There are 2 bases, $B$ and $C$ that are bases for $\Bbb R^2$. Let $B=\{b_1, b_2\}$ and $C=\{c_1,c_2\}$. Find the change of coordinates matrix from $B$ to $C$.
$b_1= \left[ \begin{matrix} 7 \\ 5 \end{matrix} \right]$ $b_2= \left[ \begin{matrix} -3 \\ -1 \end{matrix} \right]$
$c_1= \left[ \begin{matrix} 1 \\ -5 \end{matrix} \right]$ $c_2= \left[ \begin{matrix} -2 \\ 2 \end{matrix} \right]$
Let $ [b_1]_C = \left[ \begin{matrix} x_1 \\ x_2 \end{matrix} \right]$ and $ [b_2]_C = \left[ \begin{matrix} y_1 \\ y_2 \end{matrix} \right]$
$\left[ \begin{matrix} 1 && -2\\ -5 && 2 \end{matrix} \right] \left[ \begin{matrix} x_1 \\ x_2 \end{matrix} \right] = \left[ \begin{matrix} 7 \\ 5 \end{matrix} \right]$
$\left[ \begin{matrix} 1 && -2\\ -5 && 2 \end{matrix} \right] \left[ \begin{matrix} y_1 \\ y_2 \end{matrix} \right] = \left[ \begin{matrix} -3 \\ -1 \end{matrix} \right]$
Then by using row operations you can find $[b_1]_C$ and $[b_2]_C$.
Thus the change of coordinates matrix from B to C would be $ \left[ \; [b_1]_C \, \, [b_2]_C \; \right]$
Why are you able to find $[b_1]_C$ and $[b_2]_C$ by doing this?
Why is it that you can augment the matrix with both $[b_1]_C$ and $[b_2]_C$ and it works as well for solving both of these?
Edit:
I just realized why $[b_1]_C$ and $[b_2]_C$ can be obtained from the calculations from above by using this equation learned from an earlier lesson.
$ x = P_B [x]_B $
$x$ can be considered to be $b_1$ while $P_B$ can be seen as $P_C$ which would simply be $C$ as it's the change of coordinates matrix from C to the standard basis in $\Bbb R^n$. And of course $ [x]_B $ can be considered as $[b_1]_C$.