Solution 3b 8

Suppose $V$ and $W$ are finite-dimensional with $\dim V \ge \dim W \ge 2$ .
Show that $\{T \in \mathcal L(V,W) : T\text{ is not surjective}\}$ is not a subspace of $\mathcal L(V,W)$ .

Like 3b/7 it isn't closed under addition.

Let $v_1,\dots,v_m$ be a basis for $V$ and $w_1,\dots,w_n$ be a basis for $W$ . Define

\begin{aligned} T(a_1v_1+\dots+a_nv_n+\dots+a_mv_m) &= a_2w_2 + \dots + a_nw_n \\ S(a_1v_1+\dots+a_nv_n+\dots+a_mv_m) &= a_1w_1 \end{aligned}

Neither $T$ nor $S$ span $W$ , but $(T + S)$ does span $W$ .