I am given the set $B=[(2,3),(1,2)]$ and $C=[(2,1),(1,1)]$. $L: \mathbb R^2\to \mathbb R^2$ is the linear mapping such that $[x]_B = [L(x)]_C$ (like coordinate vector stuff).
I am told to find $L[(3,5)]$ and $L[(x1,x2)]$. Where should I begin? I'm not told to find $[L(x)]_C$ explicitly or anything so I'm very confused. Please help.
By using the given definition, $$[L(3,5)]_C=[(3,5)]_B\ .$$ Solving the equations $$\lambda_1(2,3)+\lambda_2(1,2)=(3,5)$$ we get $\lambda_1=1$, $\lambda_2=1$ and so $$[L(3,5)]_C=(1,1)\ .$$ By the meaning of a coordinate vector, $$L(3,5)=1(2,1)+1(1,1)=(3,2)\ .$$ Try the other one for yourself. It is exactly the same in principle, you will just have to be careful since you are dealing with unspecified $x_1,x_2$ instead of specific numbers $3,5$.