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