[線代] N(A) = {0}, nonsingular

[mqazz1 (無法顯示)]

let A be an n*n matrix

prove that the following statements are equivalent


1.N(A) = {0}

2.A is nonsingular

3.for each b屬於R^n, the system Ax=b has a unique solution


請問這要怎麼證呢?

謝謝

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◆ From: 61.228.31.6
※ 編輯: mqazz1          來自: 61.228.31.6          (04/17 23:39)
推 yueayase    :我不清楚定義是從什麼地方出發,不過你可以參考:
→ yueayase    :Linear Algebra with Applications by Steven J.Leon
→ yueayase    :他應該會直接證給你看
推 powerkshs   :N(A) = {0}→Ａ is 1-1，因為有限維，所以1-1→onto
→ powerkshs   :所以A is nonsingular
→ powerkshs   :因為A is nonsingular，所以A可逆，可逆→有唯一解
→ powerkshs   :應該就可以了