Definition:Vector Space/Division Ring

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Let $\struct {K, +_K, \times_K}$ be a division ring.

Let $\struct {G, +_G}$ be an abelian group.

Let $\struct {G, +_G, \circ}_K$ be a unitary $K$-module.

Then $\struct {G, +_G, \circ}_K$ is a vector space over $K$ or a $K$-vector space.

That is, a vector space is a unitary module whose scalar ring is a division ring.

Vector Space Axioms

The vector space axioms consist of the abelian group axioms:

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

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


Let $V$ be a vector space.

Any element $v$ of $V$ is called a vector.

Zero Vector

The identity of $\struct {G, +_G}$ is usually denoted $\bszero$, or some variant of this, and called the zero vector:

$\forall \mathbf a \in \struct {G, +_G, \circ}_R: \bszero +_G \mathbf a = \mathbf a = \mathbf a +_G \bszero$

Note that on occasion it is advantageous to denote the zero vector differently, for example by $e$, or $\bszero_V$ or $\bszero_G$, in order to highlight the fact that the zero vector is not the same object as the zero scalar.

Also known as

A vector space is also sometimes called a linear space, especially when discussing the real vector space $\R^n$.

Some go further and refer to a linear vector space

The notation $\struct {G, +_G, \circ, K}$ can also be seen for this concept.

Also see