# Definition:Vector Space

## Contents

## Definition

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.

If $\times_K$ is commutative, then $\struct {K, +_K, \times_K}$ is by definition a field.

In that case, the scalar ring of $\struct {G, +_G, \circ}_K$ is called the scalar field of $\struct {G, +_G, \circ}_K$.

### Vector Space Axioms

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

### 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 $\mathbf 0$, or some variant of this, and called the **zero vector**.

Note that on occasion it is advantageous to denote the **zero vector** differently, for example by $e$, or $0_V$ or $0_G$, 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 defined as

Some sources insist that $\struct {K, +_K, \times_K}$ needs to be a field, not just a division ring, for this definition to be valid.

## 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.

## Sources

- 1955: John L. Kelley:
*General Topology*... (previous) ... (next): Chapter $0$: Algebraic Concepts - 1961: I.M. Gel'fand:
*Lectures on Linear Algebra*(2nd ed.) ... (previous) ... (next): $\S 1$: $n$-Dimensional vector spaces: Definition $1$ - 1964: Iain T. Adamson:
*Introduction to Field Theory*... (previous) ... (next): $\S 1.4$ - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 26$ - 1969: C.R.J. Clapham:
*Introduction to Abstract Algebra*... (previous) ... (next): Chapter $7$: Vector Spaces: $\S 32$. Definition of a Vector Space - 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous): $\S 8$: Vector spaces and matrices - 1974: Robert Gilmore:
*Lie Groups, Lie Algebras and Some of their Applications*... (previous) ... (next): Chapter $1$: Introductory Concepts: $1$. Basic Building Blocks: $4$. LINEAR VECTOR SPACE - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {B}.26$: Extensions of the Complex Number System. Algebras, Quaternions, and Lagrange's Four Squares Theorem - 2017: Amol Sasane:
*A Friendly Approach to Functional Analysis*: Chapter $1$: Normed and Banach spaces