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I am having trouble understanding what effect elementary matrix operations like a row reduction has on a linear transformation. It seems to me that it would change the either the input basis or the output basis or both depending on the operation.

I understand that elementary matrix operations are themselves linear transformations. But it would seem to me that after applying them, you are no longer dealing with the same linear transformation that you started with, at least (as far as I can tell) in terms of basis.

If the basis of the matrix of a linear transformation does change after an elementary matrix operation, then how would you show that?

My thoughts so far: from Linear Algebra Done Right, if $T\in\mathcal{L}(V,W)$ and $v_1,...,v_n$ is a basis of $V$, $w_1,...,w_m$ is a basis of $W$ then

\begin{equation} \mathcal{M}\left(T\left(v_k\right)\right) = A_{1,k}w_1 + ... + A_{m,k}w_m \end{equation}

and

\begin{equation} \mathcal{M}\left(T,\left(v_1,...,v_n\right),\left(w_1,...,w_m\right)\right) = \begin{pmatrix} A_{1,1} & ... & A_{1,k} & ... & A_{1,n} \\ \vdots & ... & \vdots & ... & \vdots \\ A_{j,1} & ... & A_{j,k} & ... & A_{j,n} \\ \vdots & ... & \vdots & ... & \vdots \\ A_{m,1} & ... & A_{m,k} & ... & A_{m,n} \\ \end{pmatrix} \end{equation}

There does not seem to me to be a simple answer to what happens to the input basis if a scaled row is added to another row.

A couple of answers deal with this but provide no detailed explanation in terms of the definition of the matrix of linear transformation:

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