notation for real matrices Is this a valid notation for real $m \times n$ matrices: $\mathbb{R}^{m,n}$. $m$ and $n$ are known.
If it is not, then what would be the right notation for the set of such matrices?
 A: Some people use $\mathbb R^{m \times n}$ to denote $m \times n$ matrices over the reals. Though this notation is perhaps not standard, I like it because:


*

*It resembles the usual English phrase "$m \times n$ matrix of reals" used to describe these matrices. (Admittedly, the notation $M^{m \times n}(\mathbb R)$ suggested by Sasha conveys the same idea equally easily.) 

*When $n$ is $1$ (i.e., for a column vector), the notation $\mathbb R^{m \times 1}$ is close to the standard notation $\mathbb R^m$ for column vectors, which is nice. On the other hand, the notation $\mathbb R^{1\times m}$ standing for a row vector looks equally similar to $\mathbb R^{1 \times m}$. So one must be careful if one distinguishes between row and column vectors. 
A: I have seen many different notations in use but there is unfortunately no standard. I like to use $\mathcal{M}^m_n(\mathbb{F})$ to denote all matrices $m \times n$ over a given field $\mathbb{F}$. It is then consistent, for a given $A \in \mathcal{M}^m_n(\mathbb{F})$, to denote the element at row $i$ column $j$ by $a^i_j$. This representation (in many cases) facilitates the summation convention (when defining multiplication, expanding by bases, etc). Moreover, it is also natural to denote the $i^{th}$ row vector of $A$ by $A^i$ and the $j^{th}$ column vector of $A$ by $A_j$ 
A: A similar notation, not mentioned above, is $\mathrm{Mat}_{m,n}(\mathbb{R})$. Also I suppose it should be mentioned that in the case $m = n$, one tends to only put in a simple sub- or superscript and write something like $\mathrm{Mat}_n(\mathbb{R})$, $M_n(\mathbb{R})$ or $\mathcal{M}_n(\mathbb{R})$.
In different contexts, other notations arise depending on the relevant structure on the space of matrices. For example, when talking about Lie algebras, another way of denoting the space of $n \times n$-matrices is $\mathfrak{gl}_n(\mathbb{R})$.
