MATH135 Study Guide - Midterm Guide: Fermat Number, Xu, Contraposition

8, Recall GL (n, R) is the group consisting of all matrices, X, with real number entires and det(X) 0. Determine if the following are subgroups of GL(n, R). If they are, prove that you are correct. If they are not, explain why. Hint: the following properties of determinants might be heipful. det(AB) = det(A) det(B) and det (A-1)-det(A) (a) The set of all n × n matrices with real number entires and determinant 1. (b) The set of all n x n matrices with real number entires and determinant -1. (c) The set of all n × n matrices with real number entires and integer determinants.

Which of the following rules are operations on the indicated set? (z designates the set of the integers the rational numbers, and R the real numbers.) For each rule which is not an operation, explain why not. Example a*b = a + b/ab, on the set z. Solution This is not an operation on z. There are integers a and b such that {a + b)/ab is not an integer. example, 2 + 3/2 middot 3 = 5/6 is not an integer.) Thus, z is not closed under *. a * b = squareroot |ab|, on the set Q. a * b = a ln b, on the set {x elementof R: x > 0}. a * b is a root of the equation x^2 - a^2 b^2 = 0, on the set R. Subtraction, on the set z. Subtraction, on the set {n elementof r. greaterthanorequalto 0}. a * b = |a - b, on the set {n elementof r. greaterthanorequalto 0}.

Consider the function f : R3 times R3 rightarrow R defined by where b,c in R. Let Determine b and c such that Then either prove directly from the definition that f, using these values for b and c, is an inner product or give a counterexample to show that it is not. Let V be a finite-dimensional inner product space. Prove that for all Let P2(R) have the inner product p(x), q(x) = p(-1) + p(0)q(0) + p(1)q(1). Verify that S = {x2 + x + 1, x2 + 3x - 2, 19x2 - 3x - 14} is an orthogonal set under this inner product. Write each of 1, x and x2 as linear combinations of the elements of S. Let B and C be orthonormal bases of Rn. Prove that the change of coordinates matrixCPB is an orthogonal matrix. Let p1(x) = x, p2(x) = x2, and p3(x) = 1. Then {p1(x).p2(x),p3(x)} is a basis of P2(R). Use the Gram-Schmidt procedure on this basis, with the vectors in the given order, to find an orthonormal basis for P2(R) under the inner product p(x),q(x)) = p{0)q(0) + p(1)q(1) + p(2)q(2). Repeat part a) using the order p1(x) = x2, p2(x) = 1, and p3(x) = x.