The matrix

$$A=\left(\begin{array}{rr} 1 & -1 \\ 2 & 2 \\ -1 & 1 \end{array}\right)$$

represents a linear map $\Phi: \mathbb{R}^{2} \rightarrow \mathbb{R}^{3}$ with respect to the bases

$$B=\left\{\left(\begin{array}{l} 1 \\ 1 \end{array}\right),\left(\begin{array}{r} 1 \\ -1 \end{array}\right)\right\} \quad, \quad C=\left\{\left(\begin{array}{l} 1 \\ 1 \\ 0 \end{array}\right),\left(\begin{array}{l} 0 \\ 1 \\ 0 \end{array}\right),\left(\begin{array}{l} 0 \\ 1 \\ 1 \end{array}\right)\right\}$$

Find the matrix $A^{\prime}$ that represents $\Phi$ with respect to the bases

$$B^{\prime}=\left\{\left(\begin{array}{l} 0 \\ 2 \end{array}\right),\left(\begin{array}{l} 2 \\ 0 \end{array}\right)\right\} \quad, \quad C^{\prime}=\left\{\left(\begin{array}{r} 1 \\ 0 \\ -1 \end{array}\right),\left(\begin{array}{l} 0 \\ 1 \\ 0 \end{array}\right),\left(\begin{array}{l} 1 \\ 2 \\ 1 \end{array}\right)\right\}$$