Compute the change of basis matrix

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

B = { }

C = { }

How can I compute the change of basis matrix ?

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:

=

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 as

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