--- $\textbf{Exercise 23 Commentary:}$ This exercise provides an upper bound on the dimension of the range of a composition of two linear maps $S: V \to W$ and $T: U \to V$, in terms of the dimensions of the individual ranges. Specifically, it shows that $\text{dim range}(ST) \leq \min\{\text{dim range}(S), \text{dim range}(T)\}$. The proof leverages two key observations: (1) $\text{range}(ST) \subseteq \text{range}(S)$, since the range of a composition is a subset of the range of the outer map, and (2) $ST$ can be viewed as a linear map from $U$ to $\text{range}(S)$, allowing the Fundamental Theorem of Linear Maps to be applied. $\textbf{Exercise 23 Examples:}$ 1) Let $V = \mathbb{R}^4, U = \mathbb{R}^3, W = \mathbb{R}^2$, and let $S: V \to W$ and $T: U \to V$ be given by: $S\begin{pmatrix}x_1\\x_2\\x_3\\x_4\end{pmatrix} = \begin{pmatrix}x_1+x_2\\x_3+x_4\end{pmatrix}, \quad T\begin{pmatrix}y_1\\y_2\\y_3\end{pmatrix} = \begin{pmatrix}y_1\\y_2\\y_3\\0\end{pmatrix}$ Then $\text{dim range}(S) = 2$, $\text{dim range}(T) = 3$, and $\text{dim range}(ST) = 2$, which satisfies the inequality. 2) Let $V = \mathcal{P}_3(\mathbb{R}), U = \mathcal{P}_2(\mathbb{R}), W = \mathbb{R}^2$, and let $S: V \to W$ and $T: U \to V$ be differentiation maps. Then $\text{dim range}(S) = 2$, $\text{dim range}(T) = 3$, and $\text{dim range}(ST) = 2$, which satisfies the inequality. 3) Let $V = \mathbb{F}_2^4, U = \mathbb{F}_2^3, W = \mathbb{F}_2^2$, and let $S$ and $T$ be linear maps with matrices $\begin{pmatrix}1&0&0&1\\0&1&1&0\end{pmatrix}$ and $\begin{pmatrix}1&0&1\\0&1&1\\1&1&0\\0&0&0\end{pmatrix}$ respectively. Then $\text{dim range}(S) = 2$, $\text{dim range}(T) = 3$, and $\text{dim range}(ST) = 2$, which satisfies the inequality. $\textbf{Implications and Applications:}$ This dimension inequality provides important insights into the behavior of the ranges of composite linear maps, which arise frequently in applications such as signal processing, control theory, and the study of dynamical systems. The inequality can be used to derive upper bounds on the ranks of matrix products, which is crucial for developing efficient algorithms in numerical linear algebra and the analysis of large-scale linear systems. It also plays a role in the study of matrix decompositions, such as the singular value decomposition, and the development of low-rank approximation techniques for matrices and linear operators. From a theoretical perspective, this result sheds light on the relationship between the ranges of linear maps and their compositions, which is fundamental for understanding the algebraic and geometric properties of these maps and their representations as matrices or operators.