--- $\textbf{Exercise 4 Commentary:}$ This exercise computes the matrix representation of the differentiation operator $D: \mathcal{P}_3(\mathbb{R}) \to \mathcal{P}_2(\mathbb{R})$ with respect to specific bases of the domain and codomain. Specifically, it finds bases such that the matrix of $D$ takes the simple diagonal form $\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\end{pmatrix}$ This example illustrates how a linear operator between polynomial spaces can have a particularly nice matrix representation when working with the right monomial bases. More generally, it shows how the choice of bases can dramatically impact the form of matrix representations. $\textbf{Exercise 4 Examples:}$ 1) With respect to the bases $\{x^3, x^2, x, 1\}$ for $\mathcal{P}_3(\mathbb{R})$ and $\{3x^2, 2x, 1\}$ for $\mathcal{P}_2(\mathbb{R})$, the matrix of $D$ is: $\begin{pmatrix}0&0&0&1\\0&0&2&0\\0&3&0&0\end{pmatrix}$. 2) Let $D': \mathcal{P}_4(\mathbb{R}) \to \mathcal{P}_3(\mathbb{R})$ be differentiation. With respect to the bases $\{x^4,x^3,x^2,x,1\}$ for $\mathcal{P}_4(\mathbb{R})$ and $\{4x^3,3x^2,2x,1\}$ for $\mathcal{P}_3(\mathbb{R})$, the matrix of $D'$ is: $\begin{pmatrix}1&0&0&0&0\\0&1&0&0&0\\0&0&1&0&0\\0&0&0&1&0\end{pmatrix}$. 3) Let $U = \text{span}\{1, x, x^3\}$ and $V = \text{span}\{1, x\}$, viewed as subspaces of $\mathcal{P}_3(\mathbb{R})$ and $\mathcal{P}_2(\mathbb{R})$ respectively. Let $D_U: U \to V$ be the restriction of $D$. With respect to the given bases, the matrix of $D_U$ is: $\begin{pmatrix}0&1&0\\0&0&0 \end{pmatrix}$. $\textbf{Implications and Applications:}$ This type of basis representation is very useful in fields like numerical analysis, where polynomial bases are ubiquitous for approximating functions. Having a simple diagonal matrix representation for differentiation greatly facilitates numerical computations. It also connects to the theory of differential equations, where choices of bases can simplify studying linear ODEs. --- $\textbf{Exercise 4 Implications and Applications (continued):}$ This exercise illustrates how the choice of bases can significantly impact the matrix representation of a linear operator, and how carefully selecting bases can reveal inherent structures or simplify computations. In the context of numerical analysis and scientific computing, polynomial bases are frequently used for approximating functions, solving differential equations, and modeling physical phenomena. The simple diagonal matrix representation obtained in this exercise for the differentiation operator facilitates efficient numerical computations and provides insights into the behavior of differential operators. For instance, in the finite element method, which is widely used for solving partial differential equations in various engineering and scientific applications, the differentiation operator plays a crucial role. By carefully choosing appropriate polynomial bases, one can obtain sparse or structured matrix representations for the differential operators, leading to more efficient algorithms and better numerical stability. Furthermore, this exercise connects to the broader theory of matrix canonical forms and the study of matrix invariants under changes of basis. The ability to transform a matrix representation into a simpler or more structured form is essential for understanding the inherent properties of the linear operator and developing efficient computational techniques. In addition to its applications in numerical analysis and scientific computing, this exercise also has implications in the study of differential equations and control theory. The matrix representation of differential operators is fundamental in the analysis and numerical solution of ordinary and partial differential equations, as well as in the design and analysis of control systems. Overall, this exercise highlights the importance of basis choices in linear algebra and the connections between matrix representations, computational efficiency, and the understanding of mathematical structures in various fields of study.