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Problem 3. Let V be a vector space over a field F. Prove that -(-v) = v for all v € V.

Problem 3. Let V be a vector space over a field F. Prove that -(-v) = v for all v € V.

Advanced Engineering Mathematics
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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**Problem 3.** Let \( V \) be a vector space over a field \( \mathbb{F} \). Prove that \(-(-v) = v\) for all \( v \in V \). This problem involves demonstrating a property of vector spaces related to the additive inverse. In any vector space, each vector \( v \) has a unique additive inverse \(-v\) such that \( v + (-v) = 0 \). The statement \(-(-v) = v\) asserts that taking the additive inverse twice yields the original vector. Your task is to provide a formal proof of this property.
Transcribed Image Text:**Problem 3.** Let \( V \) be a vector space over a field \( \mathbb{F} \). Prove that \(-(-v) = v\) for all \( v \in V \). This problem involves demonstrating a property of vector spaces related to the additive inverse. In any vector space, each vector \( v \) has a unique additive inverse \(-v\) such that \( v + (-v) = 0 \). The statement \(-(-v) = v\) asserts that taking the additive inverse twice yields the original vector. Your task is to provide a formal proof of this property.
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