# Definition:Vector Space Axioms

## Definition

The vector space axioms are the defining properties of a vector space.

Let $\struct {G, +_G, \circ}_K$ be a vector space over $K$ where:

$G$ is a set of objects, called vectors.
$+_G: G \times G \to G$ is a binary operation on $G$
$\struct {K, +, \cdot}$ is a division ring whose unity is $1_K$
$\circ: K \times G \to G$ is a binary operation

The usual situation is for $K$ to be one of the standard number fields $\R$ or $\C$.

The vector space axioms consist of the abelian group axioms:

 $(V \, 0)$ $:$ Closure Axiom $\displaystyle \forall \mathbf x, \mathbf y \in G:$ $\displaystyle \mathbf x +_G \mathbf y \in G$ $(V \, 1)$ $:$ Commutativity Axiom $\displaystyle \forall \mathbf x, \mathbf y \in G:$ $\displaystyle \mathbf x +_G \mathbf y = \mathbf y +_G \mathbf x$ $(V \, 2)$ $:$ Associativity Axiom $\displaystyle \forall \mathbf x, \mathbf y, \mathbf z \in G:$ $\displaystyle \paren {\mathbf x +_G \mathbf y} +_G \mathbf z = \mathbf x +_G \paren {\mathbf y +_G \mathbf z}$ $(V \, 3)$ $:$ Identity Axiom $\displaystyle \exists \mathbf 0 \in G: \forall \mathbf x \in G:$ $\displaystyle \mathbf 0 +_G \mathbf x = \mathbf x = \mathbf x +_G \mathbf 0$ $(V \, 4)$ $:$ Inverse Axiom $\displaystyle \forall \mathbf x \in G: \exists \paren {-\mathbf x} \in G:$ $\displaystyle \mathbf x +_G \paren {-\mathbf x} = \mathbf 0$

together with the properties of a unitary module:

 $(V \, 5)$ $:$ Distributivity over Scalar Addition $\displaystyle \forall \lambda, \mu \in K: \forall \mathbf x \in G:$ $\displaystyle \paren {\lambda + \mu} \circ \mathbf x = \lambda \circ \mathbf x +_G \mu \circ \mathbf x$ $(V \, 6)$ $:$ Distributivity over Vector Addition $\displaystyle \forall \lambda \in K: \forall \mathbf x, \mathbf y \in G:$ $\displaystyle \lambda \circ \paren {\mathbf x +_G \mathbf y} = \lambda \circ \mathbf x +_G \lambda \circ \mathbf y$ $(V \, 7)$ $:$ Associativity with Scalar Multiplication $\displaystyle \forall \lambda, \mu \in K: \forall \mathbf x \in G:$ $\displaystyle \lambda \circ \paren {\mu \circ \mathbf x} = \paren {\lambda \cdot \mu} \circ \mathbf x$ $(V \, 8)$ $:$ Identity for Scalar Multiplication $\displaystyle \forall \mathbf x \in G:$ $\displaystyle 1_K \circ \mathbf x = \mathbf x$