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

Suppose

By linearity this is the same as saying

Since $T$ is injective $\text{null }T = \{0\}$ so this implies

Since $v_1,\dots,v_m$ are independent $a_1 = \dots = a_m = 0$ completing the proof.