Definition:Elementary Operation/Column

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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 $i, j \in \closedint 1 n: i \ne j$:

\((\text {ECO} 1)\)   $:$   \(\displaystyle \kappa_i \to \lambda \kappa_i \)    For some $\lambda \in K_{\ne 0}$, multiply column $i$ by $\lambda$             
\((\text {ECO} 2)\)   $:$   \(\displaystyle \kappa_i \to \kappa_i + \lambda \kappa_j \)    For some $\lambda \in K$, add $\lambda$ times column $j$ to column $i$             
\((\text {ECO} 3)\)   $:$   \(\displaystyle \kappa_i \leftrightarrow \kappa_j \)    Interchange columns $i$ and $j$             

Also defined as

The order of presentation of the elementary matrix operations, either row or column, may vary according to the source.

Some sources use the Greek letter $\rho$ to enumerate the rows, and $\kappa$ to enumerate the columns, and jocularly remind us that the name rho of the letter $\rho$ is pronounced row.

Also see

  • Results about elementary column operations can be found here.