(a) Find the coordinate vectors $[\mathbf{x}]_B$ and $[\mathbf{x}]_C$ of $\mathbf{x}$ with respect to the bases $\mathcal{B}$ and $\mathcal{C}$, respectively.
(b) Find the change-of-basis matrix $P_{\mathcal{C} \leftarrow \mathcal{B}}$ from $\mathcal{B}$ to $\mathcal{C}$.
(c) Use your answer to part (b) to compute $[\mathbf{x}]_C$, and compare your answer with the one found in part (a).
(d) Find the change-of-basis matrix $P_{\mathcal{B} \leftarrow \mathcal{C}}$ from $\mathcal{C}$ to $\mathcal{B}$.
(e) Use your answers to parts (c) and (d) to compute $[\mathbf{x}]_B$, and compare your answer with the one found in part (a).
$\mathbf{x} = \begin{bmatrix} 3 \\ 1 \\ 5 \end{bmatrix}$ and $\mathcal{B} = \left\{ \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} \right\}$ and $\mathcal{C} = \left\{ \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} \right\}$
