I know how to do this when the identity matrix is involved but not otherwise. For example can I get a step-by-step process of how this is done:

Let B = [u1, u2] and B' = [v1, v2] in R2 where

u1 = [ 2 3] u2 = [ 4 -1] v1 = [ 1 3] v2 = [-1 -1]

a) Find transition matrix from B' to B

and

b) Find transition matrix from B to B'