--- $\textbf{Exercise 26 Commentary:}$ This exercise provides a characterization of when the range of a linear map $S: V \to W$ is contained in the range of another linear map $T: V \to W$. The condition is that there exists a linear operator $E: V \to V$ such that $S = TE$. The proof first shows that if $S = TE$, then $\text{range}(S) \subseteq \text{range}(T)$, since for any $v \in V$, we have $Sv = T(Ev) \in \text{range}(T)$. This establishes the sufficiency of the condition. Conversely, if $\text{range}(S) \subseteq \text{range}(T)$, the proof constructs a linear operator $E: V \to V$ by defining it on a basis of $V$ such that $S = TE$. This shows that the range containment condition is also necessary for the existence of such an $E$. $\textbf{Exercise 26 Examples:}$ 1) Let $V = \mathbb{R}^3, W = \mathbb{R}^2$, and define $S(x,y,z) = (x+2y, 2y+3z), T(x,y,z) = (x,y)$. Then $S = TE$, where $E(x,y,z) = (x+2y, 2y+3z, 0)$. 2) Let $V = \mathcal{P}_3(\mathbb{R}), W = \mathcal{P}_2(\mathbb{R})$, and let $S, T: V \to W$ be differentiation maps. Then $S = TE$, where $E: V \to V$ is the identity map. 3) Let $V = \mathbb{F}_2^3, W = \mathbb{F}_2^2$, and define $S(x,y,z) = (x+y,y+z), T(x,y,z) = (x,y)$. Then there does not exist $E: V \to V$ with $S = TE$, since $(1,1,0) \in \text{range}(S)$ but $(1,1,0) \notin \text{range}(T)$. $\textbf{Implications and Applications:}$ This characterization has important applications in the study of linear transformations and their matrix representations, as well as in the development of efficient algorithms for computing ranges and ranks of matrices. The result provides a way to analyze the relationship between the ranges of two linear maps, which is crucial for understanding their algebraic and geometric properties, as well as their behavior under compositions and other operations. It also plays a role in the study of matrix decompositions and canonical forms, as well as in 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 ranges of linear maps and their matrix representations, which is fundamental for understanding the algebraic structures and representations associated with these maps. It also connects to the theory of generalized inverses and the study of linear systems with singular or ill-conditioned coefficient matrices.