Let be such that Prove that thenAE Mmxn AT A= I n. m> n AE Mmxn AT A= I n. m> n
7. (10 points) Let Sym(Z) = \f : Z Z : f bijective) be the set of bijective functions from Z to Z. (Sym(Z),o) is a group, where o denotes the composition of functions. Let g: Z Z be the function 8(n) = {-1 nodd n+1 neven (a) Prove that g € Sym(Z). (b) Find the order of g. Heat: gog - composition of functions
Let A and B be n × m, and m × n matrices over F respectively. Prove that rn ) = det(In-AB) = det(I,n-BA). In det A
Let A and B be n × m, and m × n matrices over F respectively. Prove that rn ) = det(In-AB) = det(I,n-BA). In det A
1. Prove that there are no Let m, n E Z with m, n > 3 and gcd(m, n) of mn. primitive roots
1. Prove that there are no Let m, n E Z with m, n > 3 and gcd(m, n) of mn. primitive roots
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...
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)...
Let U ? Rmxn. Prove that if UTI-In, then n < m.
(a) Let G be a graph with order n and size m. Prove that if (n-1) (n-2) m 2 +2 2 then G is Hamiltonian. (b) Let G be a plane graph with n vertices, m edges and f faces. Using Euler's formula, prove that nmf k(G)+ 1 where k(G) is the mumber of connected components of G.
(a) Let G be a graph with order n and size m. Prove that if (n-1) (n-2) m 2 +2 2 then...
A1. Let M be an R-module and let I, J be ideals in FR (a) Prove that Ann(I +J) -Ann(I) n Ann(J). (b) Prove that Ann(InJ)2 Ann(I) + Ann(J). (c) Give an example where the inclusion in (b) is strict. (d) If R is commutative ald unital and I, J are cornaximal (that is, 1 +J-(1)), prove that Ann(InJ) Ann(I)+Ann(J).
0 and 0, and let a E Z. Prove that [a],m C [a]n if and only if n | Let m,EN with m TT
0 and 0, and let a E Z. Prove that [a],m C [a]n if and only if n | Let m,EN with m TT