\begin{definition}[Definition:Matrix Space]
Let $m, n \in \Z_{>0}$ be (strictly) positive integers.
Let $S$ be a set.
The '''$m \times n$ matrix space over $S$''' is defined as the set of all $m \times n$ matrices over $S$, and is denoted $\map {\MM_S} {m, n}$.
Thus, by definition:
:$\map {\MM_S} {m, n} = S^{\closedint 1 m \times \closedint 1 n}$
If $m = n$ then we can write $\map {\MM_S} {m, n}$ as $\map {\MM_S} n$.
\end{definition}