Re: [理工] [線代] 子空間證明
※ 引述《armopen (考個沒完)》之銘言：
: ※ 引述《ofd168 (大色狼來襲)》之銘言：
: : Let L:V→W be linear transformation and let T be a subspace of W
: : The inverse image of T denote L^-1(T) is defined by
: : L^-1(T) = { v 屬於 V | L(v) 屬於 T}
: : Show that L^-1(T) is subspace of V.
: : 很直覺會成立 但是怎麼證明呢?
: : 感謝各位大大了
: 這種問題有一個直覺的反推證明方法，題目就是證明
: Let u, v in L^(-1)(T) and α in F (a field). Then αu + v in L^(-1)(T).
: i.e. L(αu + v) in T <=> α L(u) + L(v) in T (since T is linear).
: This is trivial since L(u), L(v) in T and T is a subspace of W.
推文的最後一位網友說的是外行話！ 將我上面提到的證法寫下來就是完整證明！
不懂證明如何思考就不要說些外行話，用一般人所謂嚴謹的證法重寫如下：
Let u, v in L^(-1)(T) and α in F (a field). Then L(u), L(v) in T.
Since T is a subspace of W, αL(u) + L(v) in T. Therefore, L(αu + v) in T
since T is linear. That is, αu + v in L^(-1)(T) and hence L^(-1)(T) is a
subspace of V.
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◆ From: 114.37.176.167
※ 編輯: armopen 來自: 114.37.176.167 (09/08 00:29)
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