2. (D5) Let n = o(a) and assume that a =bk. Prove that <a >=<b> if and only if n and k are relatively prime.
Problem 3. Prove that if bn + B and B < 0, there is an N E N such that for all n > N, bn < B/2.
5. If a, b E R, prove that abl < (a + b^).
Prove, or give a counter example to disprove the following statements. a) b) We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image
prove n^3 >=(n+1)^2 for all n>=2 Step by step answer would be appreciated. Prove that the following statement is true oo give counter example, h3> (htig for all nya
8. (a) Prove that if p and q are prime numbers then p2 + pq is not a perfect square. (b) Prove that, for every integer a and every prime p, if p | a then ged(a,pb) = god(a,b). Is the converse of this statement true? Explain why or why not. (c) Prove that, for every non-zero integer n, the sum of all (positive or negative) divisors of n is equal to zero. 9. Let a and b be integers...
Problem 5 Diagonalize B and compute XA*X-1 to prove this formula for Be, (sections 6.1, 6.2) Bk=15+ 5+-4k has 0 41, Compute also , end sin 0 4 Problem 5 Diagonalize B and compute XA*X-1 to prove this formula for Be, (sections 6.1, 6.2) Bk=15+ 5+-4k has 0 41, Compute also , end sin 0 4
Problem 1. Let A be an m x m matrix. (a) Prove by induction that if A is invertible, then for every n N, An is invertible. (b) Prove that if there exists n N such that An is invertible, then A is invertible. (c) Let Ai, . . . , An be m x m matrices. Prove that if the product Ai … An is an invertible matrix, then Ak is invertible for each 1 < k< n. (d)...
9. Prove that the function f(x) = ax+b is uniformly continuous on R by directly applying the e, 8 definition of uniform continuity.
Problem 5 (a) Let A be an n × m matrix, and suppose that there exists a m × n matrix B such that BA = 1- (i) Let b є Rn be such that the system of equations Ax b has at least one solution. Prove that this solution must be unique. (ii) Must it be the case that the system of equations Ax = b has a solution for every b? Prove or provide a counterexample. (b) Let...