Suppose $V$ is finite-dimensional with $\dim V > 0$ and suppose $W$ is infinite-dimensional. Prove that $\mathcal L(V,W)$ is infinite-dimensional.

Uhh, obviously?

Let $v \in V$ and let $w_1,\dots$ be a sequence of vectors in $W$ such that $w_1,\dots,w_m$ is independent for all $m$ (see 2a/14 for the existance of such a sequence).

Define $T_j(v) = w_j$ and define the rest of $T_j$ so that $T_j \in \mathcal L(V,W)$. To show $T_1,\dots$ is an independent sequence in $\mathcal L(V,W)$ consider

But

Implies each $a_j = 0$ since $w_1,\dots,w_m$ are independent. Thus $T_1,\dots$ is an independent sequence implying $\mathcal L(V,W)$ is infinite-dimensional by 2a/14.