Suppose $T \in \mathcal L(V,W)$ and $v_1,\dots,v_m$ is a list of vectors in $V$ such that $Tv_1,\dots,Tv_m$ is a linearly independent list in $W$. Prove that $v_1,\dots,v_m$ is linearly independent.

We are given that

By linearity this is the same as saying

Which implies $v_1,\dots,v_m$ is linearly independent, as if it wasn't we could write $0 = a_1v_1+\dots+a_mv_m$ with some $a_j\ne 0$ and have $T(0) = 0$.