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:

\((\text V 0)\)   $:$   Closure Axiom      \(\ds \forall \mathbf x, \mathbf y \in G:\) \(\ds \mathbf x +_G \mathbf y \in G \)      
\((\text V 1)\)   $:$   Commutativity Axiom      \(\ds \forall \mathbf x, \mathbf y \in G:\) \(\ds \mathbf x +_G \mathbf y = \mathbf y +_G \mathbf x \)      
\((\text V 2)\)   $:$   Associativity Axiom      \(\ds \forall \mathbf x, \mathbf y, \mathbf z \in G:\) \(\ds \paren {\mathbf x +_G \mathbf y} +_G \mathbf z = \mathbf x +_G \paren {\mathbf y +_G \mathbf z} \)      
\((\text V 3)\)   $:$   Identity Axiom      \(\ds \exists \mathbf 0 \in G: \forall \mathbf x \in G:\) \(\ds \mathbf 0 +_G \mathbf x = \mathbf x = \mathbf x +_G \mathbf 0 \)      
\((\text V 4)\)   $:$   Inverse Axiom      \(\ds \forall \mathbf x \in G: \exists \paren {-\mathbf x} \in G:\) \(\ds \mathbf x +_G \paren {-\mathbf x} = \mathbf 0 \)      


together with the properties of a unitary module:

\((\text V 5)\)   $:$   Distributivity over Scalar Addition      \(\ds \forall \lambda, \mu \in K: \forall \mathbf x \in G:\) \(\ds \paren {\lambda + \mu} \circ \mathbf x = \lambda \circ \mathbf x +_G \mu \circ \mathbf x \)      
\((\text V 6)\)   $:$   Distributivity over Vector Addition      \(\ds \forall \lambda \in K: \forall \mathbf x, \mathbf y \in G:\) \(\ds \lambda \circ \paren {\mathbf x +_G \mathbf y} = \lambda \circ \mathbf x +_G \lambda \circ \mathbf y \)      
\((\text V 7)\)   $:$   Associativity with Scalar Multiplication      \(\ds \forall \lambda, \mu \in K: \forall \mathbf x \in G:\) \(\ds \lambda \circ \paren {\mu \circ \mathbf x} = \paren {\lambda \cdot \mu} \circ \mathbf x \)      
\((\text V 8)\)   $:$   Identity for Scalar Multiplication      \(\ds \forall \mathbf x \in G:\) \(\ds 1_K \circ \mathbf x = \mathbf x \)      


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


Sources

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