3. (60 points) Let P2(F) be the vector space of polynomials in :1: of degree at most 2 with coefcients in a eld F. Dene a function 5' : P2 (F) + P2 (F) by SUM) = fth) f'(1~") + 1'01?)- [i) Find the matrix A for S with respect to the standard basis ,8 = {1,55, 32} for P2(F). (ii) Show that A is invertible and nd .44. (iii) Using the results of parts (i) and (ii), solve the equation 32f\"(m) - f'(:r:) + at) :.- a0 + (113: + azmz