※ 引述《mqazz1 (无法显示)》之铭言： : 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属於ker(L) if and only if [V] 属於 N(A) : E : (b) w属於L(V) if and only if [w] is in the column space of A : F : 请问这两题应该怎麽证呢? : 我翻过手上书好像没看到类似的证明@@ : 有没有好心的高手可以指点一下 谢谢!! (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) 有些符号我弄得不是很好,所以可能有些问题.... --

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