--- $\textbf{Exercise 16 Commentary:}$ This exercise characterizes matrices of rank 1 in terms of a simple factorization condition. It shows that an $m \times n$ matrix $A$ has rank 1 if and only if there exist vectors $c \in \mathbb{F}^m$ and $d \in \mathbb{F}^n$ such that $A_{j,k} = c_jd_k$ for all entries of $A$. The proof leverages the fundamental connection between the rank of a matrix and the dimensions of its row and column spaces. Matrices of rank 1 have their rows lying in a 1-dimensional subspace, which allows them to be expressed as scalar multiples of a single non-zero row vector $c$. Similarly, their columns lie in a 1-dimensional subspace, which means they are scalar multiples of a single non-zero column vector $d$. $\textbf{Exercise 16 Examples:}$ 1) The matrix $A = \begin{pmatrix}1&2\\3&6\end{pmatrix}$ has rank 1, since $A_{1,1} = 1 \cdot 1, A_{1,2} = 1 \cdot 2, A_{2,1} = 3 \cdot 1, A_{2,2} = 3 \cdot 2$, where $c = (1, 3)$ and $d = (1, 2)$. 2) The matrix $A = \begin{pmatrix}0&0\\0&0\end{pmatrix}$ has rank 0, but it vacuously satisfies the factorization condition with $c = 0$ and $d$ arbitrary, or $c$ arbitrary and $d = 0$. 3) The matrix $A = \begin{pmatrix}1&0&-1\\2&0&-2\\3&0&-3\end{pmatrix}$ has rank 1, since $A_{j,1} = j, A_{j,2} = 0, A_{j,3} = -j$ for $j = 1, 2, 3$, where $c = (1, 2, 3)$ and $d = (1, 0, -1)$. 4) The matrix $A = \begin{pmatrix}1&2&3\\4&5&6\end{pmatrix}$ does not have rank 1, since its rows are linearly independent. $\textbf{Implications and Applications:}$ Rank 1 matrices arise frequently in applications and play an important role in matrix analysis and numerical linear algebra. They correspond to simple outer products of vectors, which encode fundamental operations in areas like computer graphics, machine learning, and signal processing. The rank 1 characterization also connects to the study of matrix decompositions (e.g., singular value decomposition) and low-rank matrix approximations, which have numerous applications in data analysis, compression, and dimensionality reduction. From a theoretical perspective, rank 1 matrices generate the class of finite rank matrices, which form an algebraic structure closely related to tensor products of vector spaces. --- $\textbf{Exercise 16 Implications and Applications (continued):}$ The characterization of matrices of rank 1 provided in this exercise has important implications and applications in various fields, including data analysis, machine learning, signal processing, and the study of low-rank matrix approximations. In data analysis and machine learning, rank-1 matrices often arise in the context of principal component analysis (PCA) and singular value decomposition (SVD). PCA is a widely used technique for dimensionality reduction and data compression, where the goal is to represent high-dimensional data using a lower-dimensional subspace. The rank-1 characterization provides insights into the structure of the principal components, which are often expressed as outer products of vectors. In signal processing, rank-1 matrices play a crucial role in the analysis and processing of signals that can be represented as outer products of time-varying vectors. This includes applications such as beamforming, array signal processing, and source separation techniques. The study of low-rank matrix approximations is an active area of research with numerous applications in data compression, image processing, and recommender systems. The rank-1 characterization provides a building block for constructing low-rank approximations of matrices by expressing them as sums of rank-1 matrices. This decomposition is directly related to the SVD and is used in various matrix completion and approximation algorithms. Furthermore, the rank-1 characterization has connections to the study of tensor products and the representation theory of finite-dimensional algebras. Rank-1 matrices generate the class of finite rank matrices, which form an algebraic structure closely related to tensor products of vector spaces. This connection has applications in areas such as quantum computing, where tensor products play a fundamental role in describing the state of quantum systems. Overall, the characterization of rank-1 matrices provided in this exercise is a fundamental result in matrix analysis that has wide-ranging implications and applications in various fields, including data analysis, machine learning, signal processing, low-rank matrix approximations, and the study of algebraic structures related to tensor products.