Let *V* be a vector space over a field *k*, and *q* : *V* `->` *k* a quadratic form on *V*. The Clifford Algebra C(*q*) is a unital associative algebra over *k* together with a linear map *i* : *V* `->` C(*q*) defined by the following universal property:

for every associative algebra *A* over *k* with a linear map *j* : *V* `->` *A* such that for every *v* in *V* we have *j*(*v*)^{2} = *q*(*v*)1 (where 1 denotes the multiplicative identity of *A*), there is a unique algebra homomorphism
*f* : C(*q*) `->` *A* such that the following diagram commutes

V ----> C(q) | / | / Exists and is unique | / v v Ai.e. such that

The Clifford algebra exists and can be constructed as follows: take the tensor algebra T(V) and mod out by the ideal generated by

*v*tensor*v*-*q*(*v*) 1.

Let

*B*(*u*,*v*) =*q*(*u*+*v*) -*q*(*u*) -*q*(*v*)

*uv*+*vu*=*B*(*u*,*v*)1

The Clifford algebra C(*q*) is filtered by subspaces

- dim C(
*q*) = 2^{dim V}.

If *V* has finite even dimension, the field is algebraically closed and the quadratic form is non degenerate, the Clifford algebra is central simple. Thus by the Artin-Wedderburn theorem it is (non canonically) isomorphic to a matrix algebra. It follows that in this case C(*q*) has an irreducible representation of dimension 2^{dim(V)/2} which is unique up to nonunique isomorphism. This is the (in)famous *spinor representation*, and its vectors are called spinors.

If dim *V* is odd ......

In case the field *k* is the field of real numbers the Clifford algebra of a quadratic form of signature *p*,*q* is usually denoted C(*p*,*q*).
These real Clifford algebras have been classified as follows...

The Clifford algebra is important in physics. Physicists usually consider the Clifford algebra to be spanned by matices γ_{1},...,γ_{n} which have the property that

- γ
_{i}γ_{j}+ γ_{j}γ_{i}= 2 η_{i,j}