Definition:Matrix Space

Let $$m, n \in \mathbb{Z}_+$$, and 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 $$\mathcal {M}_{S} \left({m, n}\right)$$.

Thus, by definition, $$\mathcal {M}_{S} \left({m, n}\right) = S^{\left[{1 \,. \, . \, m}\right] \times \left[{1 \,. \, . \, n}\right]}$$.

If $$m = n$$ then we can write $$\mathcal {M}_{S} \left({m, n}\right)$$ as $$\mathcal {M}_{S} \left({n}\right)$$.