dimension (vector space)

Let V be a vector spaceMathworldPlanetmath 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 setMathworldPlanetmath v1,,vn. We then call the natural numberMathworldPlanetmath n the dimension of V.

Next, let UV a subspacePlanetmathPlanetmath. The dimension of the quotient vector spaceMathworldPlanetmath 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