hey, friend, we have formula to find the transition matrix from b' to B. B is the new place you want to come there, so put B first in |B|B'| and make them become
| I | some new matrix| . the last "some new matrix" is matrix transition from B' to B, denoted by P_(B' to B) and if you want to have the matrix transition from B to B', just take P inverse.
To compute [W]_B , you must do something like W = C1*u1 + C2 * u2 = (3,-5)
(actually, it's not hard , hopefully you know how to get this)
after have [w]B, use inverse transition matrix to calculate [W}B' by putting into the formula P^- * [W]B
hopefully you understand my English