Answer
$\begin{bmatrix} 1 & 5 \\ 0 & 1 \end{bmatrix}$
Work Step by Step
The $\mathcal{B}$-matrix of a transformation with standard matrix representation $A$ is given by the formula $A_{\mathcal{B}} = P^{-1} A P$, where $P$ is the matrix whose columns form the basis $\mathcal{B}$. (This is obtained from the formula $A = PDP^{-1}$ given in the text and solving for $D$, which in this case is not necessarily diagonal.)
\begin{align*}
A_{\mathcal{B}} &= P^{-1} A P = \frac{1}{5} \begin{bmatrix} 2 & -1 \\ 1 & 2 \end{bmatrix} \begin{bmatrix} 3 & 4 \\ -1 & -1 \end{bmatrix} \begin{bmatrix} 2 & 1 \\ -1 & 2 \end{bmatrix} \\
&= \frac{1}{5} \begin{bmatrix} 7 & 9 \\ 1 & 2 \end{bmatrix} \begin{bmatrix} 2 & 1 \\ -1 & 2 \end{bmatrix} \\
&= \frac{1}{5} \begin{bmatrix} 5 & 25 \\ 0 & 5 \end{bmatrix} \\
&= \begin{bmatrix} 1 & 5 \\ 0 & 1 \end{bmatrix}.
\end{align*}
