2. Let A E Mmxn (R). a) Prove that Ax . y = x . Aly for all x E R", y ERm. [3] b ) Suppose that v E Col(A) and that w E Null(A"). Prove that V . W = 0. [3] c) Prove that if w . v = 0 for all v E Col(A) then w E Null(AT). Deduce that Null(AT) = Col(A). [4] d) Let {v1, ..., Vy} be a linearly independent set of vectors in R' and { w1, ...,Ws} another set of linearly independent vectors in R'. Prove that if vi.wj = 0 for all i, j then { V1, . . ., Vr, W1, . .., Ws) is a linearly independent set of vectors in R'. [3] e) Prove that if {v1, ..., v/} C RM is a basis for Col(A) and (WI, . .., Ws} CRM is a basis for Null(A" ) then r t s = m (you may use the Rank-Nullity Theorem and/ or Theorem 3.4.8) and {V 1 , . . . , Vr, W1, . .., Ws) is a basis for m. [4]