Injectivity is equivalent to surjectivity in finite dimensions

Suppose $V$ is finite-dimensional and $T \in \mathcal L(V)$. Then the following are equivalent:

1. $T$ is invertible
2. $T$ is injective
3. $T$ is surjective

Extended

Suppose $T \in \mathcal L(V,W)$ where $\dim V = \dim W$. Then there exists an isomorphism $S \in \mathcal L(W,V)$ such that $ST \in \mathcal L(V)$. Then (by above) the following are equvalent:

1. $ST$ is invertible
2. $ST$ is injective
3. $ST$ is surjective

Since $S$ is invertible, it's easy to see we can remove $S$ from the conditions above, completing the proof.