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In real vector spaces, Ω is a hyperplane iff Ω={x : f(x)=α} for some f≠0.
Let be a real vector space.
Proposition: A subset is a hyperplane if and only if there exist a nonzero linear functional and a scalar such that
Context: One direction uses the decomposition of codimension-one subspaces (see codimension-one decomposition). The other direction uses that has codimension one (see codimension of kernels).
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. 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 subspace with codim(L) . If , then:
Proposition:
where denotes the direct sum.