推 mqazz1 :謝謝!! 05/30 09:46
※ 引述《mqazz1 (無法顯示)》之銘言：
: let V and W be vector spaces with ordered bases E and F, respectively
: if L:V->W is a linear transformation and A is the matrix representing L
: relative to E and F, show that
: (a) v屬於ker(L) if and only if [V] 屬於 N(A)
: E
: (b) w屬於L(V) if and only if [w] is in the column space of A
: F
: 請問這兩題應該怎麼證呢?
: 我翻過手上書好像沒看到類似的證明@@
: 有沒有好心的高手可以指點一下 謝謝!!
(a) If v ∈ Ker(L) => L(v) = 0 => Av = 0 => [v] ∈ N(A)
E
If [v] ∈ N(A) => Av = 0 => L(v) = 0 => v ∈ Ker(L)
E
(b) If w ∈ L(V) => Av = w => a v1 + a v + ... a v = w, where
1 2 2 n n
a is the ith column of A for 1 <= i <= n
i
So, w ∈ Rank(A) (column space of A)
If w ∈ Rank(A) (column space of A) => Av = w for some v ∈ V
=> L(v) = w. So, w ∈ L(V)
有些符號我弄得不是很好,所以可能有些問題....
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