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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

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



"vector space" is owned by djao.
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See Also: module, vector, Euclidean vector, vector subspace

Other names:  linear space
Also defines:  zero vector

Attachments:
zero vector in a vector space is unique (Theorem) by matte
$\lambda v = 0$ if and only if $\lambda =0$ or $v$ is the zero vector in a vector space (Theorem) by aoh45
zero vector space (Definition) by drini
Cartesian product of vector spaces (Definition) by Mathprof
new vector spaces from old ones (Topic) by matte
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Cross-references: vectors, ring, module, operations, division ring, field
There are 457 references to this entry.

This is version 12 of vector space, born on 2001-10-19, modified 2006-07-22.
Object id is 364, canonical name is VectorSpace.
Accessed 55246 times total.

Classification:
AMS MSC: 15-00 (Linear and multilinear algebra; matrix theory :: General reference works )
 20-00 (Group theory and generalizations :: General reference works )
 13-00 (Commutative rings and algebras :: General reference works )
 16-00 (Associative rings and algebras :: General reference works )
Pending Errata and Addenda
None.
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Discussion
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Metric spaces that cannot be made vector spaces by anon678 on 2008-01-19 15:53:48
Is there some notion of what kind of metric spaces can be turned into vector spaces by finding suitable basis vectors? In other words, given a metric space, is there a test that can say no vector space can exist that conforms to the distance function?
[ reply | up ]
vector space by Dinesh on 2005-10-27 06:05:59
Vector Space
[ reply | up ]
What vector space is not a metric space? by liugj002 on 2004-10-07 22:59:28
What vector space is not a metric space? Anyone can provide an example?

Thank you

George
[ reply | up ]
Is the noncommutative version worth mentioning? by waj on 2004-04-26 04:19:20
Hungerford's _Algebra_ defines vector spaces as unitary modules over division rings, which means funky, noncommutative things can happen (you can have a "left" vector space which isn't a "right" vector space, for example). Knowing next to nothing about noncommutative algebra, I don't know if such beasts are useful enough to talk about. Anyone know?

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