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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

  1. 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

  2. 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?

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    If you understand what the relationship between a matrix and a linear transformation given bases of $V$ and $W$ is, this should be pretty straightforward to work out yourself. Otherwise perhaps you should brush up on that aspect first.2011-06-29

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