1(a) (b) (C) (d) [1 pt] Suppose T : P2 ) 142\" is linear. If it is OlletOOHE, what is the rank of T? [1 pt] Suppose T : M'gxg ) P2 is linear. If it is onto, what is the nullity of T? l 2 pts] Suppose T : Pg ) M'gxg is linear. If it is onetoone. prove that it is onto. [2 pts] Suppose T : [C > P3(C) is linear (the eld is C). What is the smallest the nullity could be? . Let S : V > H' and T : U > V be linear transformations. [3 pts] Prove that if S o T is onetoone, then so is T. [3 pts] Prove that if S o T is onto, then so is S. [2 pts] Suppose T : R4 > R3 and S : 1R3 > R4. Explain why 3 o T cannot be onto. . For this question let .3 = {322,33,1}, '3' = {$2 + 2.3: + 3,22 + 2: + l} and .3\" = {22:2 + :11: + 1, 1132, 2:1.\" + 1}. You can assume these are all bases of P2. On Question 5 of Assignment 3 you found that 1 u 1 2 1 (J P3931 = U 1 1 P3+31r= 1 U 2 1 2 :3 1 1 (J 1 1/2 3/4 1/4 1/2 1/2 -5/4 P3q_3= 1/2 1/4 1/4 ,Pgrxx: 1/2 1/2 3/4 1/2 3/4 1/4 3/2 1/2 5/4 Let D : P2 ) 'Pg be the derivative d/dzr. [2 pts] Compute the matrix [EH/3H3 however you want. [3 pts] Compute the matrix [Db/+3\" in the usual way, columnbycolumn, by applying D to the basis vectors of ,8\" and nding the coordinate vectors of the outputs with respect to the basis .3'. [3 pts] Compute the matrix [Db/+3\" a different way, using [D] [3+3 and the change ofbasis matrices give above