Elementary Column Operations as Matrix Multiplications/Corollary

Theorem
Let $\mathbf X$ and $\mathbf Y$ be two $m \times n$ matrices that differ by exactly one elementary column operation.

Then there exists an elementary column matrix $\mathbf E$ of order $n$ such that:


 * $\mathbf X \mathbf E = \mathbf Y$

Proof
Let $e$ be the elementary column operation such that $e \paren {\mathbf X} = \mathbf Y$.

Then this result follows immediately from Elementary Column Operations as Matrix Multiplications:


 * $e \paren {\mathbf X} = \mathbf X \mathbf E = \mathbf Y$

where $\mathbf E = e \paren {\mathbf I}$.