--- $\textbf{Exercise 22 Commentary:}$ This exercise establishes an important dimension inequality for the null space of the composition of two linear maps $S: V \to W$ and $T: U \to V$. Specifically, it shows that $\text{dim null}(ST) \leq \text{dim null}(S) + \text{dim null}(T)$. The proof constructs a linear map $R: \text{null}(ST) \to V$ whose range is contained in $\text{null}(S)$, and then uses the Fundamental Theorem of Linear Maps to relate the dimensions of the domain, codomain, and null space of $R$. This allows the desired inequality to be deduced. $\textbf{Exercise 22 Examples:}$ 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 $\text{dim null}(S) = 2$, $\text{dim null}(T) = 0$, and $\text{dim null}(ST) = 0$, 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 $\text{dim null}(S) = 1$, $\text{dim null}(T) = 0$, and $\text{dim null}(ST) = 1$, 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 $\text{dim null}(S) = 2$, $\text{dim null}(T) = 1$, and $\text{dim null}(ST) = 3$, which satisfies the inequality. $\textbf{Implications and Applications:}$ This dimension inequality has important implications in the study of linear transformations and their compositions. It provides a way to bound the "degree of non-injectivity" of a composite map in terms of the degrees of non-injectivity of the individual maps. This is useful in analyzing the behavior of linear operators in areas like differential equations, control theory, and signal processing, where compositions of linear maps arise frequently. The inequality also connects to the study of matrix ranks and nullities, and the development of efficient algorithms for computing these quantities. It can be used to derive upper bounds on the ranks of matrix products, which is important in numerical linear algebra and the analysis of large-scale linear systems. From a theoretical perspective, this result sheds light on the interplay between the null spaces of linear maps and their compositions, which is crucial for understanding the algebraic and geometric properties of these maps and their representations as matrices or operators.