Subspace is an affine set containing the origin. Every affine set is a
translation of some subspace.
Proposition
(Hyperplane representation)Hyperplanes
are sets of the form
.
Proof
Subspaces of dimension
are orthogonal complements of vectors. Hyperplanes are translations of such
subspaces.
Intersection of affine sets is an affine set. Hence, we introduce the affine
hull as follows.
Definition
(Affine hull). The affine hull of the set
is
.
