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. ${\displaystyle n}$ ${\displaystyle n.}$

definition

If a vector space is over any field and is a subspace of , then the codimension of in becomes through ${\displaystyle V}$ ${\displaystyle U}$ ${\displaystyle V}$ ${\displaystyle U}$ ${\displaystyle V}$

${\displaystyle \mathrm {codim} (U,V)=\dim(V/U),}$

thus defined as the dimension of the factor space . ${\displaystyle V/U}$

properties

• It always applies

${\displaystyle \dim U+\mathrm {codim} (U,V)=\dim V.}$

Is finite dimensional, so it is ${\displaystyle V}$

${\displaystyle \mathrm {codim} (U,V)=\dim V-\dim U.}$

• Is a complementary space of in , i.e. H. , so is ${\displaystyle W}$ ${\displaystyle U}$ ${\displaystyle V}$ ${\displaystyle U\oplus W=V}$

${\displaystyle \mathrm {codim} (U,V)=\dim W.}$

• If there are two subspaces, then it always holds ${\displaystyle U_{1},U_{2}\subseteq V}$

${\displaystyle \mathrm {codim} (U_{1}\cap U_{2},V)\leq \mathrm {codim} (U_{1},V)+\mathrm {codim} (U_{2},V).}$

• If there are subspaces, then ${\displaystyle U,W\subseteq V}$

${\displaystyle \mathrm {codim} (U\cap W,W)=\mathrm {codim} (U,U+W)\leq \mathrm {codim} (U,V).}$

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 ).