Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question
    
You don't "lose" anything just by changing your point of view, so I don't see the problem. –  Jack M Aug 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 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 at 13:38
1  
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 at 21:12
    
Tim, what are your $A$ and $B$? –  whacka Aug 14 at 21:18

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.