These make no sense to me.

Consider a basis F for R^2 = \{ (1,1), (0,1)\}, and E is the standard basis.

Now I'm told that if I want to construct a transition matrix, P, from E \to F, I just place the vectors in F as columns in the matrix, I.e.:

P_{E\to F}= \begin{bmatrix}1&0\\1&1\end{bmatrix}.

Now here is the problem: My understanding is that the product, Px where x is a column vector relative to E should give me x' relative to F. (Since that's presumably what E \to F means, I hope).

Ok, so let x = \begin{bmatrix}2\\3\end{bmatrix}, then

Px=\begin{bmatrix}1&0\\1&1\end{bmatrix}\begin{bmat  rix}2\\3\end{bmatrix}=\begin{bmatrix}2\\5\end{bmat  rix} = x'.

So presumably x' is now x transitioned to F by the matrix P.

But: If we use the classic way to find x relative to F, we do:

(2,3)=a(1,1)+b(0,1)

\implies 2=a, ~3 = a + b so (a, b) = (2, 1) relative to F.

The two are contradictory!