A map from R^n to R^m is a matrix transformation if and only if it is linear

Statement

Let $T \colon \mathbb{R}^n \to \mathbb{R}^m$ be a function. There is an $m \times n$ matrix $A$ with $T(x) = Ax$ for all $x \in \mathbb{R}^n$ if and only if $T$ is a linear map. That is, $T(cv) = cT(v)$ and $T(v+w) = T(v) + T(w)$ for all scalars $c$ and all vectors $v, w$.

When this holds, the matrix $A$ is the one whose $j$-th column is $T(e_j)$, where $e_1, \dots, e_n$ are the standard basis vectors of $\mathbb{R}^n$. So $A = [\,T(e_1)\ \ T(e_2)\ \cdots\ T(e_n)\,]$.