These make no sense to me.

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

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

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

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

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

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

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

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

$\displaystyle (2,3)=a(1,1)+b(0,1)$

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

The two are contradictory!