Let A1, A2, ...An Prove : P(Un k=1 Ak) = P9A1) + P(A1c ...... Problem 4.Let A1, A2, . . . , An be events. Prove
44. a.Let A and B be two 2 × 2 matrices,Let Tr denote the trace and det denote the determinant. Prove that Tr(AB)-Tr(BA) and det(AB) - det(BA). b. If A is any matrix in SLa(R), prove that det ((-A-t +1 where t = Tr(A). 44. a.Let A and B be two 2 × 2 matrices,Let Tr denote the trace and det denote the determinant. Prove that Tr(AB)-Tr(BA) and det(AB) - det(BA). b. If A is any matrix in SLa(R), prove...
(Exercise 4.13, reordered) Given a series ΣΧί ak, let 8,-Ση-i ak. Σχί ak is Cesaro summable if S1 + 82 +... +Sn lim n-+o converges. (a) Give an example of a series Σ00i ak that is Cesaro sum mable but not convergent (b) Prove that if 1 ak converges, then it is Cèsaro summable. Hint: Say the sequence of partial sums sn → L. Try to prove that =1 8k → L by showing and then splitting the latter sum...
4. Let A be a square matrix. Assume that Ak = 0 for some positive integer k. Then prove that a) 1-A is is invertible b) (1 - A)-1 = 1 + A + A + A + ................ + Ak-1 (This question is printed wrong in the text book, 10th edition. If you have this book, correct it)
(d) Show that if L E Mn is upper triangular, th LL, and argue that IAgIP-lAollF, where IIA]IF、/tr(ATA) represents the Frobenius norm of A, and tr(A)-Σ.1 A" is the trace of A. (e) Assu me that an upper triangular matrix L has the block structure し11 し12 0 In with the size of the Ln blook being m × m. Let A-LTL, and λ = LLT. Show that tr(A (1 : m, 1 : m))-tr(A(1 : m, 1 : m))...
Problem 1 Let {ak} and {bk} be sequences of positive real numbers. Assume that lim “k = 0. k+oo bk 1. Prove that if ) bk converges, so does 'ak k=1 k=1 2. If ) bk diverges, is it necessary that ) ak diverges? k=1 k=1
4. Consider R2x2 with inner product (A, B) tr(ATB), and let V CR2x2 be the subspace 1 1 1 0 This is consisting of upper-triangular matrices. Let B= a basis for V. (You do not need to prove this.) (a) (8 points) Use the Gram-Schmidt procedure on 3 to find an orthonormal basis for V. Find projy (B) (b) (4 points) Let B= 4. Consider R2x2 with inner product (A, B) tr(ATB), and let V CR2x2 be the subspace 1...
3. Prove valid by a deductive proof: 1. S (TR) 2. R R 3. (V S)-(W T)/ .. V D~W 4. Prove valid by a deductive proof: 1. (B. L)VT 2. (BVC) (~LO M) 3.~M /.. T 5. Prove valid by a deductive proof: 1. E.(FVG) 2. (E.G)(HVI) 3. (~HV I)(E . F) /.. H= I
2. a. Define that the vectors α, α 2, ,Ak are linearly independent. b. Prove that α, α 2, ,Ak are linearly independent if α 0 and for every 0 < i k one has that α, κ α, > , αϊ-1 3. Find a linear system with real coefficients for which the span of 2. a. Define that the vectors α, α 2, ,Ak are linearly independent. b. Prove that α, α 2, ,Ak are linearly independent if α...
OTO (7) (a) Let T = (a1, ..., ak) be a k-cycle in Sn, and let o E Sn. Prove that is the k-cycle (o(a), o(az),..., 0(ak)) (b) Let o,t e Sn. Prove that if t is a product of r pairwise disjoint cycles of lengths k1,..., kr, respectively, where kit..., +kr = n, then oto-1 is also a product of r pairwise disjoint cycles of lengths k1,..., kr. (c) Let T1 and T2 be permutations in Sn. Prove that...