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.
Best Answer
When $W = V$ you generally choose the same basis for $W$ as for $V$ when you change bases. (The point of doing this is, for example, so that you can sensibly use the corresponding matrix representation to take powers or exponentials of the corresponding linear transformation and to find eigenvalues and eigenvectors, etc.) When $W \neq V$ you are free (if you want) to change bases both in $V$ and in $W$, but I don't think people generally call the corresponding equivalence relation similarity. Similarity is very much a $W = V$ kind of phenomenon.
(Exercise: show that up to a change of basis in $V$ and in $W$, any linear transformation is uniquely determined by the dimension of its range.)