--- $\textbf{Exercise 25 Commentary:}$ This exercise characterizes when the null space of a linear map $T: V \to W$ is contained in the null space of another linear map $S: V \to W$. The condition is that there exists a linear operator $E: W \to W$ such that $T = ES$. The proof first shows that if $T = ES$, then $\text{null}(T) \subseteq \text{null}(S)$, since if $Sv = 0$, then $Tv = E(Sv) = 0$. This establishes the sufficiency of the condition. Conversely, if $\text{null}(S) \subseteq \text{null}(T)$, the proof constructs a linear operator $E: W \to W$ by defining it on the range of $S$ as $E(Sv) = Tv$, and extending it to all of $W$ using a result from an earlier exercise. This shows that the null space containment condition is also necessary for the existence of such an $E$. $\textbf{Exercise 25 Examples:}$ 1) Let $V = \mathbb{R}^3, W = \mathbb{R}^2$, and define $S, T: V \to W$ by $S(x,y,z) = (x+y, y+z), T(x,y,z) = (2x+3y, 4y+5z)$. Then $T = ES$, where $E: W \to W$ is given by $E(a,b) = (2a+3b, 4b+5(b-a))$. Note that $\text{null}(S) = \{(x,-x,0) \mid x \in \mathbb{R}\} \subseteq \text{null}(T) = \{(0,0,0)\}$. 2) Let $V = \mathcal{P}_2(\mathbb{R}), W = \mathbb{R}^3$, and define $S: V \to W$ by $S(a+bx+cx^2) = (a,b,c), T: V \to W$ by $T(a+bx+cx^2) = (2a+b, 3b+c, 4c)$. Then $T = ES$, where $E: W \to W$ is given by $E(x,y,z) = (2x+y, 3y+z, 4z)$. Note that $\text{null}(S) = \{0\} \subseteq \text{null}(T) = \{0\}$. 3) Let $V = \mathbb{R}^2, W = \mathbb{F}_3^2$, and define $S(x,y) = (x,y), T(x,y) = (2x,y)$. Then there does not exist $E: W \to W$ with $T = ES$, since $\text{null}(S) = \{(0,0)\} \not\subseteq \text{null}(T) = \{(0,y) \mid y \in \mathbb{F}_3\}$. $\textbf{Implications and Applications:}$ This characterization has important applications in the study of linear transformations and their matrix representations. It provides a way to analyze the relationship between the null spaces of two linear maps, which is crucial for understanding their algebraic and geometric properties, as well as their behavior under compositions and other operations. The result also plays a role in the development of efficient algorithms for computing null spaces and ranks of matrices, as well as in the study of matrix decompositions and canonical forms. It connects to the theory of generalized inverses and the analysis of linear systems with singular or ill-conditioned coefficient matrices. From a theoretical perspective, this characterization sheds light on the interplay between the null spaces of linear maps and their matrix representations, which is fundamental for understanding the algebraic structures and representations associated with these maps.