In the solution, it simply multiplied $C$ and $P$ respectively.
I don't understand this because the problem states $P$ is the change of coordinate matrix from $B$ to $C$, so I thought that I would need to take the inverse of $P$ and left multiply it with $C$ to find $B$.
I'd appreciate explaining how to go about solving this and why the solution was right multiplication the $B \to C$ matrix.
Thanks
In order for $P$ to be the change of coordinate matrix from $B$ to $C$, it needs to hold that the columns of $P$ are the coordinates of the basis vectors of $B$ regarding $C$.
In particular: it must hold that
$$\begin{pmatrix} 1 \\ -3 \\ 4 \end{pmatrix} = \lambda_1^1\boldsymbol{v_1} + \lambda_2^1\boldsymbol{v_2}+\lambda_3^1\boldsymbol{v_3} = \lambda_1^1 \begin{pmatrix} -2 \\ 2 \\ 3 \end{pmatrix} + \lambda_2^1 \begin{pmatrix} -8 \\ 5 \\ 2 \end{pmatrix} + \lambda_3^1 \begin{pmatrix} -7 \\ 2 \\ 6 \end{pmatrix} $$
$$\begin{pmatrix} 2 \\ -5 \\ 6 \end{pmatrix} = \lambda_1^1\boldsymbol{v_1} + \lambda_2^1\boldsymbol{v_2}+\lambda_3^1\boldsymbol{v_3} = \lambda_1^2 \begin{pmatrix} -2 \\ 2 \\ 3 \end{pmatrix} + \lambda_2^2 \begin{pmatrix} -8 \\ 5 \\ 2 \end{pmatrix} + \lambda_3^2 \begin{pmatrix} -7 \\ 2 \\ 6 \end{pmatrix} $$
$$\begin{pmatrix} -1 \\ 0 \\ 1 \end{pmatrix} = \lambda_1^3\boldsymbol{v_1} + \lambda_2^3\boldsymbol{v_2}+\lambda_3^3\boldsymbol{v_3} = \lambda_1^3 \begin{pmatrix} -2 \\ 2 \\ 3 \end{pmatrix} + \lambda_2^3 \begin{pmatrix} -8 \\ 5 \\ 2 \end{pmatrix} + \lambda_3^3 \begin{pmatrix} -7 \\ 2 \\ 6 \end{pmatrix} $$
where $$ \boldsymbol{u_i} = \begin{pmatrix} \lambda_1^i \\ \lambda_2^i \\ \lambda_3^i \end{pmatrix}$$