# 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.