An Introduction to Real Clifford Algebras and Their Classification

Real Clifford algebras are associative, unital algebras that arise from a pairing of a finite-dimensional real vector space and an associated nondegenerate quadratic form. Herein, all the necessary mathematical background is provided in order to develop some of the theory of real Clifford algebras. This includes the idea of a universal property, the tensor algebra, the exterior algebra, and Z2-graded algebras. Clifford algebras are defined by means of a universal property and shown to be realizable algebras that are nontrivial. The proof of the latter fact is fairly involved and all details of proof are given. A method for creating a basis of any Clifford algebra is given. We conclude by giving a classification of all real Clifford algebras as various matrix algebras.