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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


[1] N. Bourbaki, "Elements of mathematics. Algebra: Algebraic structures. Linear algebra" , 1 , Addison-Wesley (1974) pp. Chapt.1;2 (Translated from French)
[2] N. Bourbaki, "Elements of mathematics. Differentiable and analytic manifolds" , Addison-Wesley (1966) (Translated from French)
[3] M. Golubitsky, "Stable mappings and their singularities" , Springer (1973)


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:
This article was adapted from an original article by V.E. GovorovA.F. Kharshiladze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article