--- $\textbf{Exercise 3 Commentary:}$ This exercise verifies two fundamental properties of matrix addition and scalar multiplication, namely that $\mathcal{M}(S+T) = \mathcal{M}(S) + \mathcal{M}(T)$ and $\mathcal{M}(\lambda T) = \lambda \mathcal{M}(T)$, where $\mathcal{M}(T)$ denotes the matrix representation of a linear map $T$ with respect to given bases. These properties show that the matrix operations of addition and scalar multiplication are compatible with the algebraic operations of sum and scalar multiplication of linear maps. This compatibility is crucial for developing a robust theory of matrix algebra that accurately captures linear mappings between vector spaces. $\textbf{Exercise 3 Examples:}$ 1) Let $S, T: \mathbb{R}^3 \to \mathbb{R}^2$ be the linear maps given by: $S\begin{pmatrix}x\\y\\z\end{pmatrix}$ = $\begin{pmatrix}x+y\\2z\end{pmatrix}$, \quad T$\begin{pmatrix}x\\y\\z\end{pmatrix}$ = $\begin{pmatrix}2x-z\\y+z\end{pmatrix}$. With respect to the standard bases, we have: $\mathcal{M}(S)$ = $\begin{pmatrix}1&1&0\\0&0&2\end{pmatrix}$, \quad $\mathcal{M}(T)$ = $\begin{pmatrix}2&0&-1\\0&1&1\end{pmatrix}$. One can verify that $\mathcal{M}(S+T) = \mathcal{M}(S) + \mathcal{M}(T) = \begin{pmatrix}3&1&-1\\0&1&3\end{pmatrix}$, illustrating the result for addition. 2) Let $V = \mathcal{P}_2(\mathbb{R})$, $W = \mathbb{R}^3$, and define $T: V \to W$ by $T(a + bx + cx^2) = (a, b, c)$. Let $\lambda = 2$. With respect to the standard bases, we have $\mathcal{M}(T) = \begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}$. One can check that $\mathcal{M}(\lambda T) = \begin{pmatrix}2&0&0\\0&2&0\\0&0&2\end{pmatrix} = 2\mathcal{M}(T)$, illustrating the scalar multiplication property. $\textbf{Implications and Applications:}$ These properties are fundamental for developing a rich theory of matrix algebra and computational linear algebra. They ensure that matrix operations accurately model the linear algebraic operations, allowing us to translate problems about linear maps into problems about matrix algebra. This is the key reason why matrix algebra plays such a pivotal role in physics, engineering, computer graphics, machine learning, and many other fields. --- $\textbf{Exercise 3 Examples (continued):}$ 4) Let $S: \mathbb{R}^4 \to \mathbb{R}^3$ and $T: \mathbb{R}^4 \to \mathbb{R}^3$ be linear maps with matrix representations: $\mathcal{M}(S) = \begin{pmatrix} 1 & 0 & 0 & 2\\ 0 & 1 & 0 & -1\\ 0 & 0 & 1 & 0 \end{pmatrix}, \quad \mathcal{M}(T) = \begin{pmatrix} 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 1 & 0 & 0 & 1 \end{pmatrix}$ Then $\mathcal{M}(S + T) =$$\begin{pmatrix} 1 & 1 & 0 & 2\\ 0 & 1 & 1 & -1\\ 1 & 0 & 1 & 1 \end{pmatrix}$ = $\mathcal{M}(S) + \mathcal{M}(T)$. 5) Let $T: \mathcal{P}_3(\mathbb{R}) \to \mathcal{P}_2(\mathbb{R})$ be the linear map defined by $T(a + bx + cx^2 + dx^3) = (a + b) + (b + c)x + (c + d)x^2$. 6) With respect to the standard bases, we have $\mathcal{M}(T) =$$ \begin{pmatrix} 1 & 1 & 0 & 0\\ 0 & 1 & 1 & 0\\ 0 & 0 & 1 & 1 \end{pmatrix}$. If $\lambda = 2$, then $\mathcal{M}(\lambda T) = 2\mathcal{M}(T) =$$ \begin{pmatrix} 2 & 2 & 0 & 0\\ 0 & 2 & 2 & 0\\ 0 & 0 & 2 & 2 \end{pmatrix}$. $\textbf{Implications and Applications (continued):}$ The linearity properties of matrix addition and scalar multiplication are fundamental in developing a coherent theory of matrix algebra and computational linear algebra. They ensure that matrix operations accurately model the algebraic operations on linear maps, allowing us to translate problems about linear maps into problems about matrix algebra. These properties are also essential for the study of matrix groups and algebras, which are algebraic structures built from matrices and play a crucial role in various areas of mathematics, physics, and engineering. For instance, the linearity properties are necessary for defining matrix Lie groups and Lie algebras, which are fundamental tools in the study of symmetries, differential equations, and quantum mechanics. Furthermore, the linearity properties are crucial for the development of efficient algorithms for matrix computations, such as matrix-vector multiplication, matrix factorizations, and the solution of linear systems. These algorithms often rely on exploiting the linearity properties to reduce computational complexity and achieve better numerical stability and accuracy. In summary, the linearity properties of matrix addition and scalar multiplication are essential for the development of a rich theory of matrix algebra and its applications in various fields, ranging from pure mathematics to computational science and engineering.