Re: [理工] [線代]-線性映射
※ 引述《CCZR (阿翔)》之銘言：
: 1.Let a be a basis for a finite dimensional vector space V and
: T(a)={T(v),v屬於a} If T is a linear operator on V then T is
: onto if T(a) is a basis for V.
: 答案是true 但是他只有說他是onto沒說是one to one? 怎麼確定
: 他映射出來的會是基底?
: 2.兩個方陣A B
: rank(AB)=rank(A) if and only if B是非奇異
: 有辦法證明嗎? 解答都只舉反例
1.
T:V->V
if T is onto ->rank(T)=dim(V)
->N(T)=0
->It's one-to-one
2.
(1)rank(AB)=rank(A)
<=min{rank(A),rank(B)}
只能知道rank(B)>rank(A)
並不能保證B是full rank
(2)if B is full rank
rank(A)
=rank(ABB^-1)
<=min(rank(AB),rank(B^-1))
<=rank(AB)
rank(AB)<=min(rank(A),rank(B))
<=rank(A)
Rank(AB)<=rank(A) ...(1)
Rank(AB)=>rank(A) ...(2)
=>Rank(AB)=Rank(A)
所以這是 B is full rank =>rank(AB)=rank(A)
但是推不回來,所以不是if and only if
反例也算證明的一種,因為要否定它是對的.只要能找一個錯的case就行了.
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◆ From: 123.193.214.165
※ 編輯: iyenn 來自: 123.193.214.165 (11/03 10:07)
※ 編輯: iyenn 來自: 123.193.214.165 (11/03 17:12)
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