Definition:Vector Space

Definition

Let $\left({K, +_K, \times_K}\right)$ be a division ring.

Let $\left({G, +_G}\right)$ be an abelian group.

Let $\left({G, +_G, \circ}\right)_K$ be a unitary $K$-module.

Then $\left({G, +_G, \circ}\right)_K$ 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 $\left({K, +_K, \times_K}\right)$ is by definition a field.

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

Vector Space Axioms

The vector space axioms consist of the abelian group axioms:

\((G0):\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \forall \mathbf x, \mathbf y \in G\)	\(:\)	\(\displaystyle \mathbf x +_G \mathbf y \in G\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Closure Axiom
\((G1):\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \forall \mathbf x, \mathbf y, \mathbf z \in G\)	\(:\)	\(\displaystyle \left({\mathbf x +_G \mathbf y}\right) +_G \mathbf z = \mathbf x +_G \left({\mathbf y +_G \mathbf z}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Associativity Axiom
\((G2):\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \exists \mathbf 0 \in G: \forall \mathbf x \in G\)	\(:\)	\(\displaystyle \mathbf 0 +_G \mathbf x = \mathbf x = \mathbf x +_G \mathbf 0\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Identity Axiom
\((G3):\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \forall \mathbf x \in G: \exists \left({-\mathbf x}\right) \in G\)	\(:\)	\(\displaystyle \mathbf x +_G \left({-\mathbf x}\right) = \mathbf 0\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Inverse Axiom
\((C):\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \mathbf x, \mathbf y \in G\)	\(:\)	\(\displaystyle \mathbf x +_G \mathbf y = \mathbf y +_G \mathbf x\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Commutativity Axiom

together with the properties of a unitary module:

$\forall \lambda \in \mathbb F: \forall \mathbf x, \mathbf y \in G: \lambda \circ \left({\mathbf x +_G \mathbf y}\right) = \lambda \circ \mathbf x +_G \lambda \circ \mathbf y$

$\forall \lambda, \mu \in \mathbb F: \forall \mathbf x \in G: \left({\lambda + \mu}\right)\circ \mathbf x = \lambda \circ \mathbf x + \mu \circ \mathbf x$

$\forall \lambda, \mu \in \mathbb F: \forall \mathbf x \in G: \lambda \circ \left({\mu \circ \mathbf x}\right) = \left({\lambda \cdot \mu}\right) \circ \mathbf x$

$\forall \mathbf x \in G: 1_{\mathbb F} \circ \mathbf x = \mathbf x$

Vector

Let $V$ be a vector space.

Any element $v$ of $V$ is called a vector.

Zero Vector

The identity of $\left({G, +_G}\right)$ 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 $\R^n$.

Also defined as

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

Also see

• Scalar Field
• The axioms for a vector space are listed here.

• 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

• Iain T. Adamson: Introduction to Field Theory (1964)... (previous)... (next): $\S 1.4$
• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 26$
• C.R.J. Clapham: Introduction to Abstract Algebra (1969)... (previous)... (next): $\S 7.32$