Independence of Vector Space Axioms

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SUMMARY

The discussion centers on the independence of the commutativity axiom in vector spaces, specifically whether it can be derived from the other vector space axioms. The participant struggles to find a proof that demonstrates commutativity using the remaining axioms and questions the existence of non-abelian groups that satisfy the vector space properties. The underlying field is specified as R, and the equation 2·(x+y) = 2·x + 2·y is referenced as a relevant axiom.

PREREQUISITES
  • Understanding of vector space axioms, including closure, associativity, and distributivity.
  • Familiarity with the properties of abelian and non-abelian groups.
  • Knowledge of field theory, particularly the field of real numbers (R).
  • Basic proficiency in mathematical proofs and logic.
NEXT STEPS
  • Research the properties of abelian groups and their implications for vector spaces.
  • Study the axioms of vector spaces in detail, focusing on the role of commutativity.
  • Explore examples of non-abelian structures and their characteristics in relation to vector spaces.
  • Investigate the implications of the equation 2·(x+y) = 2·x + 2·y in the context of vector space axioms.
USEFUL FOR

Mathematics students, educators, and researchers interested in abstract algebra, particularly those studying vector spaces and group theory.

jgens
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Homework Statement



Determine whether the commutativity of (V,+) is independent from the remaining vector space axioms.

Homework Equations



N/A

The Attempt at a Solution



I am having a really hard time with this problem. Off the top of my head I could not think of any way to prove commutativity using the other axioms. On the other hand, I cannot think of any non-abelian groups (V,+) with a map R x V -> V that satisfies all the desired properties.

If someone could get me pointed in the right direction, it would be appreciated.

Edit: I should clarify that the underlying field is R
 
Physics news on Phys.org
What can you deduce from

2\cdot (x+y)=2\cdot x+2\cdot y

which is one of the axioms.
 

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