Find a transition matrix from bases?

[proctortom]

Homework Statement

I have 2 bases, a = {1, x, x^2} and b = {-2 - 2x + 3x^2 , 1 + 2x - x^2 , -1 - x + 2x^2} of P2.

Find the transition matrix Pab.

How is this done??

Homework Equations

Since this is Linear Algebra, there aren't really any relevant "Equations" as such. More logic based. Right?

The Attempt at a Solution

I am quite muddled. Best I could get was to make [v]s = [1; 1; 1] (Thats a vertical matrix of 1s)

Not quite sure where to go from here.

 

[Dr. T]

The transition matrix is computed with coordinates. For example, the coordinates of the vector 'x^2' in the ordered basis 'a' are (0, 0, 1). Now, write this in the coordinates in the basis 'b.' This can be done by solving a system of equations: x^2 = u_1 b_1 + u_2 b_2 + u_3 b_3 where u_i is an unknown coefficient (to be solved) and b_1, b_2, b_3 are elements in the ordered basis 'b.' Once solved for u_1, u_2, and u_3, the coordinates of x^2 in the ordered basis 'b' are (u_1, u_2, u_3). This gives the first column of the transition matrix. Do a similar thing for the remaining elements in the ordered basis 'a.'