---
COMMENTARY:
$\textbf{Exercise 1 Commentary:}$ This exercise shows that for any linear transformation $T: V \to W$, the matrix representation of $T$ with respect to any choice of bases must have at least $\text{dim range}(T)$ non-zero entries. The key idea is that if the matrix had fewer than $\text{dim range}(T)$ non-zero columns, then the dimension of the range would be less than $\text{dim range}(T)$, which is a contradiction.
This result has important implications in numerical linear algebra and the study of matrix ranks/nullities. It provides a simple way to get a lower bound on the rank of a matrix just by counting its non-zero entries, without performing complicated row reduction. It is also useful in analyzing the sparsity patterns of matrices arising from discretizations of differential equations.
$\textbf{Exercise 1 Examples:}$
1) Let $T: \mathbb{R}^4 \to \mathbb{R}^3$ be the linear map with standard matrix representation $\begin{pmatrix}
1 & 0 & 2 & 1\\
0 & 1 & 0 & 0\\
3 & 0 & 0 & 2
\end{pmatrix}$.
Since $\text{rank}(T) = \text{dim range}(T) = 3$, the matrix must have at least 3 non-zero entries, which is indeed the case.
2) Let $V = \mathcal{P}_3(\mathbb{R})$ and $W = \mathcal{P}_2(\mathbb{R})$, and let $T: V \to W$ be the differentiation map. Then relative to the standard basis of $V$ and the basis $\{1, x, x^2\}$ for $W$, the matrix of $T$ is $\begin{pmatrix}
0 & 1 & 0 & 0\\
0 & 0 & 2 & 0\\
0 & 0 & 0 & 3
\end{pmatrix}$,
which has exactly 3 non-zero entries, matching $\text{dim range}(T) = 3$.
3) Let $V = M_{2,2}(\mathbb{R})$ 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 relative to the standard bases, the matrix of $T$ is $\begin{pmatrix}
1&0&0&0\\
0&0&1&0\\
0&1&0&0\\
0&0&0&1
\end{pmatrix}$ which has exactly 4 non-zero entries, matching $\text{dim range}(T) = 4$.
$\textbf{Implications and Applications:}$ This result is useful in numerical linear algebra for estimating matrix ranks, studying sparse matrix computations, and designing efficient algorithms. It also arises in coding theory when studying parity check matrices of linear codes. More broadly, it illustrates the deep connections between the algebraic properties of linear maps and the combinatorial/numerical properties of their matrix representations.
---
$\textbf{Exercise 1 Commentary (continued):}$ In addition to its implications in numerical linear algebra and the study of matrix ranks/nullities, this result also has applications in the analysis of sparse matrices and the development of efficient algorithms for solving large-scale linear systems.
In many applications, such as finite element methods, graph theory, and network analysis, the matrices involved are often very large but have a sparse structure (i.e., many entries are zero). In such cases, the number of non-zero entries in the matrix can provide valuable information about the rank and the computational complexity of various matrix operations.
This result allows us to quickly estimate a lower bound on the rank of a sparse matrix simply by counting its non-zero entries, without performing expensive computational procedures like Gaussian elimination or SVD decomposition. This information can then be used to design more efficient algorithms tailored to the specific sparsity pattern of the matrix.
Furthermore, the result also connects to the study of matrix factorizations and low-rank approximations, which are essential tools in data compression, machine learning, and signal processing. By understanding the relationship between the number of non-zero entries and the rank, we can develop better strategies for computing and approximating low-rank matrix factorizations, which can lead to significant computational savings in these applications.
$\textbf{Exercise 3 Examples (continued):