vector space

Let $F$ be a field (or, more generally, a division ring). A vector space $V$ over $F$ is a set with two operations, $+:V\\times V\\longrightarrow V$ and $\\cdot:F\\times V\\longrightarrow V$ , such that

1.
$(\\mathbf{u}+\\mathbf{v})+\\mathbf{w}=\\mathbf{u}+(\\mathbf{v}+\\mathbf{w})$ for all $\\mathbf{u},\\mathbf{v},\\mathbf{w}\\in V$
2.
$\\mathbf{u}+\\mathbf{v}=\\mathbf{v}+\\mathbf{u}$ for all $\\mathbf{u},\\mathbf{v}\\in V$
3.
There exists an element $\\mathbf{0}\\in V$ such that $\\mathbf{u}+\\mathbf{0}=\\mathbf{u}$ for all $\\mathbf{u}\\in V$
4.
For any $\\mathbf{u}\\in V$ , there exists an element $\\mathbf{v}\\in V$ such that $\\mathbf{u}+\\mathbf{v}=\\mathbf{0}$
5.
$a\\cdot(b\\cdot\\mathbf{u})=(a\\cdot b)\\cdot\\mathbf{u}$ for all $a,b\\in F$ and $\\mathbf{u}\\in V$
6.
$1\\cdot\\mathbf{u}=\\mathbf{u}$ for all $\\mathbf{u}\\in V$
7.
$a\\cdot(\\mathbf{u}+\\mathbf{v})=(a\\cdot\\mathbf{u})+(a\\cdot\\mathbf{v})$ for all $a\\in F$ and $\\mathbf{u},\\mathbf{v}\\in V$
8.
$(a+b)\\cdot\\mathbf{u}=(a\\cdot\\mathbf{u})+(b\\cdot\\mathbf{u})$ for all $a,b\\in F$ and $\\mathbf{u}\\in V$

Equivalently, a vector space is a module $V$ over a ring $F$ which is a field (or, more generally, a division ring).

The elements of $V$ are called vectors, and the element $\\mathbf{0}\\in V$ is called the zero vector of $V$ .

Title	vector space
Canonical name	VectorSpace
Date of creation	2013-03-22 11:49:10
Last modified on	2013-03-22 11:49:10
Owner	djao (24)
Last modified by	djao (24)
Numerical id	17
Author	djao (24)
Entry type	Definition
Classification	msc 16-00
Classification	msc 13-00
Classification	msc 20-00
Classification	msc 15-00
Classification	msc 70B15
Synonym	linear space
Related topic	Module
Related topic	Vector2
Related topic	Vector
Related topic	VectorSubspace
Defines	zero vector