How to Find Transition Matrices and Vectors in Ordered Bases

[Benzoate]

Homework Statement

Consider ordered bases B={(1,2),(1,1)} and C={w(1),w(2)} for R^2. Suppose P, a 2 X 2 matrix, P=((2,1),(5,3)) (2,1) and (5,3) being vectors in matrix P. P is the transition matrix from B to C.

a)Find the transition matrix from C to B
b) Find w(1) and w(2)
c) if b =(1,2)((1,2) is a vector.) , read as with subscript B, use P to find c also read as with subscript c. Also find vector u.

Homework Equations

The Attempt at a Solution

a) Is just taking the inverse of P. P^1-= ((3,-1),(-5,2))
b) w(1)=2*v(1)+v(2)=2*(1,2)+(1,1)=(3,5)
w(2)= 5*v(1)+3*v(2)=5*(1,2)+3*(1,2)=(8,13). v(1) and v(2) are vectors in ordered base B by the way.
c) b=(1,2)=> v(1)+2*v(2)=(3,4)=u. Not sure how to find c , but I think c = P^1-

 

[Benzoate]

No one has replied yet. Are the symbols I used to represent my matrices readable.