# Definition:Vector Space

## Definition

### Definition 1

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

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 field.

### Definition 2

Let $\struct {K, +_K, \times_K}$ be a field whose unity is $1_K$.

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

Let $\struct {\map {\mathrm {End} } G, +, \circ}$ be the endomorphism ring of $\struct {G, +_G}$ such that $I_G$ is the identity mapping.

Let $\cdot: \struct {K, +_K, \times_K} \to \struct {\map {\mathrm {End} } G, +, \circ}$ be a ring homomorphism from $K$ to $\map {\mathrm {End} } G$ which maps $1_K$ to $I_G$.

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

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

### Vector

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 defined as

Some sources take a more general view of how to define a **vector space**, and allow the scalar ring to be a division ring:

### Vector Space over Division Ring

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.

## 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.

## Examples

### Arrows through Point in $3$ D Space

The set $\mathbf V$ of all arrows through a given point in ordinary $3$-dimensional Euclidean space forms a vector space whose scalar field is the set of real numbers $\R$.

Thus $\mathbf V$ is itself a vector space of $3$ dimensions.

## Also see

- Results about
**vector spaces**can be found here.

As a vector space is also a unitary module, all the results which apply to modules, and to unitary modules, also apply to vector spaces.