= = = (20 points) As usual, we write P, for the vector space of polynomials with degree less than or equal to 1. A basis for P, is B = {P1, P2}, where p = -1 and P2 = 1-t. Let p(t) = lo + ait be an element of P1, and consider the linear transformation T: P1 P1 given by d T(p) = dip+p= a1 +p. p dt P (a) (8 pts) Find the coordinate vectors (T (P1)] and [T (P2)]B of p and p2 in the basis B. (b) 2 pts) Use the coordinate vectors from part (a) to determine if T(p1) and T (p2) are linearly independent or linearly dependent. (c) (6 pts) Find the matrix of T in the basis B. 2 = [ - - (d) (4 pts) Consider a polynomial q P, whose coordinate vector is [q]b Using the matrix you found in part (c), find the coordinates (T(m)]b, and then use this result to express the polynomial itself, q(t), as a function of t