# A matrix as a point in $\mathbb{R}^{nm}$

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?

• I think that last line is exactly what it represents. – Khallil Jul 23 '15 at 21:13
• this is just an abstract way of saying that in a matrix, there are $m \times n$ coefficients. To understand matrices, thinking of them as a $\mathbb{R}^{nm}$ vector is certainly not the good way ! Just keep the array representation. – Tlön Uqbar Orbis Tertius Jul 23 '15 at 21:14
• yes. That's what it means. So if we were doing it by rows we could identify the point $p=(1,1,2,2)\in \mathbb{R}^4$ by the matrix $$p=\begin{pmatrix} 1 & 1 \\ 2 & 2 \end{pmatrix}$$. So $n\times m$ matrices are a subset of $F^{m\times n}$ space. Or you can think of them as linear transformations $F^m\rightarrow F^n$ (or $F^n\rightarrow F^m$ if you multiply on the other side). – Eoin Jul 23 '15 at 21:15
• are you familiar with term isomorphism? Then you can think of the vector space of matrices and the corresponding euclidean vectorspace and they are isomorphic – user190080 Jul 23 '15 at 21:18

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 $\mathbb{R}^{2\times 3}$