\begin{definition}[Definition:Elementary Operation/Column]
Let $\mathbf A = \sqbrk a_{m n}$ be an $m \times n$ matrix over a field $K$.
The '''elementary column operations''' on $\mathbf A$ are operations which act upon the columns of $\mathbf A$ as follows.
For some $i, j \in \closedint 1 n: i \ne j$:
{{begin-axiom}}
{{axiom | n = \text {ECO} 1
        | t = For some $\lambda \in K_{\ne 0}$, multiply column $i$ by $\lambda$
        | m = \kappa_i \to \lambda \kappa_i
}}
{{axiom | n = \text {ECO} 2
        | t = For some $\lambda \in K$, add $\lambda$ times column $j$ to column $i$
        | m = \kappa_i \to \kappa_i + \lambda \kappa_j
}}
{{axiom | n = \text {ECO} 3
        | t = Interchange columns $i$ and $j$
        | m = \kappa_i \leftrightarrow \kappa_j
}}
{{end-axiom}}
\end{definition}