--- $\textbf{Exercise 7 Commentary:}$ This exercise is the "dual" of Exercise 6, where now the basis of the codomain $W$ is fixed instead of the domain basis. It shows that for any linear map $T: V \to W$, there exists a basis of $V$ such that the first row 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 similar to Exercise 6, but now we start by finding a vector $u_1 \in V$ such that $Tu_1$ is not in the span of $\{w_2, \ldots, w_m\}$, and use this to construct a basis $\{u_1, \ldots, u_n\}$ of $V$. $\textbf{Exercise 7 Examples:}$ 1) Let $V = \mathbb{R}^3$ and $W = \mathbb{R}^2$ with basis $\{(1,0), (0,1)\}$, and let $T: V \to W$ be given by $T(x,y,z) = (x+y, x-z)$. Choosing the basis $\{(1,1,0), (0,1,-1), (0,0,1)\}$ for $V$ yields the matrix representation $\begin{pmatrix}1&0&0\\1&-1&0\end{pmatrix}$. 2) Let $V = \mathcal{P}_3(\mathbb{R})$ and $W = \mathcal{P}_2(\mathbb{R})$ with basis $\{1, x, x^2\}$, and let $T: V \to W$ be differentiation. Choosing the basis $\{1, x^2, x^3, x\}$ for $V$ yields the matrix representation $\begin{pmatrix}0&0&0&1\\0&0&2&0\\0&3&0&0\end{pmatrix}$. 3) Let $V = \mathbb{F}_4^3$ and $W = \mathbb{F}_4^2$ with basis $\{(1,0), (0,1)\}$, and let $T: V \to W$ be given by the matrix $\begin{pmatrix}1&2&1\\3&1&2\end{pmatrix}$. Choosing the basis $\{(1,2,0), (0,1,1), (0,0,1)\}$ for $V$ yields the matrix representation $\begin{pmatrix}1&0&0\\0&1&1\end{pmatrix}$. $\textbf{Implications and Applications:}$ Similar to Exercise 6, this result is very useful in numerical linear algebra and the study of matrix canonical forms. It provides a way to systematically simplify the structure of matrix representations by eliminating non-zero entries in the first row. This can lead to more efficient algorithms and a deeper understanding of the inherent properties of linear maps. It also connects to the theory of matrix normal forms and matrix invariants under similarity transformations. --- $\textbf{Exercise 7 Implications and Applications (continued):}$ Similar to Exercise 6, 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 LU decomposition, which is a fundamental matrix factorization technique used in various numerical linear algebra algorithms. The LU decomposition represents a matrix as the product of a lower triangular matrix (L) and an upper triangular matrix (U). The first step in computing the LU decomposition often involves transforming the matrix into a specific form where the first row has a particular structure, which can be achieved using the technique described in this exercise. Another application arises in the study of matrix pencils and the computation of the Kronecker canonical form, similar to the applications discussed for Exercise 6. The ability to obtain a matrix representation with a particular structure in the first row can facilitate the computation of the Kronecker canonical form and the analysis of the properties of the associated differential-algebraic equation or dynamical 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 row 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.