30. let r {(a, a2, ag) : a; e r be a vector space over r. then which of the following is not a subspace of r? (a) w = {(0, a2, as): a2, ag e r (b) w {(a, a2, ay

30. Let R {(a, a2, ag) : a; E R be a vector space over R. Then which of the following is not a subspace of R? (A) W = {(0, a2, as): a2, ag E R (B) W {(a, a2, ay ): a 20 (C) W {(a), az, as ) :aq t ag + ag = 0 (D) None of these 31. Let Wi and W, be subspaces of a vector space V. Then which one is not always a subspace of V ? (A) W W2 (B) Wi U W2 (C) Wi+W2 (D) None of these 32. Let Vbe a vector space over a field F and W be a subset of V. Then W is a subspace of V iff W is closed under (A) vector addition (B) scalar multiplication (C) Both (A) and (B) (D) None of these 33. The characteristic roots ofa Hermitian matrix are: (A) either pure imaginary or zero (B) all real (C) all complex (D) all pure imaginary 4. The characteristic roots of a square matrix: 0 1 2 A =1 0 -1 2-1 0 are given by : (A) 2,-1,-1 (B) 2,-1+ V3, - 1-3 (C) 1,-2,1 (D) None of these H-3)-MAT(O) (8)