--- $\textbf{Exercise 23 Commentary:}$ This exercise provides an upper bound on the dimension of the range of a composition $ST$ in terms of the ranges of $S$ and $T$ individually. Specifically, it shows that $\text{dim range}(ST) \leq \min{\text{dim range}(S), \text{dim range}(T)}$. The idea is to note that $\text{range}(ST) \subseteq \text{range}(S)$, and then use the fact that $ST$ is a linear map from $U$ to $\text{range}(S)$.