PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (213)
Orphanage
Unclass'd (1)
Unproven (473)
Corrections (40)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: Low Entry average rating: No information on entry rating
dimension (vector space) (Definition)

Let $ V$ be a vector space over a field $ K$. We say that $ V$ is finite-dimensional if there exists a finite basis of $ V$. Otherwise we call $ V$ infinite-dimensional.

It can be shown that every basis of $ V$ has the same cardinality. We call this cardinality the dimension of $ V$. In particular, if $ V$ is finite-dimensional, then every basis of $ V$ will consist of a finite set $ v_1,\ldots, v_n$. We then call the natural number $ n$ the dimension of $ V$.

Next, let $ U\subset V$ a subspace. The dimension of the quotient vector space $ V/U$ is called the codimension of $ U$ relative to $ V$.

In circumstances where the choice of field is ambiguous, the dimension of a vector space depends on the choice of field. For example, every complex vector space is also a real vector space, and therefore has a real dimension, double its complex dimension.



Anyone with an account can edit this entry. Please help improve it!

"dimension (vector space)" is owned by rmilson. [ full author list (2) ]
(view preamble)

View style:

See Also: dimension

Also defines:  dimension, codimension, finite-dimensional, infinite-dimensional

Attachments:
dimension formulae for vector spaces (Feature) by yark
Log in to rate this entry.
(view current ratings)

Cross-references: real, complex, quotient vector space, subspace, natural number, finite set, cardinality, basis, finite, field, vector space
There are 168 references to this entry.

This is version 10 of dimension (vector space), born on 2002-06-01, modified 2006-09-06.
Object id is 2993, canonical name is Dimension2.
Accessed 19784 times total.

Classification:
AMS MSC: 15A03 (Linear and multilinear algebra; matrix theory :: Vector spaces, linear dependence, rank)
Pending Errata and Addenda
None.
[ View all 8 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add derivation | add example | add (any)