Question

线性代数证明题（英文） 求高手！！！

Let T : Rm → Rn be a linear transformation.
(1) Prove that T is injective if and only if for every linearly independent set {v1, . . . ,vk} in Rm, the

set {T(v1), . . . , T(vk)} is linearly independent in Rn.
(2) Prove that T is surjective if and only if for every set of vectors {v1, . . . ,vk} that span Rm, the set {T(v1), . . . , T(vk)} spans Rn.
(3) Use parts (1) and (2) to fill in each blank with 4 words or less, and prove the statement: T is invertible if and only if for every set of vectors {v1, . . . ,vk} ——（填空）that Rm, the set{T(v1), . . . , T(vk)} also——（填空） Rn.

求证明过程，中英文皆可！！！

Answer 1

我线性代数忘光了 就帮你第一题吧 真的是很基本的证明 自己找本书 网上下载份答案 绝对搞的定
左证右
是单射的话 如果k1v1+k1v2+...+knvn=0，等价于T(k1v1+k1v2+...+knvn)=0，等价于k1T(v1)+...knT(vn)=0，所以在Rn中的也是线性无关 其实就是用到了单射的充要条件是f(0)=0

追问

能简单说说第二问第三问的思路么？