, since . As a special case, if some in satisfies , then . Properties Relation to the exterior algebra Given a vector space , one can construct the exterior algebra , whose definition is independent of any quadratic form on . It turns out that if does not have characteristic then there is a natural isomorphism between and considered as vector spaces (and there exists an isomorphism in characteristic two, which may not be natural). This is an algebra isomorphism if and only if . One can thus consider the Clifford algebra as an enrichment (or more precisely, a quantization, cf. the Introduction) of the exterior algebra on with a multiplication that depends on (one can still define the exterior product independently of ). The easiest way to establish the isomorphism is to choose an orthogonal basis for and extend it to a basis for as described above. The map is determined by Note that this only works if the basis is orthogonal. One can show that this map is independent of the choice of orthogonal basis and so gives a natural isomorphism. If the characteristic of is , one can also establish the isomorphism by antisymmetrizing. Define functions by where the sum is taken over the symmetric group on elements, . Since is alternating it induces a unique linear map . The direct sum of these maps gives a linear map between and . This map can be shown to be a linear isomorphism, and it is natural. A more sophisticated way to view the relationship is to construct a filtration on . Recall that the tensor algebra has a natural filtration: , where contains sums of tensors with order . Projecting this down to the Clifford algebra gives a filtration on . The associated graded algebra is naturally isomorphic to the exterior algebra . Since the associated graded algebra of a filtered algebra is always isomorphic to the filtered algebra as filtered vector spaces (by choosing complements of in for all ), this provides an isomorphism (although not a natural one) in any characteristic, even two. Grading In the following, assume that the characteristic is not 2. Clifford algebras are Z2-graded algebras (also known as superalgebras). Indeed, the linear map on V defined by (reflection through the origin) preserves the quadratic form Q and so by the universal property of Clifford algebras extends to an algebra automorphism