Deriving the Commutativity of Addition in a Vector Space

One intriguing aspect of vector spaces is the commutativity of their addition operation, often seen as a given property. However, can this commutativity be derived from the other axioms that define a vector space? This article explores how the commutativity of addition can be logically derived, thanks to the properties of modules over a ring.

Understanding the Axioms of Vector Spaces

A vector space is a mathematical construct that combines a field of scalars (or a ring, in the case of modules) with a set of vectors (or module elements) that satisfy a series of axioms. These axioms are designed to ensure that the operations defined on the set of vectors behave consistently and naturally.

Deriving Commutativity from Distributivity and Associativity

Let's consider a module M over a ring R, where the distributive and associative properties of scalar multiplication and addition are given. Suppose M is a monoid with respect to addition, where the addition of any element to itself yields the zero element, i.e., 0u 0 for all u ∈ M. This property is crucial and sets the stage for our derivation.

Initial Assumptions and Notations

Let's denote the scalar multiplication of r in R and vector u in M as r·u, and the addition of vectors u and v as u v. We assume distributive and associative properties for both scalar multiplication and addition in M.

Step-by-Step Derivation

Starting from the assumption that 0u 0 for all u ∈ M, let's expand the expression 1 · (u v) and (u v) · 1 in two different ways:

1 · (u v) u v (by distributivity)

(u v) · 1 u v (by distributivity)

From these expansions, we see that the operation of multiplying by 1 and adding elements do not alter the elements, suggesting a consistent behavior under scalar multiplication and addition.

Simplifying the Expression

Now, let's consider the expression -1 · (u v) and (u v) · -1:

-1 · (u v) -1 -1 (by distributivity)

(u v) · -1 -1 -1 (by distributivity)

Since we know that 0 0 for any u in M, we can equate the two expressions:

-1 -1 -1 -1 (simplification)

Final Step in the Derivation

Now, let's consider the expression u v and v u. By the properties of the module, we can argue that:

-1(u v) -1u -1v (additive inverse)

-1(v u) -1v -1u (additive inverse)

Given that addition is both left and right cancellable, we can conclude that if u v w v and v u w v, then u w and v w. Thus, we can assert that:

u v v u (distributivity and cancellability)

Conclusion

The commutativity of addition in a vector space (or module over a ring) can indeed be derived from the distributive and associative properties of scalar multiplication and addition, along with the property that 0u 0 for all u in the module. This derivation does not require the existence of a division ring, but instead relies on the cancellability of addition and the given axioms of the module.

Related Keywords

vector space commutativity of addition module over a ring