So I have the bases S=(1,0),(0,1) and C=(3,1),(5,2)

and a transformation matrix (in terms of S) (0,1;1,0) . To find this transformation matrix in terms of C I did this;

(basis S in terms of C)(transformation in terms of S)(basis C in terms of S)

I got matrix (-13,21;8,13) if this is the right thing to do, how can I prove it?

Cheers.