Suppose $V$ and $W$ are both finite-dimensional. Prove that there exists an injective linear map from $V$ to $W$ if and only if $\dim V \le \dim W$.

First suppose there exists $T \in \mathcal L(V,W)$ such that $T$ is injective.
3.16 implies $\dim \text{null }T = 0$, then apply 3.22 to get $\dim V = \dim \text{range }T$.
Now because $\text{range }T$ is a subspace of $W$ it's dimension is less, meaning

This completes the forward direction.

Now suppose $\dim V \le \dim W$, Let $v_1,\dots,v_n$ be a basis of $V$ and let $w_1,\dots,w_m$ be a basis for $W$. Define $Tv_j = w_j$ for $1 \le j \le n$ and $Tv_j = 0$ for $n < j \le m$. Clearly $\text{null } T = \{0\}$ since each $v_j$ is mapped to a nonzero $w_j$, thus 3.16 implies $T$ is injective.