| # Dimension of hyperplanes Theorem A set in of the form where , , and are given, is an affine set of dimension . Conversely, any affine set of dimension can be represented by a single affine equation of the form , as in the above. Proof: Consider a set described by a single affine equation: with . Let us assume for example that . We can express as follows: This shows that the set is of the form , where Since the vectors are independent, the dimension of is . This proves that is indeed an affine set of dimension . The converse is also true. Any subspace of dimension can be represented via an equation for some . A sketch of the proof is as follows. We use the fact that we can form a basis for the subspace . We can then construct a vector that is orthogonal to all of these basis vectors. By definition, is the set of vectors that are orthogonal to .