dimension (vector space)

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 $v_1,\mathrm{\dots },v_n$. 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.

Title	dimension (vector space)
Canonical name	DimensionvectorSpace
Date of creation	2013-03-22 12:42:31
Last modified on	2013-03-22 12:42:31
Owner	rmilson (146)
Last modified by	rmilson (146)
Numerical id	13
Author	rmilson (146)
Entry type	Definition
Classification	msc 15A03
Related topic	dimension3
Defines	dimension
Defines	codimension
Defines	finite-dimensional
Defines	infinite-dimensional