**Problem 5.**Let \( V = \mathbb{R}^+ = \{ x \in \mathbb{R} : x > 0 \} \). Define an addition \( \oplus \) and scalar multiplication \( \otimes \) on \( V \) as follows:1. For \( x, y \in V \), \( x \oplus y = xy \), and2. For \( a \in \mathbb{R} \) and \( x \in V \), \( a \otimes x = x^a \).Show that \( V \) is a vector space under the operations \( \oplus \) and \( \otimes \).

Answered: Image /qna-images/answer/a3b1231b-11ea-41a8-8959-37c89984d90e.jpg

5. Let V = R+ = {x € R : x > 0}. Define an addition and scalar multiplication on V as follows: 1. For x, y € V, x ⇒ y = xy, and 2. For a ER and x EV, ax = xª. Show that V is a vector space under the operations and > .

5. Let V = R+ = {x € R : x > 0}. Define an addition and scalar multiplication on V as follows: 1. For x, y € V, x ⇒ y = xy, and 2. For a ER and x EV, ax = xª. Show that V is a vector space under the operations and > .

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

**Problem 5.**

Let \( V = \mathbb{R}^+ = \{ x \in \mathbb{R} : x > 0 \} \). Define an addition \( \oplus \) and scalar multiplication \( \otimes \) on \( V \) as follows:

1. For \( x, y \in V \), \( x \oplus y = xy \), and
2. For \( a \in \mathbb{R} \) and \( x \in V \), \( a \otimes x = x^a \).

Show that \( V \) is a vector space under the operations \( \oplus \) and \( \otimes \).

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