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Math Help - Nullspace and Range (Image)?

  1. #1
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    Nullspace and Range (Image)?

    Let V and W be vector spaces, let T: V --> W be linear, and let {w1, w2,...,wk} be a linearly independant subset of R(T). Prove that if S = {v1,v2,...,vk} is chosen so that T(vi) = wi for i = 1, 2 ,...,k, then S is linearly independent.

    My prof had mention about taking the image of both sides, however I still don't see where to start
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  2. #2
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    Quote Originally Posted by Qt3e_M3 View Post
    Let V and W be vector spaces, let T: V --> W be linear, and let {w1, w2,...,wk} be a linearly independant subset of R(T). Prove that if S = {v1,v2,...,vk} is chosen so that T(vi) = wi for i = 1, 2 ,...,k, then S is linearly independent.
    Say that a_1\bold{v}_1 + ... + a_k \bold{v}_k = \bold{0}.
    Thus, T(a_1\bold{v}_1 + ... + a_k \bold{v}_k) = T(\bold{0}) = \bold{0}.
    Thus, a_1T(\bold{v}_1) + ... + a_k T(\bold{v}_k) = 0.
    Thus, a_1\bold{w}_1 + ... + a_k \bold{w}_k = 0.
    Thus, a_1=...=a_k=0 since \{\bold{w}_1,...,\bold{w}_k\} is linearly independent.
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