Definition:Vector Space Axioms

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

Let $\left({G, +_G, \circ}\right)_{\mathbb F}$ be a vector space over $\mathbb F$ where:


 * $G$ is a set of objects, called vectors.


 * $+_G: G \times G \to G$ is a binary operation on $G$


 * $\left({\mathbb F, +, \cdot}\right)$ is a division ring whose unity is $1_{\mathbb F}$


 * $\circ: \mathbb F \times G \to G$ is a binary operation

The usual situation is for $\mathbb F$ 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:

Also see

 * Vector Inverse is Negative Vector
 * Vector Scaled by Zero is Zero Vector
 * Vector Inverse Unique
 * Zero Vector Unique