# Codimension

The **codimension** describes the complement to the dimension in various areas of mathematics . So in -dimensional space the sum of dimension and codimension of an object is the same.In three-dimensional space, a surface (dimension: 2) has codimension 1, a line (dimension: 1) codimension 2 and a point (dimension: 0) has Codimension 3.

## definition

If a vector space is over any field and is a subspace of , then the *codimension* of in becomes through

thus defined as the dimension of the factor space .

## properties

- It always applies

- Is finite dimensional, so it is

- Is a complementary space of in , i.e. H. , so is

- If there are two subspaces, then it always holds

- If there are subspaces, then

## Examples

A plane has dimension 2. In a three-dimensional space it has codimension 1 and in a four-dimensional space it has codimension 2. A point has codimension 1 in a straight line and codimension 2 in a plane. A hyperplane always has codimension 1, the dimension of the hyperplane is always 1 smaller than the dimension of the surrounding space.

## literature

- VE Govorov, AF Kharshiladze: Codimension . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).