Q:

Is there a theorem that says that a function between finite dimensional spaces has a matrix representation?

Just the line in the question. Is there an actual theorem that states this or is it something that people just know.
If there is such a theorem I would like to be able to cite it, that's the purpose of this question.

A:

If $f: E\to F$ is linear yes and the matrix representation is given to a basis as follow :
Let $(e_i)_{i\leq n}$ be a basis of $E$ and $(g_j)_{j\leq m}$ is a basis of $F$
then we calculate $f(e_i)$ for all $i$ and we give their expression in the basis $(g_j)$ 
and if we put :
$$
A=\left(\begin{array}{ccc}
  f(e_1) & f(e_2) & \dots & f(e_n)\\
\end{array}\right)
$$
then for all $x\in E \qquad Ax=f(x)$.
as example : if we take $f: \mathbb{R}^2 \to \mathbb{R^2}$ such that
$$
f(x,y)=(x+y,x-y)
$$
then 
$$
f(e_1)=(1,1)\\
f(e_2)=(1,-1)
$$
so 
$$
A=\left(\begin{array}{ccc}
  f(e_1) & f(e_2) \\
\end{array}\right)=\left(\begin{array}{cc}
  1 & 1\\
1 & -1
\end{array}\right)
$$
we can prove using that representation that every matrix represent a linear mapping. So for general fonction we can't have this representation. 

