1. [23 marks] Consider the linear operator T on P2 (R), defined by T (at br + cx?) = (a + 4c) + (-2b) x + (3a + 2c)x, and let 3 = {1, 1 - x, 1 -r'} be a basis for P2 (R). (a) Determine the matrix representation of T with respect to , [T],. 8= {1,1- x,1-x } = {el, ez, ea} T(1) = (1)+ (0)=+ (3)12 = (4) e, + (0) e2 + (-3) e3 T(1-x) = (1)+(2)x+ (3)x2 = (6) en + (-2) ez + (-3) es T(1-x]) = (-3) +(0)r+ (1)+2 = (-2) e1+ (0) ez + (-1) es 4 6 O 0 3