The University of Sydney School of Mathematics and Statistics Assignment 2 MATH1014: Introduction to Linear Algebra Intensive February, 2021 Lecturer: Joshua Ciappara 1. Consider the points A = (2, 0, 10), B = (−12,−5, 11), C = (6, 1, 8), D = (−17, 11, 8), and E = (7, 3, 12). (a) Find general equations for the plane P1 containing A, B, and C, and the plane P2 containing A, D, and E. (b) Give a vector form and parametric equations for the line of intersection ` of P1 and P2. (c) When two distinct, non-perpendicular planes intersect, they form an acute angle α and an obtuse angle β = pi − α; see an example below. By considering normal vectors to the planes, find cos θ, where θ is the acute angle formed by P1 and P2. 2. Let `1 be the line through the point P = (4,−7, 2) with direction vector u = [3, 2, 2], and let `2 be the line through the point Q = (1, 0,−3) with direction vector v = [8,−3,−2]. (a) Write down parametric equations for `1 and `2 in terms of parameters s ∈ R and t ∈ R, respectively. (b) State a system of linear equations involving s and t whose solution (if it exists) would give a point of intersection of `1 and `2. (c) By first reducing the associated augmented matrix to row echelon form, determine whether or not `1 and `2 intersect. 3. Let n ≥ 1 be a natural number. The totient ϕ(n) is the number of positive integers k such that 1 ≤ k ≤ n and gcd(k, n) = 1. For example, ϕ(6) = 2, because 1 and 5 are the only possible values of k. (a) Explain why ϕ(p) = p− 1 if p is prime. (b) Euler’s theorem is a more powerful version of Fermat’s little theorem. It says that aϕ(n) = 1 in Zn if a and n are coprime positive integers. Use this to calculate 22021 and 72019 in Z15, showing all working. Copyright © 2021 The University of Sydney 欢迎咨询51作业君
