These make no sense to me.
Consider a basis F for = , and E is the standard basis.
Now I'm told that if I want to construct a transition matrix, P, from , I just place the vectors in F as columns in the matrix, I.e.:
Now here is the problem: My understanding is that the product, where is a column vector relative to E should give me relative to F. (Since that's presumably what means, I hope).
Ok, so let , then
So presumably is now transitioned to F by the matrix P.
But: If we use the classic way to find relative to F, we do:
so relative to F.
The two are contradictory!