Elementary Matrix corresponding to Elementary Column Operation

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

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

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

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

where $\mathbf I$ is the unit matrix.

Let $\kappa_k$ denote the $k$th column of $\mathbf I$ for $1 \le k \le n$.

Case $(3)$: Exchange Columns
Throughout the above:
 * $E_{a b}$ denotes the element of $\mathbf E$ whose indices are $\tuple {a, b}$
 * $\delta_{a b}$ is the Kronecker delta:
 * $\delta_{a b} = \begin {cases} 1 & : \text {if $a = b$} \\ 0 & : \text {if $a \ne b$} \end {cases}$

Also see

 * Elementary Matrix corresponding to Elementary Row Operation