Definition:Vector Space Axioms

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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      \(\displaystyle \forall \mathbf x \in G:\) \(\displaystyle 1_K \circ \mathbf x = \mathbf x \)             


Also see


Sources