Columns of a matrix transformation are the images of the standard basis vectors

Statement

Let $A$ be an $m \times n$ real matrix. Let $T \colon \mathbb{R}^n \to \mathbb{R}^m$ be the map given by multiplication by a matrix, so that $T(x) = Ax$ for every $x$ in $\mathbb{R}^n$.

For $j = 1, \dots, n$, let $e_j$ be the vector of $\mathbb{R}^n$ with entry $1$ in place $j$ and entry $0$ elsewhere.

Then column $j$ of $A$ is the vector $T(e_j)$ for each $j$. In other words, $A$ is the matrix whose columns are $T(e_1), T(e_2), \dots, T(e_n)$, in this order.