Suppose $v_1,...,v_m$ is a linearly independent list, Will the list of vectors $Tv_1,...,Tv_m$ be a linearly independent list. Suppose $v_1,...,v_m$ is a linearly independent list of vectors in $V$ and $T∈L(V,W)$ is a linear map from V to W. Will the list of vector $Tv_1,...,Tv_m$ be a linearly independent list in W? If it is, please give a rigorous proof; if not, give a counter-example.
I know that if we reverse the condition and results, it's easy to prove v1,...,vm is linearly independent, but I'm get stuck in this one. 

Short Answer: No, for example let $Tv=0$ for all $v\in V$.
Long Answer: If null space of $T$ is $\{0\}$ you can conclude that $Tv_1, \dots, Tv_m$ are linearly independent. Otherwise, they are linearly dependent.
