Find the transaction matrix representing the change of coord

Find the transaction matrix representing the change of coordinates on P_3 from the ordered basis epsilon = [x, 1, x^2] to the ordered basis B = [1 - x^2, x - x^2, 2 - 2x + x^2]. Let p(t) = 1 + 4x + 7x^2. Find [p]B.

Solution

[P]_B = (X-X²) + 4 (1-X²) + 7 (2-2X+X²)

= X-X² +4 -4X²+ 14 -14X + 7X²

= 2X² - 13X + 18