Solution
= Solution
{c}
Since $S'$ is the <restriction> of $S$ to $\operatorname{im}T$,
$$
\operatorname{im}(ST)=\{S(Tu):u\in U\}
=\{S(v):v\in\operatorname{im}T\}=\operatorname{im}S'.
$$
This <image of a linear map> is contained in $\operatorname{im}S$, so $r_{ST}\leq r_S$. Also, applying $S$ to a <basis> of $\operatorname{im}T$ gives a <spanning set> for $\operatorname{im}S'$, of size $r_T$. Thus $r_{ST}\leq r_T$, and
$$
\boxed{r_{ST}\leq\min(r_S,r_T).}
$$