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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.

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You don't "lose" anything just by changing your point of view, so I don't see the problem. – Jack M Aug 14 '14 at 12:56
No, because from $\mathbb R^{nm}$, you can't tell it is two dimensional. A vector and a matrix are never the same. – Tim Aug 14 '14 at 13:35
While I see what you mean by "a matrix is two dimensional", that's a matter of how we think about them, it's not a precise mathematical quality, which makes the question hard if not impossible to answer. – Jack M Aug 14 '14 at 13:38
To elaborate, you can endow $\mathbb R^{nm}$ with a ring structure in the obvious way that makes it isomorphic to the ring of $n \times m$ real matrices, so there's absolutely no algebraic loss of information. – Shawn O'Hare Aug 14 '14 at 21:12
Tim, what are your $A$ and $B$? – whacka Aug 14 '14 at 21:18

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