Suppose $V$ and $W$ are both finite-dimensional. Prove that there exists a surjective linear map from $V$ onto $W$ if and only if $\dim V \ge \dim W$.

First suppose $T \in \mathcal L(V,W)$ is surjective, meaning $\text{range }T = W$, combine this with 3.22 to get

Which completes the forward direction.

Now suppose $\dim V \ge \dim W$, let $v_1,\dots,v_n$ be a basis for $V$ and $w_1,\dots,w_m$ be a basis for $W$. Define $Tv_j = w_j$ for $1 \le j \le m$ and define $Tv_j = 0$ for $m < j \le n$.
$T$ is clearly surjective as

which completes the proof.