A relation between sets $A_i, i = 1, \dots, n$ is defined as a subset of $\prod_i A_i$.
Given $m, n \in \mathbb N$, is a set of (some or all) $m \times n$ matrices over $\mathbb R$ considered a subset of $\mathbb R^{nm}$, and therefore a relation on $\mathbb R$?
If yes, do we lose the fact that a $m \times n$ matrix is two dimensional, by treating it as a one dimensional $nm$ vector?
How can we not losing that fact, by considering a set of (some or all) $m \times n$ matrices not just as a relation on $\mathbb R$, but as something more sophisticated? Thanks.
Note: I am ignoring the operations on the matrices and therefore any algebraic structure on the set of some matrices. just think a n by m matrix as a n by m array please.