Elementary Matrix corresponding to Elementary Row Operation

Theorem
Let $\map \MM {m, n}$ be a metric space of dimensions $m \times n$ over a field $K$.

Let $\mathbf I$ denote the unit matrix of order $m$ in $\map \MM {m, n}$.

Let $e$ be an elementary row operation on $\map \MM {m, n}$.

Let $\mathbf E$ be the elementary matrix of order $m$ defined as:
 * $\mathbf E = e \paren {\mathbf I}$

where $\mathbf I$ is the unit matrix.

Let $r_k$ denote the $k$th row of $\mathbf I$ for $1 \le k \le m$.