exercise .] 47 . VI = [ABI . AB, ]. ] of vectors . [ H . every choice of bin R~ - 1 O either Eq . ( 5 ) or Eq. ( 6 ) . ) ] |93 V 3 = A and consider $...

question 53. If A, B, and C are (nxn) matrices such that A is nonsingular and AB = 0, then prove that B = 0

exercise .]47 . VI =[ABI . .... AB, ]. ]of vectors . [ H .every choice of bin R~ - 1Oeither Eq . ( 5 ) or Eq. ( 6 ) . ) ]|93 V 3 = A and consider $ 70 . ]V 2 + V 3 ) is also linearly independent .V 2 = \150 . If the set ( VI, V2 , V3 ) of vectors in RM is linearly148 . Let's = ( VI , V 2 , V 3 ) be a set of vectors in R', where49. Let (VI, V2, V3 ) be a set of nonzero vectors in RM152 . If A and B are ( ~ X ~`) matrices such that A is non -151. Suppose that ( Vi, V2, V 3 ) is a linearly independent53 . If A, B, and Care ( n X ~ ) matrices such that A is14 . Let A = [A] . ... . An - 1] bean ( n x ( 1 - 1 ) matrix.. Suppose that C and B are ( 2 X 2 ) matrices and theis linearly independent . [Hint : Set aIV , + az V 2 +dependent , then argue that the set ( V 1 , V2 , V3 , VAJ issuch that V! V ; = 0 when i * j. Show that the setalso linearly dependent for every choice of VA in RM.V1 = Q. Show that 'S is a linearly dependent setsubset of RM. Show that the set ( V 1 , V , + V2, VIt[Hint : Write B = [BY . .... BJ and consider AB =singular and AB = O, then prove that B = O.Show that B = [A] . ... . An - 1, Ab] is singular fononsingular and AB = AC , then prove that B _ C.[ Hint : Consider A ( B - C ) and use the preceding|Hint : Exhibit a nontrivial solution for` V 3 =\\[1. 1 sturgis