Exercise 20

Prove that the image of a span equals the span of the images. That is, where $h\colon V\to W$ is linear, prove that if $S$ is a subset of $V$ then $h([S])$ equals $[h(S)]$. This generalizes Lemma 2.1 since it shows that if $U$ is any subspace of $V$ then its image $\{\hskip 1.00006pth(\vec{u})\mid\vec{u}\in U\hskip 1.00006pt\}$ is a subspace of $W$, because the span of the set $U$ is $U$.