Definition:Vector Space

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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 $\R^n$.

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

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