Independence of Vector Space Axioms

In summary, the question asks whether the commutativity of (V,+) can be proven using the other vector space axioms. The attempt at a solution suggests that the author cannot think of a way to prove commutativity using the other axioms and also cannot think of any non-abelian groups that satisfy all the desired properties. The author also mentions the axiom 2\cdot (x+y)=2\cdot x+2\cdot y and asks for guidance.
  • #1
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
 
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  • #2
What can you deduce from

[tex]2\cdot (x+y)=2\cdot x+2\cdot y[/tex]

which is one of the axioms.
 

FAQ: Independence of Vector Space Axioms

1) What are the vector space axioms?

The vector space axioms are a set of properties that define a vector space. These properties include closure under addition and scalar multiplication, existence of a zero vector and additive inverses, and distributivity and associativity of operations.

2) Why is independence of vector space axioms important?

The independence of vector space axioms is important because it ensures that the properties of a vector space are consistent and well-defined. If any of the axioms were dependent on each other, it could lead to contradictions and invalidate the entire concept of a vector space.

3) How do the vector space axioms relate to linear algebra?

The vector space axioms serve as the foundation for linear algebra, which is the study of vector spaces and their properties. These axioms allow us to define and manipulate vectors and linear transformations in a consistent and systematic manner.

4) Can the vector space axioms be generalized to other mathematical structures?

Yes, the vector space axioms can be generalized to other structures such as modules, which are similar to vector spaces but may not have all the properties of a vector space (e.g. closure under scalar multiplication). These axioms can also be used to define other algebraic structures such as rings and fields.

5) What are some real-world applications of the vector space axioms?

The vector space axioms have numerous applications in fields such as physics, engineering, and computer science. They are used to model and analyze physical systems, design efficient algorithms, and solve problems in areas such as image and signal processing, machine learning, and optimization.

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