linear transformation/linear combination question

Let B = {$\displaystyle u_1, u_2, u_3$} and B' = {$\displaystyle v_1, v_2, v_3$}

Be the two bases for $\displaystyle R^3$

given that the transition matrix from B' to B is

P = $\displaystyle \begin{bmatrix}1 & -1 & 2\\0 & 1 & 2\\3 & 0 & -1\end{bmatrix}$

how can i find the $\displaystyle u_1, u_2, u_3$ as linear combinations of $\displaystyle v_1, v_2, v_3$

Re: linear transformation/linear combination question

I could be mistaken, but I think that v1 in it's "native basis" of B' would be written just as [1 ; 0 ; 0] (a column vector). So when you left multiply v1 by the matrix P, what you get is u1. So u1 (expressed in the B' basis, ie as a linear combo of v1 v2 v2) is just the first column of P, [1; 0 ; 3]. Similarly, u2 = [-1; 1 ; 0] and u3 = [2 2 -1].

If you want to go the other way, just invert P. It is invertible.