I just had a really quick question to ask. I was reading a book on linear algebra and have just been trying to wrap my head around what exactly a matrix represents. At one point, the book said
"In a more formal sense an $m × n$ matrix $A$ can be thought of as a point in $\mathbb R^{nm}$, with the agreement that the entries are ordered into rows and columns rather than a single row or single column."
I didn't really understand what this meant. What is a point in $\mathbb R^{nm}$ from an intuitive/not so abstract perspective? Does it mean literally the vector space $\mathbb R^{n × m}$ e.g a $2 × 3$ matrix represents a point in $\mathbb R^{6}$ except instead of the 6-tuple written with 6 entries in one row or 6 entries in one column they are written in both rows and columns?
Using your example, a matrix: $$ A=\begin{bmatrix} a_{11}&a_{12}\\ a_{21}&a_{22}\\ a_{31}&a_{32}\\ \end{bmatrix} $$ is an element of a vector space that , in the standard basis, is represented as: $$ A= a_{11} \begin{bmatrix} 1&0\\ 0&0\\ 0&0\\ \end{bmatrix}+ a_{12} \begin{bmatrix} 0&1\\ 0&0\\ 0&0\\ \end{bmatrix}+ a_{21} \begin{bmatrix} 0&0\\ 1&0\\ 0&0\\ \end{bmatrix}+ a_{22} \begin{bmatrix} 0&0\\ 0&1\\ 0&0\\ \end{bmatrix}+ a_{31} \begin{bmatrix} 0&0\\ 0&0\\ 1&0\\ \end{bmatrix}+ a_{32} \begin{bmatrix} 0&0\\ 0&0\\ 0&1\\ \end{bmatrix} $$
so it can be represented by a vector $[a_{11},a_{12},a_{21},a_{22},a_{31},a_{32}]$ in this space that is isomorphic to $\mathbb{R}^{2\times 3}$