Problem 44) Prove: n!> 2" for n24. Problem 45) Prove by induction: For n>0·AT- i=1
Prove each problem, prove by induction 1)Statement 2 Statement: 3 (n-1)n 2forn 2 1
Problem # 1: Prove that Pr(AB)-1-Pr(A, 1B). Problem#2: Show that if E and F are independent, then E ' and F are independent.
Problem #2 (a) Prove that 2.At At At where A E Rnxn (b) If (λί,ui), i-1, 2, . .. , n, are the eigenvalue-eigenvector pairs of A Rnxn, what are the eigenvalues and eigenvectors of e? Prove your answer Problem #2 (a) Prove that 2.At At At where A E Rnxn (b) If (λί,ui), i-1, 2, . .. , n, are the eigenvalue-eigenvector pairs of A Rnxn, what are the eigenvalues and eigenvectors of e? Prove your answer
2 x Problem 2. Prove that f(x) = is a bijection frorn [1,2] to [0, 1].
3n-1 is not Cauchy sequence. 3(-1)*+* 5n-27 2. Problem 2 (10 pts.) Prove that sequence an = (-1)"+1 %3D
Prove that x*-(1, 1/2-1) is optimal for the optimization problem (1/2)xTPx + qTr + r -1 xi<1, i-1,2,3, minimize subject to where 13 12-2 22.0 P-12 176 14.5 2 6 12 13.0 Prove that x*-(1, 1/2-1) is optimal for the optimization problem (1/2)xTPx + qTr + r -1 xi
Problem 3 1. Find the values of (379) and (4725). 2. Prove that for any m > 2, (m) is even. 3. Prove that if (371) - 36(n) then 3|n. Hint: Try proving the contrapositive. 4. Suppose that a =b (mod m), a = b (mod n), and ged(m, n) = 1. Prove that a = b (mod mn). 5. Use Euler's Theorem and the method of successive squaring to find 56820 (mod 2444). That is, find the canonical residue...
Problem 5 1. Find the values of (379) and (4725). 2. Prove that for any m > 2, (m) is even. 3. Prove that if (371) - 36(n) then 3|n. Hint: Try proving the contrapositive. 4. Suppose that a =b (mod m), a = b (mod n), and ged(m, n) = 1. Prove that a = b (mod mn). 5. Use Euler's Theorem and the method of successive squaring to find 56820 (mod 2444). That is, find the canonical residue...
Problem 1. Prove that the composition of injective linear maps, when it is defined, yields injective linear map an Problem 2. Prove that if V = span(v1....,) and fe L(V,W) is surjec- Problem 1. Prove that the composition of injective linear maps, when it is defined, yields injective linear map an Problem 2. Prove that if V = span(v1....,) and fe L(V,W) is surjec-