Find the change of coordinates matrices from $B$ to $C$ and from $C$ to $B$.

Question

Let $V$ be the set of all points on the plane $x − 2y − 3z = 0$. You need not verify that $V$ is a vector space. Both $B = \left\{\left( \begin{smallmatrix} 2\\ 1\\0\end{smallmatrix} \right), \left( \begin{smallmatrix} 3\\0\\1 \end{smallmatrix} \right)\right\}$ and $C = \left\{\left( \begin{smallmatrix} 1\\-1\\1 \end{smallmatrix} \right), \left( \begin{smallmatrix} 5\\1\\1 \end{smallmatrix} \right)\right\}$ are bases for $V$. Find the change of coordinates matrices from $B$ to $C$ and from $C$ to $B$.

Answer 1

Hint: Express vectors of $B$ in terms of vectors of $C$ and vice-versa. The coefficients will give you the change of coordinates matrices.