We take to be the field for a linear space (otherwise known simply as a vector space, but following Hestenes I will reserve the word for the space of first grade elements) of dimention . The outer product (the exterior product, or the wedge product) is defined such that the graded algebra (exterior algebra of Hermann Grassmann) of multivectors is generated. The geometric algebra is the algebra generated by the geometric product (which is to be thought of as more fundamental) with (for all multivectors )
- Distributivity over the addition of multivectors: and
- Contraction for any "vector" (a grade one element) is a scalar (real number)
The connection between Clifford algebras and quadratic forms come from the distinctive contraction property. This rule also gives the space a metric defined by the naturally derived inner product. It is to be noted that in geometric algebra in all its generality there is no restriction whatsoever on the value of the scalar, it can very well be negative, even zero (in that case, the possibility of an inner product is ruled out if you require ).
One more useful example to convince yourself is to consider , and to generate , one instance of geometric algebra specifically dubbed spacetime algebra by Hestenes (not without reason!). Electromagnetic field tensor, in this context, becomes just a bivector where the imaginary unit, not surprizingly, is the volume element, giving an example of the geometric reinterpretation of the traditional "tricks", making them meaningful. Boosts in this Lorenzian metric space have the same expression as rotation in Euclidean space, where is of course the bivector generated by the time and the space directions involved, whereas in the Euclidean case it is the bivector generated by the two space directions, strengthening the "analogy" to almost identity.