Change of Coordinate Matrix question. I have this question and the wording is very confusing, I dont understand how to answer it. Any help will be greatly appreciated. I have tried answering it and I just dont know where to begin.

EDIT: I'm not just looking for an answer, I genuinely want to understand how it works. Thank you.
 A: To find the target matrix for b, we have to solve $$b_1=\alpha_1 c_1+\beta_1 c_2\\\ b_2=\alpha_2 c_1+\beta_2 c_2 $$ or equivalently: 
$$
 \left\{
        \begin{array}{ll}
            (4,~4)=\alpha_1 (2,~2)+\beta_1 (-2,~2)\to \left\{
        \begin{array}{ll}
            4=2\alpha_1-2\beta_1 \\
            4=2\alpha_1 +2\beta_1
        \end{array}
    \right.\\
            (8,~4)=\alpha_2 (2,~2)+\beta_2 (-2,~2)\to \left\{
        \begin{array}{ll}
            8=2\alpha_2-2\beta_2 \\
            4=2\alpha_2 +2\beta_2
        \end{array}
    \right.
        \end{array}
    \right.
$$
A: $(a)$ If you express the vectors $b_1$ and $b_2$ in terms of $c_1$ and $c_2$ i.e. you find $\alpha,\beta,\gamma, \delta$ s.t.
$$b_1=\alpha c_1+\beta c_2\quad\text{and}\quad b_2=\gamma c_1+\delta c_2$$
then the matrix
$$P=\left(\begin{matrix}\alpha&\gamma\\
\beta&\delta\end{matrix}\right)$$
is the change matrix from the basis $C$ to the basis $B$
$(b)$ Redo the same work or you can also inverse the matrix $P$ to find the change matrix from the basis $B$ to the basis $C$.
