My book (Tensor Geometry - Poston & Dodson) says the following:
I just can't seem to agree with this result!If is a basis for X, and is an isomorphism, then is a basis for Y.
If is a basis for X and is an isomorphism, the change of basis matrix is exactly the matrix .
After hours of tearing my hair out I have come up with the following argument...Please point out where I've gone wrong...
For some basis , some vector and its representation in the coords.
for some other basis where is the change of basis matrix from to coordinates.
We also know the coordinates of in the coords using the basis:
(*) and (**) combine to give
This seems like a nice neat result to me, but if as it is in the book, we have
which is the required result.....
I have tried some basic examples with actual numbers and the results support what I have here... Unless I have some fundamental misunderstanding of all this and what it is supposed to mean, which is quite possible...