Let $V$ and $W$ be finite-dimensional vector spaces and let $T:V \rightarrow W$ be a linear transformation between them. I have read that
Performing an elementary row operation on the matrix that represents $T$ is equivalent to performing a corresponding change of basis in the range of $T$, and
Performing an elementary column operation on the matrix that represents $T$ is equivalent to performing a corresponding change of basis in the domain of $T$
Admittedly, this is a rather vague formulation but it's all I have. My question is: Can anyone either explain, or provide a reference to, a precise statement of the relationship between change of basis operations and elementary matrices as described above?