Problem
Suppose $V$ is finite-dimensional with $\dim V > 0$, and suppose $W$ is infinite-dimensional. Prove that $\mathcal{L}(V,W)$ is infinite-dimensional
Solution
Let $w_1, w_2, \dots$ be an infinite sequence of linearly independent vectors in $W$ which exists as per P2A14. If we let $v_1, \dots, v_n$ be a basis of $V$, then by theorem 3.5 we can construct an infinite sequence of linear maps in $\mathcal{L}(V,W)$ by letting $T_i \in \mathcal{L}(V,W)$ be defined by $T_i v_j = w_{i+j-1}$. The sequence $T_1, T_2, \dots$ is linearly independent by the linear independence of $w_1, w_2, \dots$ - *Proof:*
If we let $a_1, \dots, a_m \in \mathbb{F}$ such that $a_1T_1 + \dots + a_mT_m = 0$, where the right hand side is the $0$ map on $V$. Then: $$a_1w_1 + \dots + a_mw_m = a_1T_1v_1 + \dots + a_mT_mv_1$$ $$(a_1T_1 + \dots + a_mT_m)(v_1) = 0(v_1) = 0$$ Implying $a_1 = \dots = a_m = 0$ by the linear independence of $T_1, \dots, T_m$ for any $m \in \mathbb{N}$. P2A14 therefore implies that $\mathcal{L}(V,W)$ is infinite-dimensional.