what is the precise definition of matrix? Let $F$ be a field. 
To me, saying 'Matrix is a rectangular array of elements of $F$' seems extremely terse. What kind of set is a rectangular array of elements of $F$? 
Precisely, $(F^m)^n \neq (F^n)^m \neq F^{n\times m}$. I wonder which is the precise definition for $M_{n\times m}(F)$?
 A: As a sequence can be viewed as a function, a matrix can be viewed as a function $A:\{1,\ldots,m\}\times \{1,\ldots,n\}\to F$.
A: 
What kind of set is a rectangular array of elements of F?

I want to emphasize that this question is not relevant for working with matrices. We describe a matrix as a rectangular array of elements of F because rectangular arrays are something we understand well and can easily work with. (although many people would have trouble with $0 \times n$ rectangular arrays)
Furthermore, while knowledge that rectangular arrays can be represented set-theoretically is occasionally useful, the specific way to do so is less important.
That said, if somebody demanded that I choose a specific representation of $M_{n \times m}(F)$ as a set for set-theoretic purposes, I would probably select $F^{n \times m}$. (where I've assumed we've chosen the usual representation of integers as sets)
However, if someone asked me to do so for the purposes of algebra, my incredulity would require me to insist that the person explain his application, expecting with near certainty to be able to easily point out that it is, in fact, either not relevant at all, or that the application is not actually to do algebra.
A: You can encode an $n\times m$ matrix over $\mathbb F$ as an element of $(\mathbb F^m)^n$. I can encode it as an element of $(\mathbb F^n)^m$. Someone else can encode it as an element of $\mathbb F^{n\times m}$. How does it matter? In the end, all you care about is being able to talk about the $(i,j)$th entry of the matrix unambiguously, and all of the different encodings provide that.
Similarly, you can ask what the precise definition of an ordered pair is in terms of set theory, but you will get different answers depending on who you ask.
A: The precise definition of $M_{n,m}(F)$ is $F^{nm}$, plus some operations.
A: In an engineering class where we don't do things very rigorously at all, we defined an $n\times m$ matrix as a linear function from $\mathbb{R}^m$ to $\mathbb{R}^m$.
