# Difference between revisions of "Codimension"

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− | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> | + | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Algebra: Algebraic structures. Linear algebra" , '''1''' , Addison-Wesley (1974) pp. Chapt.1;2 (Translated from French) {{MR|0354207}} {{ZBL|}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> N. Bourbaki, "Elements of mathematics. Differentiable and analytic manifolds" , Addison-Wesley (1966) (Translated from French) {{MR|0205211}} {{MR|0205210}} {{ZBL|}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> M. Golubitsky, "Stable mappings and their singularities" , Springer (1973) {{MR|0341518}} {{ZBL|0294.58004}} </TD></TR></table> |

## Revision as of 16:56, 15 April 2012

The codimension (or quotient or factor dimension) of a subspace of a vector space is the dimension of the quotient space ; it is denoted by , or simply by , and is equal to the dimension of the orthogonal complement of in . One has

If and are two subspaces of of finite codimension, then and are also of finite codimension, and

The codimension of a submanifold of a differentiable manifold is the codimension of the tangent subspace of the tangent space at . If and are finite-dimensional, then

If and are differentiable manifolds, if is a submanifold of and if is a differentiable mapping transversal to , then

The codimension of an algebraic subvariety (or an analytic subspace) of an algebraic variety (analytic space) is the difference

#### References

[1] | N. Bourbaki, "Elements of mathematics. Algebra: Algebraic structures. Linear algebra" , 1 , Addison-Wesley (1974) pp. Chapt.1;2 (Translated from French) MR0354207 |

[2] | N. Bourbaki, "Elements of mathematics. Differentiable and analytic manifolds" , Addison-Wesley (1966) (Translated from French) MR0205211 MR0205210 |

[3] | M. Golubitsky, "Stable mappings and their singularities" , Springer (1973) MR0341518 Zbl 0294.58004 |

#### Comments

The codimension of a subspace of a vector space is equal to the dimension of any complement of in , since all complements have the same dimension (as the orthogonal complement).

**How to Cite This Entry:**

Codimension.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Codimension&oldid=18855