Codimension

The dimension of the quotient space X/L for a subspace L⊂X.

Codimension

Let $X$ be a vector space and let $L\subset X$ be a linear subspace . The codimension of $L$ in $X$ is

\operatorname{codim}(L):=\dim(X/L),

where $X/L$ is the quotient space of $X$ by $L$ .

Codimension measures “how many independent linear constraints” define $L$ . Codimension one subspaces play a special role in the geometry of hyperplanes .

Examples:

In $\mathbb{R}^n$ , a hyperplane through the origin has codimension $1$ .
If $L=\{0\}$ and $\dim X=n<\infty$ , then $\operatorname{codim}(L)=n$ .