Our book gives this problem:
Find the $\mathcal{B}$-matrix for the transformation $\vec{x} \rightarrow A\vec{x}$ when the basis $\mathcal{B} = \{ \vec{b}_1, \vec{b}_2 \}$, where $A = \left[\begin{array}{cc} 3 & 4 \\ -1 & -1 \\ \end{array} \right]$, $\vec{b}_1 = \left[\begin{array}{cc} 2 \\ -1 \\ \end{array} \right]$, and $\vec{b}_2 = \left[\begin{array}{cc} 1 \\ 2 \\ \end{array} \right]$.
From what I understand, it's asking us to find the matrix for the same exact transformation as $A$, except relative to to the given bases. I can't figure out where to go from here, though... any thoughts?