--- $\textbf{Exercise 6 Commentary:}$ This exercise considers a linear map $T: V \to W$ where the basis of $V$ is fixed, and shows that there exists a basis of $W$ such that the first column of the matrix representation $\mathcal{M}(T)$ contains at most one non-zero entry (which can be made to be 1 if desired). The idea is to take the first non-zero vector $Tv_1$ in the image of the basis vectors $\{v_1, \ldots, v_n\}$, and use it to start constructing a basis $\{w_1, \ldots, w_m\}$ of $W$. $\textbf{Exercise 6 Examples:}$ 1) Let $V = \mathbb{R}^3$ with basis $\{(1,0,0), (0,1,0), (0,0,1)\}$, and let $T: V \to \mathbb{R}^2$ be given by $T(x,y,z) = (x+y, x-z)$. Choosing the basis $\{(1,1), (1,-1)\}$ for $\mathbb{R}^2$ yields the matrix representation $\begin{pmatrix}1&1&0\\1&0&-1\end{pmatrix}$. 2) Let $V = \mathcal{P}_2(\mathbb{R})$ with basis $\{1, x, x^2\}$, and let $T: V \to \mathbb{R}^3$ be given by $T(a + bx + cx^2) = (a, b, c)$. Choosing the basis $\{(1,0,0), (0,1,0), (0,0,1)\}$ for $\mathbb{R}^3$ yields the matrix representation $\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}$. 3) Let $V = \mathbb{F}_2^4$ with basis $\{(1,0,0,0), (0,1,0,0), (0,0,1,0), (0,0,0,1)\}$, and let $T: V \to \mathbb{F}_2^3$ be given by the matrix $\begin{pmatrix}1&0&1&1\\0&1&1&0\\1&1&0&1\end{pmatrix}$. Choosing the basis $\{(1,0,1), (0,1,1), (1,1,0)\}$ for $\mathbb{F}_2^3$ yields the matrix representation $\begin{pmatrix}1&1&0&0\\0&1&1&0\\0&0&0&1\end{pmatrix}$. $\textbf{Implications and Applications:}$ This result is useful in numerical linear algebra for developing efficient algorithms, especially those involving LU or QR factorizations of matrices. It also connects to the theory of matrix canonical forms and matrix invariants under changes of basis. More broadly, it illustrates how carefully choosing bases can simplify and reveal the inherent structure in matrix representations of linear maps. --- $\textbf{Exercise 6 Implications and Applications (continued):}$ The result presented in this exercise has important implications in the development of efficient algorithms for computing matrix ranks, null spaces, and ranges, as well as in the study of matrix decompositions and canonical forms. One of the key applications of this result is in the context of the QR decomposition, which is a fundamental matrix factorization technique used in various numerical linear algebra algorithms. The QR decomposition represents a matrix as the product of an orthogonal matrix (Q) and an upper triangular matrix (R). The first step in computing the QR decomposition involves transforming the matrix into a specific form where the first column has a single non-zero entry, which can be achieved using the technique described in this exercise. Another application arises in the study of matrix pencils, which are matrix polynomials of the form A + λB, where A and B are matrices and λ is a scalar parameter. Matrix pencils arise in the analysis of differential-algebraic equations, control theory, and the study of dynamical systems. The ability to obtain a matrix representation with a particular structure in the first column can facilitate the computation of the Kronecker canonical form of a matrix pencil, which is a powerful tool for analyzing the properties of the system. Furthermore, this result can be used in the development of efficient algorithms for computing matrix ranks and null spaces, particularly in the context of sparse or structured matrices. By transforming the matrix into a form where the first column has a specific structure, one can leverage computational techniques tailored to this structure, leading to more efficient algorithms and better numerical stability. In addition to its applications in numerical linear algebra, this result also has connections to the theory of matrix canonical forms and the study of matrix invariants under changes of basis. The ability to transform a matrix into a specific form while preserving its fundamental properties is a crucial step in understanding the inherent structure of the matrix and developing efficient computational techniques.