## Change of basis for linear transformation

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.