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Every nonempty affine set is parallel to a unique subspace L=Ω−Ω.
Let be a vector space and let be a nonempty affine set.
Proposition: The set is parallel to a unique linear subspace , and this subspace is
Context: The subspace captures the "direction" of the affine set. It is the appropriate linear object used to define hyperplanes as affine sets of codimension one.
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. 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.
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.