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!