I'm trying to understand a little better change of basis matrices and how they relate to determining if two matrices are similar.
Given finite vector spaces $V,W$ such that $\textrm{dim} V=\text{dim} W$ and a linear transformation $T:W\rightarrow V$ and ordered bases $V_B$ and $W_B$.
Now my book only covers the case where $W=V$ and defines similar matrices such that an invertible change of basis matrix $M$ exists such that:
$[M]_{W_B}^{V_B}[T]_{W_B}[M]^{W_B}_{V_B}=[T]_{V_B}$
Now how does this work when $W=V$? Are the columns of the change of basis matrix still just the basis vectors of $W$ according to their coordinates in $V$? Or does some type of mapping of $W_B$ to $V_B$ have to be done first? I'm asking because I kind of was looking at a change of basis matrix as a special case where the transformation is the identity transformation.
I hope this question makes sense.