Codimension

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).