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

As a sequence can be viewed as a function, a matrix can be viewed as a function $A:\\{1,\ldots,m\\}\times \\{1,\ldots,n\\}\to F$.