Definition:Vector Space/Definition 2

Definition

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 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

• Equivalence of Definitions of Vector Space over Field

Sources

• 1955: John L. Kelley: General Topology ... (previous) ... (next): Chapter $0$: Algebraic Concepts