Let $B = \{\vec{b_1}, \vec{b_2}\}$ and $C = \{\vec{c_1}, \vec{c_2}\}$ be bases for the same vector space $V$. Suppose $\vec{b_1} = 6\vec{c_1} - 2\vec{c_2}$ and $\vec{b_2} = 9\vec{c_1} - 4\vec{c_2}$.
(a) Find the change of coordinates matrix from $B$ to $C$.
(b) Find $[\vec{x}]_C$ for $\vec{x} = -3\vec{b_1} + 2\vec{b_2}$.
Let $B = \{\vec{b_1}, \vec{b_2}\}$ and $C = \{\vec{c_1}, \vec{c_2}\}$ be bases for $\mathbb{R}^2$. Find the change of coordinates matrix from $B$ to $C$ and the change of coordinates matrix from $C$ to $B$.
$\vec{b_1} = \begin{bmatrix} 7 \ 5 \end{bmatrix}$, $\vec{b_2} = \begin{bmatrix} -3 \ -1 \end{bmatrix}$, $\vec{c_1} = \begin{bmatrix} 1 \ -5 \end{bmatrix}$, $\vec{c_2} = \begin{bmatrix} -2 \ 2 \end{bmatrix}$
