VectorSpace2

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vector space

(Definition)

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

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

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 $V$ .

This entry is a copy of the GNU FDL vector space article from PlanetMath. Author of the original article: djao. History page of the original is here

"vector space" is owned by bloftin.

See Also: Hilbert space

Cross-references: vectors, module, operations, field

This is version 1 of vector space, born on 2009-07-12.
Object id is 817, canonical name is VectorSpace2.
Accessed 580 times total.

Classification:

Physics Classification:	02.10.Ud (Linear algebra)
	02.10.Xm (Multilinear algebra)
	02.10.Yn (Matrix theory)
	02.10.Hh (Rings and algebras)

Pending Errata and Addenda

None.

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