Please help me with question 1

Page ( ms 0 ZOOM + Proof 1 (10 points): For an n X n matrix A, show that the following statements are equivalent. a) A is non-singular. b) The columns ofA are linearly independent. c) Rank A = n. d) Nullity A = 0. Proof 2 (10 points): If V1, V2,and V3 are vector spaces; F:V1>V2 and G:V2>V3 are linear transformations, then H: V1>V3 dended by: H07) = G(F(1J)) v 6 V1 is also a linear transformation. Note: To show a map is a linear transformation, you need to show HaH (17) +BH (w) f or v,w 6 V1 and a, [2 are scalars. Proof 3 (10 points): LetA and B are non-singularn X n matrices. a) Show that MA) = N(A2). Hint: let x E N(A), show that x is also in N(A2), and vice versa. b) Show that if A and B has the same rank, then the rank of A2 must be equal to the rank of 32