Similar Matrices and Change of Basis

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.

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For distinct vector spaces there is no such thing as a change-of-basis matrix. Recall that (depending on how it is taught) either (1) the columns of the change-of-basis matrix express the vectors of one basis in coordinates on the other basis, or (2) a change-of-basis matrix describes the identity transformation using one of the bases at departure and the other at arrival. Neither description makes sense if you have a pair of basis of different vector spaces (in (2) because there is no identity transformation between them). – Marc van Leeuwen Jun 20 '12 at 8:11

1 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.)

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I didn't quite understand the exercise. What do you mean by "up to a change of basis"? – Robert S. Barnes Jun 21 '12 at 6:30
@Robert: I mean that there is an equivalence relation on linear transformations $V \to W$ where two transformations are equivalent if there is a matrix that represents both of them using appropriate choices of bases in $V$ and $W$. The equivalence classes of this equivalence relation are precisely the linear transformations having range a fixed dimension. – Qiaochu Yuan Jun 21 '12 at 6:53