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 $M_{n\times m}(F)$?

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 $\mathbb{R}^m$.
