Editing Clifford algebra (section)

== Basis and dimension ==

Since {{math|''V''}} comes equipped with a quadratic form&nbsp;{{math|''Q''}}, in characteristic not equal to {{math|2}} there exist [[Basis (linear algebra)|bases]] for {{math|''V''}} that are [[orthogonal]]. An [[orthogonal basis]] is one such that for a symmetric bilinear form
<math display="block">\langle e_i, e_j \rangle = 0 </math> for <math> i\neq j</math>, and <math display="block">\langle e_i, e_i \rangle = Q(e_i).</math>

The fundamental Clifford identity implies that for an orthogonal basis
<math display="block">e_i e_j = -e_j e_i</math> for <math>i \neq j</math>, and <math display="block">e_i^2 = Q(e_i).</math>

This makes manipulation of orthogonal basis vectors quite simple. Given a product <math>e_{i_1}e_{i_2}\cdots e_{i_k}</math> of ''distinct'' orthogonal basis vectors of {{math|''V''}}, one can put them into a standard order while including an overall sign determined by the number of [[Transposition (mathematics)|pairwise swaps]] needed to do so (i.e. the [[Parity of a permutation|signature]] of the ordering [[permutation]]).

If the [[dimension (linear algebra)|dimension]] of {{math|''V''}} over {{math|''K''}} is {{math|''n''}} and {{math|{{mset|''e''<sub>1</sub>, ..., ''e''<sub>''n''</sub>}}}} is an orthogonal basis of {{math|(''V'', ''Q'')}}, then {{math|Cl(''V'', ''Q'')}} is [[Free module|free over {{math|''K''}}]] with a basis
<math display="block">\{e_{i_1}e_{i_2}\cdots e_{i_k} \mid 1\le i_1 < i_2 < \cdots < i_k \le n\text{ and } 0\le k\le n\}.</math>

The empty product ({{math|1=''k'' = 0}}) is defined as being the multiplicative [[identity element]]. For each value of {{math|''k''}} there are [[Binomial coefficient|{{math|''n'' choose ''k''}}]] basis elements, so the total dimension of the Clifford algebra is
<math display="block">\dim \operatorname{Cl}(V, Q) = \sum_{k=0}^n \binom{n}{k} = 2^n.</math>