Accelerated Mathematics

MAST10008 Accelerated Mathematics 1 Version 2021 1. Given a matrix M in RREF, we refer to the positions of its leading ones as the leading one configuration of M . For example, there are exactly 4 distinct leading one configurations for 2 × 2 matrices, namely ] ] [ ] [ ] [ [ 1 0 1 ∗ 0 1 0 0 0 1 0 0 0 0 0 0 Here we denote by ∗ an entry that can take any real value. (a) Write down the list of distinct leading one configurations for 2 × 3 matrices. Explain briefly but clearly why your list is complete. (b) Write down the list of distinct leading one configurations for 3 × 3 matrices. Mathematics and Statistics 1 2 University of Melbourne MAST10008 Accelerated Mathematics 1 Version 2021 (c) Find (in terms of n) the number of distinct leading one configurations for 2 × n matrices, where n ≥ 2. Prove that your answer is correct (preferably using induction). (d) Find (in terms of n) the number of distinct leading one configurations for n×n matrices, where n ≥ 2. No proof required for this part. Mathematics and Statistics 2 3 University of Melbourne MAST10008 Accelerated Mathematics 1 Version 2021 2. Let Mn×n denote the set of n × n matrices. (a) If A ∈ Mn×n is invertible and X ∈ Mn×n is such that AX = 0, show that X = 0. (b) If A ∈ Mn×n is not invertible, show that there exists a nonzero X ∈ Mn×n such that AX = 0. (c) Let A ∈ Mn×n be fixed and consider the function f : Mn×n → Mn×n defined by f (X) = AX Prove that f is injective if and only if A is an invertible matrix. Mathematics and Statistics 3 4 University of Melbourne MAST10008 Accelerated Mathematics 1 Version 2021 (d) Fix an integer n ≥ 1 and consider the function g : Mn×n → R defined by g(X) = (det X)2021 Is g injective? Surjective? Bijective? Explain your answers. Mathematics and Statistics 4 5 University of Melbourne MAST10008 Accelerated Mathematics 1 Version 2021 3. Consider the following subset of N: S := {6n − 1 | n ∈ N} = {5, 11, 17, 23, 29, 35, . . . } (a) Show that every s ∈ S must be divisible by some prime p ∈ S. (You may assume the Fundamental Theorem of Arithmetic is true.) Mathematics and Statistics 5 6 University of Melbourne MAST10008 Accelerated Mathematics 1 Version 2021 (b) Consider the following proof of the statement “The set S contains infinitely many prime numbers”: Proof. Suppose S contains only finitely many prime numbers, and list all of them: p1 = 6n1 − 1, p2 = 6n2 − 1, . . . , pk = 6nk − 1 Define the number s = 6p1 p2 . . . pk − 1 Clearly s ∈ S, and by part (a) it must be divisible by some prime p ∈ S. But s is not divisible by p1 , p2 , . . . , pk and these are all the primes in S, contradiction. i. Is the statement “The set S contains infinitely many prime numbers” true if we replace 6n − 1 in the definition of S by 6n + 2? Or 6n + 3? Or 6n + 4? Explain your reasoning. ii. Try to modify the proof given above to prove the statement “The set T contains infinitely many prime numbers”, where T := {6n + 1 | n ∈ N} = {7, 13, 19, 25, . . . } Discuss your success or lack thereof. Mathematics and Statistics 6 7 University of Melbourne
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