--- $\textbf{Exercise 5 Commentary:}$ This exercise shows that for any linear transformation $T: V \to W$ between finite-dimensional vector spaces, there exist bases of $V$ and $W$ such that the matrix representation $\mathcal{M}(T)$ takes a very special "triangular" form. Specifically, the non-zero entries of $\mathcal{M}(T)$ occur only on the diagonal and possibly the entries above the diagonal. This triangular form arises from first choosing bases of the null space of $T$ and its complement, and then choosing bases of the range of $T$ and its complement in $W$. Reordering these bases appropriately yields the desired matrix structure. $\textbf{Exercise 5 Examples:}$ 1) Let $T: \mathbb{R}^3 \to \mathbb{R}^2$ be the linear map given by $T(x,y,z) = (x+y, x-z)$. Then with respect to the bases $\{(1,1,0), (1,0,1), (0,1,-1)\}$ for $\mathbb{R}^3$ and $\{(1,1), (1,-1)\}$ for $\mathbb{R}^2$, the matrix of $T$ takes the form $\begin{pmatrix}2&1&0\\0&0&-2\end{pmatrix}$. 2) Let $V = \mathcal{P}_3(\mathbb{R})$ and $W = \mathcal{P}_2(\mathbb{R})$, and let $T: V \to W$ be differentiation. Then with respect to the bases $\{1, x^2, x^3, x\}$ for $V$ and $\{1, x, x^2\}$ for $W$, the matrix of $T$ takes the form $\begin{pmatrix}0&0&0&1\\0&0&2&0\\0&3&0&0\end{pmatrix}$. 3) Let $V = M_{2,3}(\mathbb{R})$ and $W = \mathbb{R}^4$, and let $T: V \to W$ be the linear map that sends a matrix to the vector obtained by stacking its columns. Then with respect to the bases $\begin{pmatrix}1&0&0\\0&0&0\end{pmatrix}, \begin{pmatrix}0&1&0\\0&0&0\end{pmatrix}, \begin{pmatrix}0&0&1\\0&0&0\end{pmatrix}, \begin{pmatrix}0&0&0\\1&0&0\end{pmatrix}, \begin{pmatrix}0&0&0\\0&1&0\end{pmatrix}, \begin{pmatrix}0&0&0\\0&0&1\end{pmatrix}$ for $V$ and the standard basis for $W$, the matrix of $T$ takes the form $\begin{pmatrix}1&0&0&0&0&0\\0&1&0&0&0&0\\0&0&1&0&0&0\\0&0&0&1&0&0\end{pmatrix}$. $\textbf{Implications and Applications:}$ Having a triangular matrix representation is very useful for studying properties of linear maps and developing efficient numerical algorithms. For example, triangular matrices have very simple formulas for computing determinants, ranks, and null spaces. Triangular forms also facilitate solving linear systems via back-substitution. More broadly, this result connects to the theory of matrix canonical forms and the study of matrix invariants under change of basis. --- $\textbf{Exercise 5 Implications and Applications (continued):}$ The triangular matrix representation obtained in this exercise has significant implications in numerical linear algebra and the development of efficient algorithms for solving linear systems, computing matrix ranks and nullities, and performing matrix decompositions. One of the key advantages of triangular matrices is the ease of computing their determinants, ranks, and null spaces. For instance, the rank of a triangular matrix is simply the number of non-zero entries on the main diagonal, and the null space can be computed efficiently using back-substitution techniques. This property is particularly useful in the context of the LU decomposition, which is a fundamental matrix factorization technique used in many numerical linear algebra algorithms. The LU decomposition aims to represent a matrix as the product of a lower triangular matrix (L) and an upper triangular matrix (U). By obtaining a triangular matrix representation, one can efficiently compute the LU decomposition and subsequently solve linear systems or compute matrix inverses. Moreover, triangular matrix representations are also valuable in the study of matrix canonical forms and the development of efficient algorithms for computing matrix normal forms. Many important matrix decompositions, such as the QR decomposition and the Schur decomposition, rely on transforming matrices into triangular or quasi-triangular forms, which can be facilitated by the techniques presented in this exercise. Beyond numerical linear algebra, triangular matrix representations also find applications in the analysis of linear control systems and the study of dynamical systems governed by linear differential equations. In these contexts, the triangular structure of the system matrices can provide insights into the stability and controllability properties of the system, enabling more efficient analysis and design techniques. Overall, the ability to obtain triangular matrix representations is a powerful tool in linear algebra and its applications, enabling efficient computations, facilitating the analysis of matrix properties, and providing insights into the behavior of linear systems and dynamical processes.