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The dimension of the quotient space X/L for a subspace L⊂X.
Let be a vector space and let be a linear subspace. The codimension of in is
where is the quotient space of by .
Codimension measures "how many independent linear constraints" define . Codimension one subspaces play a special role in the geometry of hyperplanes.
A vector space over a field is a set equipped with two operations (functions) and such that for all and :
Vector spaces are the basic objects studied via linear maps and invariants such as the determinant and eigenvalues of operators.
Let be a vector space over , and let .
The set is a linear subspace of if:
With the inherited operations, is itself a vector space, and many constructions in analysis arise as subspaces (kernels, images, function spaces, etc.). Subspaces are also the building blocks for quotient spaces and direct sums.
Let be a vector space over , and let be a linear subspace.
Define a relation on by
This is an equivalence relation. The equivalence class of is denoted
The set of all equivalence classes is written
Define operations on by
These operations are well-defined (independent of representatives) and make a vector space, called the quotient vector space.
The codimension of in is defined by
Let be a vector space. An affine set is called a hyperplane if it has codimension one.
More precisely: if , it is parallel to a unique subspace (see Ω−Ω characterization). Then is a hyperplane iff .
In real vector spaces, hyperplanes are exactly level sets of nonzero linear functionals; see the level-set characterization.