Codimension

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

$$\dim L+\codim L=\dim V.$$

If $M$ and $N$ are two subspaces of $V$ of finite codimension, then $M\cap N$ and $M+N$ are also of finite codimension, and

$$\codim(M+N)+\codim(M\cap N)=\codim M+\codim N.$$

The codimension of a submanifold $N$ of a differentiable manifold $M$ is the codimension of the tangent subspace $T_x(N)$ of the tangent space $T_x(M)$ at $x\in N$. If $M$ and $N$ are finite-dimensional, then

$$\codim N=\dim M-\dim N.$$

If $M$ and $N$ are differentiable manifolds, if $L$ is a submanifold of $N$ and if $f\colon M\to N$ is a differentiable mapping transversal to $L$, then

$$\codim f^{-1}(L)=\codim L.$$

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

$$\codim Y=\dim X-\dim Y.$$

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 $L$ of a vector space $V$ is equal to the dimension of any complement of $L$ in $V$, since all complements have the same dimension (as the orthogonal complement).