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.
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