반응형
Thm_
Let A∈Mm×n,B,C∈Mn×p,D,E∈Mq×m.
Then
(a) A(B+C)=AB+AC,(D+E)A=DA+EA
(b) a(AB)=(aA)B=A(aB) (a∈F)
(c) ImA=A=AIn
(d) If V is an n-dimensional vector space with an ordered basis, then [IV]β=In.
Proofof(c)_
(ImA)ij=m∑k=1(Im)ikAkj=m∑k=1δikAkj=Aij
Coro_
Let A∈Mm×n,B1,⋯,Bk∈Mn×p,C1,⋯,Ck∈Mq×m,a1,⋯,ak∈F.
Then
A(k∑i=1aiBi)=k∑i=1aiABi=(k∑i=1aiCi)A=k∑i=1aiCiA
반응형