Definition:Vector Space Axioms

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

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


 * $\mathbf V$ is a set of objects, called vectors.


 * $+: \mathbf V \times \mathbf V \to \mathbf V$ is a binary operation on $\mathbf V$


 * $\mathbb F$ is $\R$, $\C$, or any other division ring, with a unity $1_{\mathbb F}$, whose binary operations are $+$ and $\cdot$


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

The vector space axioms consist of the four abelian group axioms:


 * $\forall \mathbf x, \mathbf y, \mathbf z \in \mathbf V: \left({\mathbf x + \mathbf y}\right) + \mathbf z = \mathbf x + \left({\mathbf y + \mathbf z}\right)$


 * $\exists \mathbf 0 \in \mathbf V: \forall \mathbf x \in \mathbf V: \mathbf x + \mathbf 0 = \mathbf x$


 * $\forall \mathbf x \in \mathbf V: \exists \left({-\mathbf x}\right) \in \mathbf V: \mathbf x + \left({-\mathbf x}\right) = \mathbf 0$


 * $\forall \mathbf x, \mathbf y \in \mathbf V: \mathbf x + \mathbf y = \mathbf y + \mathbf x$

together with the properties of a unitary module:


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


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


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


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