# 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. 1.

$(\mathbf{u}+\mathbf{v})+\mathbf{w}=\mathbf{u}+(\mathbf{v}+\mathbf{w})$ for all $\mathbf{u},\mathbf{v},\mathbf{w}\in V$

2. 2.

$\mathbf{u}+\mathbf{v}=\mathbf{v}+\mathbf{u}$ for all $\mathbf{u},\mathbf{v}\in V$

3. 3.

There exists an element $\mathbf{0}\in V$ such that $\mathbf{u}+\mathbf{0}=\mathbf{u}$ for all $\mathbf{u}\in V$

4. 4.

For any $\mathbf{u}\in V$, there exists an element $\mathbf{v}\in V$ such that $\mathbf{u}+\mathbf{v}=\mathbf{0}$

5. 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. 6.

$1\cdot\mathbf{u}=\mathbf{u}$ for all $\mathbf{u}\in V$

7. 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. 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