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1、September21,20158:8Matrices:Algebra,AnalysisandApplications-9inx6inb2108-ch04page195Chapter4InnerProductSpaces4.1InnerProductDefinition4.1.1LetF=R,CandletVbeavectorspaceoverF.Then·,·:V×V→Fiscalledaninnerproductifthefollowingconditionshold:(a)ax+by,z=ax,z
2、+by,z,foralla,b∈F,x,y,z∈V,(br)forF=Ry,x=x,y,forallx,y∈V;(bc)forF=Cy,x=x,y,forallx,y∈V;(c)x,x>0forallx∈V{0}.x:=x,xiscalledthenorm(length)ofx∈V.OtherstandardpropertiesofinnerproductsarementionedinProblems1and2.WewillusetheabbreviationIPSfori
3、nnerprod-uctspace.Inthischapter,weassumethatF=R,Cunlessstatedotherwise.Proposition4.1.2LetVbeavectorspaceoverR.IdentifyVC,calledthecomplexificationofV,withthesetofpairs(x,y),195September21,20158:8Matrices:Algebra,AnalysisandApplications-9inx6inb2108-ch04page1
4、96196Matricesx,y∈V.ThenVCisavectorspaceoverCwith√(a+−1b)(x,y):=a(x,y)+b(−y,x),foralla,b∈R,x,y∈V.IfVhasabasise1,...,enoverRthen(e1,0),...,(en,0)isabasisofVCoverC.Anyinnerproduct·,·onVoverFinducesthefollowinginnerproductonVC:√(x,y),(u,v)=x,u+y,v+−1(y,
5、u−x,v),x,y,u,v∈V.Weleavetheproofofthispropositiontothereader(Problem3).Definition4.1.3LetVbeanIPS.Then(a)x,y∈Varecalledorthogonalifx,y=0.(b)S,T⊂Varecalledorthogonalifx,y=0foranyx∈S,y∈T.(c)ForanyS⊂V,S⊥⊂VisthemaximalorthogonalsettoS.(d)x1,...,xmiscalleda
6、northonormalsetifxi,xj=δij,foralli,j=1,...,m.(e)x1,...,xniscalledanorthonormalbasisifitisanorthonormalsetwhichisabasisinV.Definition4.1.4(Gram–Schmidtalgorithm)LetVbeanIPSandS={x1,...,xm}⊂Vafinite(possiblyempty)set(m≥0).ThenS˜={e1,...,ep}istheorthonormalset(
7、p≥1)ortheemptyset(p=0)obtainedfromSusingthefollowingrecursivesteps:(a)Ifx1=0removeitfromS.Otherwisereplacex1byx1−1x1.(b)Assumethatx1,...,xkisanorthonormalsetand1≤k8、eptember21,20158:8Matrices:Algebra,AnalysisandApplications-9inx6inb2108-ch04page197InnerProductSpaces197Corollary4.1.5LetVbeanIPSandS={x1,...,xn}⊂Vbenlinearlyindependentvectors.ThentheGram–Schmi