Question 51. Suppose that {v1, v2, v3} is a linearly independent subset of Rm. Show that the set {v1, v1 + v2, v1 + v2 + v3} is also a linearly independent

47 . V1 =V 2 =a48. Let ‘S = (VI, V2, V3 ) be a set of vectors in R3,, whereV1 = Q. Show that’s is a linearly dependent setof vectors . [Hint : Exhibit a nontrivial solution foreither Eq. ( 5 ) or Eq. ( 6 ). ) ]49. Let ( VI, V2, V3) be a set of nonzero vectors in RM)such that V!Viv; = 0 when i * j. Show that the setis linearly independent . [ Hint : Set aIV, + az V 2 +a3 V 3 = A and consider Q’ 9 . ]50 . If the set ( VI, V2 , V3 ) of vectors in RM is linearlydependent , then argue that the set ( VI , V2 , V3 , VA ) isalso linearly dependent for every choice of V 4 in RM.51. Suppose that (Vi, V2, V3 ) is a linearly independentsubset of RM. Show that the set ( VI , VI + V2, V ItV 2 + V3 ) is also linearly independent .52 . If A and B are ( ~ X ~` ) matrices such that A is non -singular and AB = O, then prove that B = O.[ Hint : Write B = [ B ! . …. B.J and consider AB -[ABI . …. AB~ ]. ]53 . If A , B , and Care ( ~ X ~) matrices, such that A isnonsingular and AB = AC, then prove that B =_ C.5. Suppose that C and B are ( 2 X 2 ) matrices and theevery choice of bo in RX – 1Show that B = [A] . …. An – 1, Ab] is singular forexercise . ][ Hint : Consider A ( B – C) and use the preceding*[1. 1 STwar`$4 . Let A = [A] . …. An- 1] be an ( x X ( n2 – 1 ) ) matrix.