change of basis calculation step 2

Question

I'm practicing using this textbook I have the following $V=R^2 B = {(1,2),(3,4)}, C = {(7,3),(4,2)}$ and $v = (1,0)$... The solution shows

a) determine $[v]B$ and $[V]C$

b) find $P C\leftarrow B$ .....

I'm getting stuck on b)

For step a) we obtain $[v]B = [-1,1], [V]C = [1,-1.5]$. I get this!

Then step b) it shows: $[(1,2)]C = [-3, 5.5]$ and $[(3,4)]C = [-5,9.5]$ I'm stuck on what C is and how they obtain the values....

Answer 1

I understand how b) is calculated.

We want to find $[B_1]_C, [B_2]_C$, aka what are the component vectors of B over C?

$B_1 = (1,2),$ and $C = (7,3), (4,2)$:

$c_1 (7,3) + c_2(4,2) = (1,2)$

$c_1 = -3$... and we continue from there