1. For each of the following statements, determine whether the statement is true or false. If the statement is true, give a brief (one or two line) explanation of why the statement is true. If the statement is false, give an explicit numerical counterexample to the statement. No marks will be given without correct justification. (a) For any natural number m, the product of two non-zero numbers in Zm is non-zero. (b) There are exactly 2 solutions to 4x = 2 in Z6. (c) If TR2 R2 is a rotation about the origin such that T₁ = 12, then T = I₂ or T = −I2. (d) If v = Rn is an eigenvector of an n × n matrix A with eigenvalue 0, then rank(A) < n. (e) If x is a steady state vector for a Markov chain with transition matrix P then (P²+2P)(x) = 3x.

1. For each of the following statements, determine whether the statement is true or false. If the statement is true, give a brief (one or two line) explanation of why the statement is true. If the statement is false, give an explicit numerical counterexample to the statement. No marks will be given without correct justification. (a) For any natural number m, the product of two non-zero numbers in Zm is non-zero. (b) There are exactly 2 solutions to 4x = 2 in Z6. (c) If TR2 R2 is a rotation about the origin such that T₁ = 12, then T = I₂ or T = −I2. (d) If v = Rn is an eigenvector of an n × n matrix A with eigenvalue 0, then rank(A) < n. (e) If x is a steady state vector for a Markov chain with transition matrix P then (P²+2P)(x) = 3x.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter1: Fundamental Concepts Of Algebra

Section1.1: Real Numbers

Problem 35E

See similar textbooks