorSpacePnPndenotethesetofallpolynomialsofdegreelessthann.Theorem3.1.1IfVisavectorspaceanfxisanyElementofV,then0x=0.(2)x+y=0impliesthaty=-x.(3)(-1)x=-x.2SubspacesExample1、Let,SisasubspaceofR3.TheNullspaceofaMatrixLetAbeanm×nmatrix.LetN(A)denotethesetofallsolutionstothehomogeneoussystemAx=0.ThusTheSpanofaSetofVectorsTheorem3.2.1Ifv1,v2,…,vnareelementsofavectorspaceV,thenSpan(v1,v2,…,vn)isasubspaceofV.ExampleWhichofthefollowingarespanningsetsforR3?3LinearIndependenceConclusion:Ifv1,v2,…,vnspanavectorspaceVandoneofthesevectorscanbewrittenasalinearcombinationoftheothern-1vectors,th
