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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.



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"dimension (vector space)" is owned by rmilson. [ full author list (2) ]
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See Also: dimension

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

Attachments:
dimension formulae for vector spaces (Feature) by yark
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Cross-references: real, complex, quotient vector space, subspace, natural number, finite set, cardinality, basis, finite, field, vector space
There are 174 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 21095 times total.

Classification:
AMS MSC: 15A03 (Linear and multilinear algebra; matrix theory :: Vector spaces, linear dependence, rank)
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