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:

together with the properties of a unitary module:

Terminology
The binary operation $+_G: G \times G \to G$ is usually referred to as (vector) addition.

The element $\mathbf c = \mathbf a + \mathbf b$ of $G$ is called the (vector) sum of $\mathbf a$ and $\mathbf b$.

The identity element $\mathbf 0$ of the abelian group $\struct {G, +_G}$ is called the zero vector.

The inverse element $-\mathbf x$ of a vector $\mathbf x$ is called the negative of $\mathbf x$.

Also see

 * Vector Inverse is Negative Vector
 * Vector Scaled by Zero is Zero Vector
 * Additive Inverse in Vector Space is Unique
 * Zero Vector Unique