Monthly Archives: July 2020

Axler’s Linear Algebra Done Right, Exercise 1C-24

Problem
A function is called even if

for all . A function is called odd if

for all . Let denote the set of real-valued even functions on and let denote the set of real-valued odd functions on . Show that:

Proof. To show that is a direct sum, it is sufficient to prove that . Let . Hence, for all , which only happens for . Hence, is a direct sum.

Let . The equation means that can be written as the sum of an odd function and an even function . Or in other words, for all :

(1)

and also:

(2)

By the definition of the odd and even functions, (2) can be written as:

(3)

The system of linear equations (1) and (3) has a unique solution and . Hence, can always be uniquely written as the sum of an odd function and an even function.

Pinter’s Book of Abstract Algebra Chapter 11, Exercise D5

Problem
Let be a group and let . Prove the following:
Let , and suppose has a th root, say . Then iff and are relatively prime.

Proof. Consider the cyclic subgroup of . For and , we have , , and . However, but consists of all even numbers of . Hence . A contradiction.