1(a) [1 pt] Suppose T : P2 - M2x2 is linear. If it is one-to-one, what is the rank of T? (b) [1 pt] Suppose T : M2x2 - P2 is linear. If it is onto, what is the nullity of T? (c) [2 pts] Suppose T : P3 - M2x2 is linear. If it is one-to-one, prove that it is onto. (d) [2 pts] Suppose T : C6 - P3(C) is linear (the field is C). What is the smallest the nullity could be? 2. Let S : V - W and T : U - V be linear transformations. (a) [3 pts] Prove that if So T is one-to-one, then so is T. (b) [3 pts] Prove that if So T is onto, then so is S. (c) [2 pts] Suppose T : R4 - R3 and S : R3 - R4. Explain why So T cannot be onto. 3. For this question let B = {x2, x, 1}, B' = (x2 + 2, x + 3, x2 + x + 1} and B" = {2x2 + x + 1, x2, 2x+ 1}. You can assume these are all bases of P2. On Question 5 of Assignment 3 you found that 0 1 PB+B'= 0 1 PBB"= 1 0 2 2 0 1/2 -3/4 1/4 1/2 1/2 -5/4 PBB= -1/2 1/4 1/4 PBB" = -1/2 -1/2 3/4 1/2 3/4 -1/4 3/2 1/2 5/4 Let D : P2 -+ P2 be the derivative d/dx. (a) [2 pts] Compute the matrix [D]s+ 8 however you want. (b) [3 pts] Compute the matrix [D]s'+8" in the usual way, column-by-column, by applying D to the basis vectors of " and finding the coordinate vectors of the outputs with respect to the basis B'. (c) [3 pts] Compute the matrix [D]s'+ 8" a different way, using [D]s+-s and the change- of-basis matrices give above