Definition:Vector Space

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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      \(\ds \forall \mathbf x, \mathbf y \in G:\) \(\ds \mathbf x +_G \mathbf y \in G \)      
\((\text V 1)\)   $:$   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      \(\ds \forall \mathbf x \in G:\) \(\ds 1_K \circ \mathbf x = \mathbf x \)      


Vector

The elements of the abelian group $\struct {G, +_G}$ are called vectors.


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.


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 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

  • 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

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