Change of basis, Linear Algebra I got a basis $B = \{u,v\}$ and a change of basis matrix, $P$, that takes us from $B$ to $C$. 
$$u = (1,0,1,-1)$$
$$v = (1, 1 , 0, 1)$$
$$P = \begin{pmatrix}1& 2\\4& 3\end{pmatrix}$$
I want to figure out which vectors make up the basis $C$.
At first thought I multiplied $B$ with $P$ because I thought that would transform the basis from $B$ to $C$. However when checking the answer they used $P$ inverse to find the $C$. 
Anyone got time to explain this (hopefully) simple problem?  

The problem is as Friedrich Philipp points out. When you quoted the actual problem statement for him, it said:

Let $B$ and $C = [w_1, ..... , w_n]$ both be bases for a vector space $V$. The matrix $P = ( [w_1]B , .... , [w_n]B)$ is called the change of coordinates matrix from $C$-coordinates to $B$-corordinates and satisfies $[X]_B = P [X]_C$

But in your problem statement, you called it the change of coordinates matrix from $B$ to $C$, and treated it as such, when in fact, it goes the other way.
So if $[X]_B = P [X]_C$, then $[X]_C = P^{-1} [X]_B$. Thus the inverse.
