Suppose $v_1,\dots,v_m$ spans $V$ and $T \in \mathcal L(V,W)$. Prove that the list $Tv_1,\dots,Tv_m$ spans $\text{range }T$.

Let $w \in \text{range }T$, since it's in the range there exists a $v \in V$ such that $Tv = w$. Writing $v$ in terms of $v_1,\dots,v_m$ gives

Which by linearity implies

Thus we can write any $w \in \text{range }T$ as a linear combination of $Tv_1,\dots,Tv_m$ implying the list spans the range of $T$.