MATH 272 Final: MATH 272 Amherst S17M272Final

Define a function : Z Z such that ?(n) is the number of units in the ring Z/nZ. (a) If p is prime, then compute ?(p). (b) Compute t(6), ???'(10), and ???(15). Conjecture the value of t(pq) when p and q are prime and ptq. (e) A theoremof Eer states that if ged(a,n), the1 (mod n). Using this theorem and part (a), prove that every element of Z/pZ is a root of the polynonial ??-? ?n?/Pz1x]. In other words, prove that for any integer a the congruence a a (mod p) holds

For this problem, the norm A of a matrix A is the norm induced by the standard inner product, (A, B) = tr(BTA). Let P be an n x n orthogonal matrix and let A be an n x m matrix. Prove that PA = A . Let Q be an m x m orthogonal matrix and let A be an n x m matrix. Prove that AQ = A . Let A and B be orthogonally similar matrices, that is, assume that there exists an orthogonal matrix P with PTAP = B. Show that A = B . Let A = and Let B = Show that while there exists an invertible matrix C with B = C-l AC, there does not exist an orthogonal matrix P with B = PTAP.

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.