(a) Find the change of basis matrix $P_{\mathcal{B} \leftarrow \mathcal{C}}$.
$$
\begin{gathered}
{\left[\mathbf{c}_{1}\right]_{\mathcal{B}}=\left[\begin{array}{r}
1 \\
0 \\
-1
\end{array}\right], \quad\left[\mathbf{c}_{2}\right]_{\mathcal{B}}=\left[\begin{array}{r}
3 \\
-1 \\
1
\end{array}\right], \quad\left[\mathbf{c}_{3}\right]_{\mathcal{B}}=\left[\begin{array}{l}
1 \\
2 \\
1
\end{array}\right],} \\
P_{\mathcal{B} \leftarrow \mathcal{C}}=\left[\left[\mathbf{c}_{1}\right]_{\mathcal{B}},\left[\mathbf{c}_{2}\right]_{\mathcal{B}},\left[\mathbf{c}_{3}\right]_{\mathcal{B}}\right]=\left[\begin{array}{rrr}
1 & 3 & 1 \\
0 & -1 & 2 \\
-1 & 1 & 1
\end{array}\right] .
\end{gathered}
$$
(b) Consider the vector $\mathbf{x}=2 \mathbf{c}_{1}+\mathbf{c}_{2}-3 \mathbf{c}_{3}$. Find $[\mathbf{x}]_{\mathcal{B}}$, that is, the components of $\mathbf{x}$ in the basis $\mathcal{B}$.
$$
\begin{gathered}
{[\mathbf{x}]_{\mathcal{C}}=\left[\begin{array}{r}
2 \\
1 \\
-3
\end{array}\right], \quad[\mathbf{x}]_{\mathcal{B}}=P_{B \leftarrow \mathcal{C}}[\mathbf{x}]_{\mathcal{C}}} \\
{[\mathbf{x}]_{\mathcal{B}}=\left[\begin{array}{rrr}
1 & 3 & 1 \\
0 & -1 & 2 \\
-1 & 1 & 1
\end{array}\right]\left[\begin{array}{r}
2 \\
1 \\
-3
\end{array}\right]=\left[\begin{array}{r}
2 \\
-7 \\
-4
\end{array}\right]}
\end{gathered}
$$
