My book (Tensor Geometry - Poston & Dodson) says the following:

I just can't seem to agree with this result!

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.

so

--(*)

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

However, if

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...