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.
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$(a)$ 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 $C$.
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: $$ \left\{ \begin{array}{ll} (4,~4)=\alpha_1 (2,~2)+\beta_1 (-2,~2)\to \left\{ \begin{array}{ll} 4=2\alpha_1-2\beta_1 \\ 4=2\alpha_1 +2\beta_1 \end{array} \right.\\ (8,~4)=\alpha_2 (2,~2)+\beta_2 (-2,~2)\to \left\{ \begin{array}{ll} 8=2\alpha_2-2\beta_2 \\ 4=2\alpha_2 +2\beta_2 \end{array} \right. \end{array} \right. $$