Compute the change of basis matrix

I wish to compute the change of basis for the following:

B = {$\displaystyle \begin{bmatrix} 1\\ 1 \end{bmatrix},\begin{bmatrix} 1\\ -1 \end{bmatrix} $}

C = {$\displaystyle \begin{bmatrix} 0\\ 1 \end{bmatrix},\begin{bmatrix} 2\\ 1 \end{bmatrix} $}

How can I compute the change of basis matrix $\displaystyle P_{C \leftarrow B}$?

There are supposedly (at least) two ways to do this.

I know how to do this by expressing the vectors in B in terms of the basis C; when I do that I get the result that:

$\displaystyle P_{C \leftarrow B}$ = $\displaystyle \begin{bmatrix} 1/2 & -3/2\\ 1/2 & 1/2 \end{bmatrix}$

Is this correct at all? Although even if it is correct I'd like to be able to do it using a different approach I got described.

Supposedly I first have to transform the B coordinates to the canonical coordinates and thereafter the canonical coordinates to C coordinates, this should give $\displaystyle P_{C \leftarrow B}$ as $\displaystyle P_C^{-1}P_B$

Could someone show me how to do this for this example?