The essence of Gaussian elimination

The matrix, say, $A$, depends not only on $T$ but also on the choice of bases in $V, W$. The $i$-th column of $A$ consists of coefficients of the linear combination of the basis of $W$ which is equal to the image of the $i$-th vector of the basis of $V$.

Row transformations of $A$ correspond to changes of the basis of $W$. Say, switching two rows, corresponds to switching two elements of the basis. So the Gauss reduction gives a better basis of $W$. The new matrix corresponds to the same $T$ and the two new bases (the old basis of $V$ and the new basis of $W$).

Similarly, column transformations of $A$ correspond to changes of the basis of $V$.