矩阵论-线性变换

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1、5N(Linearmaps)ïJJian-BiaoChenMathematicandComputerSciencesSMUDefinitionandelementarypropertiesDef1.V(F),W(F)5m,AV(F)W(F)N,∀α,β∈V(F),∀k,l∈F,kA(kα+lβ)=kA(α)+lA(β)K¡AV(F)W(F)5N.Jian-BiaoChenMathematicandComputerSciencesSMUDefinitionandelementarypro

2、pertiesDef1.V(F),W(F)5m,AV(F)W(F)N,∀α,β∈V(F),∀k,l∈F,kA(kα+lβ)=kA(α)+lA(β)K¡AV(F)W(F)5N.·PL(V(F),W(F))=AAV(F)W(F)5N}L(V(F))=L(V(F),V(F))Jian-BiaoChenMathematicandComputerSciencesSMUDefinitionandelementarypropertiesDef1.V(F),W(F)5m,AV(F)W(F)

3、N,∀α,β∈V(F),∀k,l∈F,kA(kα+lβ)=kA(α)+lA(β)K¡AV(F)W(F)5N.·PL(V(F),W(F))=AAV(F)W(F)5N}L(V(F))=L(V(F),V(F))5A(0)=0PPmmA(kiαi)=kiA(αi)i=1i=1Jian-BiaoChenMathematicandComputerSciencesSMUDefinitionandelementaryproperties~f1"Nµ∀α∈V,0(α)=0∈W.2ðNµ∀α∈V

4、,I(α)=α(I∈L(V)).3A_NµA∈L(V,W),BWVN.∀α∈V,∀β∈W,B(A(α))=αA(B(β))=β.K¡A_,BA_,PA−1.KkA−1∈L(W,V).Jian-BiaoChenMathematicandComputerSciencesSMUDefinitionandelementaryproperties~f1"Nµ∀α∈V,0(α)=0∈W.2ðNµ∀α∈V,I(α)=α(I∈L(V)).3A_NµA∈L(V,W),BWVN.∀α∈V,∀

5、β∈W,B(A(α))=αA(B(β))=β.K¡A_,BA_,PA−1.KkA−1∈L(W,V).5A∈L(V,W)_⇐⇒∀β∈W,3α∈V¦A(α)=β.A∈L(V,W)_⇐⇒∀β∈W,3α∈V¦A(α)=βA(γ)=0§kγ=0.Jian-BiaoChenMathematicandComputerSciencesSMUNullspacesandrangesDef2.V,W5m,A∈L(V,W),NA={αA(α)=0,α∈V}RA={β∃α∈V,3β=A(α)}K¡NAA"m(

6、Nullspace),RAAm(Range).w,,NA´Vfm,RA´Wfm.Jian-BiaoChenMathematicandComputerSciencesSMUNullspacesandrangesDef2.V,W5m,A∈L(V,W),NA={αA(α)=0,α∈V}RA={β∃α∈V,3β=A(α)}K¡NAA"m(Nullspace),RAAm(Range).w,,NA´Vfm,RA´Wfm.5A∈L(V,W)_⇐⇒NA={0},RA=W.Jian-BiaoChen

7、MathematicandComputerSciencesSMUThedimensionformulaTh1.V,W5m,A∈L(V,W),edimV=n,KdimNA+dimRA=dimVJian-BiaoChenMathematicandComputerSciencesSMUThedimensionformulaTh1.V,W5m,A∈L(V,W),edimV=n,KdimNA+dimRA=dimVy²µξ1,ξ2,...,ξrNAÄ,r§¿VÄξ1,ξ2,...,ξr,ξr+1,.

8、..,ξn.ØJy²A(ξr+1),A(ξr+2),...,A(ξn)´RAÄ.dimNA+dimRA=r+(n−r)=dimVJian-BiaoChenMathematicandComputerSciencesSMUThedimensionformulaTh1.V,W5m,A∈L(V,W),edimV=n,KdimNA+dimRA=dimVy²µξ1,ξ2,...,ξrNAÄ,r§¿VÄξ1,ξ2,...,ξr,ξr+1,...,ξn.ØJy²A(ξr+1),A(ξr