Monthly Archives: July 2020

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

Posted on July 23, 2020 | Leave a comment

Problem
A function f:\mathbb{R} \rightarrow \mathbb{R} is called even if

    \[ f(-x) = f(x)\]


for all x \in \mathbb{R}. A function f:\mathbb{R} \rightarrow \mathbb{R} is called odd if

    \[ f(-x) = -f(x)\]


for all x \in \mathbb{R}. Let U_e denote the set of real-valued even functions on \mathbb{R} and let U_o denote the set of real-valued odd functions on \mathbb{R}. Show that:

    \[ \mathbb{R}^\mathbb{R}=U_e \oplus U_o\]



Proof. To show that U_e + U_o is a direct sum, it is sufficient to prove that U_e \cap U_o = {0}. Let f \in U_e \cap U_o. Hence, f(x) = f(-x) = -f(-x) for all x \in R, which only happens for f(x) = 0. Hence, U_e + U_o is a direct sum.

Let g \in \mathbb{R}^\mathbb{R}. The equation \mathbb{R}^\mathbb{R}=U_e \oplus U_o means that g can be written as the sum of an odd function o and an even function e. Or in other words, for all x \in \mathbb{R}:

(1)   \begin{equation*} g(x) = o(x) + e(x)\end{equation*}

and also:

(2)   \begin{equation*} g(-x) = o(-x) + e(-x)\end{equation*}


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

(3)   \begin{equation*} g(-x) = -o(x) + e(x)\end{equation*}


The system of linear equations (1) and (3) has a unique solution o(x) = (g(x) - g(-x)) / 2 and e(x) = (g(x) + g(-x)) / 2. Hence, g can always be uniquely written as the sum of an odd function and an even function. \qed

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Posted in Axler's Linear Algebra Done Right, Linear Algebra, Mathematics

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

Posted on July 23, 2020 | Leave a comment


Problem
Let G be a group and let a, b \in G. Prove the following:
Let \operatorname{ord}(a)=n, and suppose a has a kth root, say a=b^k. Then \langle a\rangle=\langle b\rangle iff k and n are relatively prime.

Proof. Consider the cyclic subgroup of \mathbb{Z}_{10}. For a = 2 and b = 1, we have a = b^2, n = 5, and \gcd(2, 5) = 1. However, \langle 1 \rangle = \mathbb{Z}_{10} but \langle 2 \rangle consists of all even numbers of \mathbb{Z}_{10}. Hence \langle a \rangle \neq \langle b \rangle. A contradiction. \qed

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Posted in Abstract Algebra, Mathematics, Pinter's Book of Abstract Algebra