Let $L: V \rightarrow W$ be an injective linear transformation.
Let $dim V , dimW = n$.
Show that L is surjective.
If $L(V)$ is the image of $V$, we can show that that $L: V \rightarrow L(V)$ is a bijection and thus that $dim L(V) = n$. And then from there maybe you can show that L(V) has to be equal to W (surjective) maybe with a proof by contradiction but I can't quite make it work...