## transition matrices...

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.