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:


 * $\forall \lambda \in \mathbb F: \forall \mathbf x, \mathbf y \in G: \lambda \circ \left({\mathbf x +_G \mathbf y}\right) = \lambda \circ \mathbf x +_G \lambda \circ \mathbf y$


 * $\forall \lambda, \mu \in \mathbb F: \forall \mathbf x \in G: \left({\lambda + \mu}\right)\circ \mathbf x = \lambda \circ \mathbf x + \mu \circ \mathbf x$


 * $\forall \lambda, \mu \in \mathbb F: \forall \mathbf x \in G: \lambda \circ \left({\mu \circ \mathbf x}\right) = \left({\lambda \cdot \mu}\right) \circ \mathbf x$


 * $\forall \mathbf x \in G: 1_{\mathbb F} \circ \mathbf x = \mathbf x$

Also see

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