Editing Clifford algebra (section)

=== Complex numbers ===
One can also study Clifford algebras on complex vector spaces. Every nondegenerate quadratic form on a complex vector space of dimension {{math|''n''}} is equivalent to the standard diagonal form
<math display="block">Q(z) = z_1^2 + z_2^2 + \dots + z_n^2.</math>
Thus, for each dimension {{math|''n''}}, up to isomorphism there is only one Clifford algebra of a complex vector space with a nondegenerate quadratic form. We will denote the Clifford algebra on {{math|'''C'''<sup>''n''</sup>}} with the standard quadratic form by {{math|Cl{{sub|''n''}}('''C''')}}.

For the first few cases one finds that
* {{math|Cl{{sub|0}}('''C''') ≅ '''C'''}}, the [[complex number]]s
* {{math|Cl{{sub|1}}('''C''') ≅ '''C''' ⊕ '''C'''}}, the [[bicomplex number]]s
* {{math|Cl{{sub|2}}('''C''') ≅ M<sub>2</sub>('''C''')}}, the [[biquaternion]]s
where {{math|M<sub>''n''</sub>('''C''')}} denotes the algebra of {{math|''n'' × ''n''}} matrices over {{math|'''C'''}}.