# Thread: Nullspace and Range (Image)?

1. ## 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

2. Originally Posted by Qt3e_M3
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.