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 $\mathbf 0$, 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 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.