Clifford Algebra - Basis and Dimension

Basis and Dimension

If the dimension of V is n and {e₁, …, e_n} is a basis of V, then the set

is a basis for Cℓ(V, Q). The empty product (k = 0) is defined as the multiplicative identity element. For each value of k there are n choose k basis elements, so the total dimension of the Clifford algebra is

Since V comes equipped with a quadratic form, there is a set of privileged bases for V: the orthogonal ones. An orthogonal basis is one such that

where ⟨·,·⟩ is the symmetric bilinear form associated to Q. The fundamental Clifford identity implies that for an orthogonal basis

This makes manipulation of orthogonal basis vectors quite simple. Given a product of distinct orthogonal basis vectors of V, one can put them into standard order while including an overall sign determined by the number of pairwise swaps needed to do so (i.e. the signature of the ordering permutation).