Matrices as Functions A friend of mine was criticized in undergrad by a Professor for saying that a matrix is a function. Now, a matrix can be represented by a linear transformation, and linear transformations by definition are functions. Is there any theoretical reason as to why a matrix can't be dubbed a function? From my understanding, they are functions. Am I missing something?

A linear transformation is a function.

Given a linear transformation $T : \mathbb{R}^n \to \mathbb{R}^m$, and a choice of bases on $\mathbb{R}^n$ and $\mathbb{R}^m$, there is an $m\times n$ matrix $A$ such that $T(x) = Ax$ for all $x \in \mathbb{R}^n$. Note, choices were made to write $T$ in this form. Note, $A$ is not the function, it is just used in the definition of the function.

In the other direction, given an $n\times m$ matrix $A$, and a choice of bases on $\mathbb{R}^n$ and $\mathbb{R}^m$, you can define a linear transformation $T : \mathbb{R}^n \to \mathbb{R}^m$ by $T(x) = Ax$ but again, $A$ is not the function, $T$ is.

Consider the analogous situation of a function $f : \mathbb{R} \to \mathbb{R}$ given by $f(x) = 2x$. Would you say that $2$ is a function? Here $2$ plays the role of the matrix $A$. This isn't so much an analogy as it is the special case $n = m = 1$.