By David Hestenes

Matrix algebra has been known as "the mathematics of upper arithmetic" [Be]. we predict the root for a greater mathematics has lengthy been to be had, yet its versatility has rarely been favored, and it has now not but been built-in into the mainstream of arithmetic. We confer with the process ordinarily known as 'Clifford Algebra', even though we desire the identify 'Geometric Algebm' advised by way of Clifford himself. Many designated algebraic structures were tailored or built to specific geometric relatives and describe geometric buildings. specifically outstanding are these algebras which were used for this function in physics, specifically, the process of advanced numbers, the quatemions, matrix algebra, vector, tensor and spinor algebras and the algebra of differential kinds. each one of those geometric algebras has a few major virtue over the others in sure functions, so not anyone of them presents an sufficient algebraic constitution for all reasons of geometry and physics. while, the algebras overlap significantly, so that they offer a number of diverse mathematical representations for person geometrical or actual ideas.

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C) =(B X A) . C + A . 65) B X (A A C) = (B X A) A C + A A (B X C). 66) leads to the expansion formula for vectors ak, B X (al A ... A a,) =(B X ad A a2 A ... Aa, + + al A(B X a2)Aa3 A ... Aa, + ... +al A ... ak) Aal A ... ilk ... Aa,. 62b) implies that the space of bivectors is closed under the commutator product. It follows that under the commutator product the bivectors make up a lie algebra, which is, as is well-known and easy to show with geometric algebra, the lie algebra of rotations in Euclidean space.

Ilk ... A an) . (an A .... A ad (al A .. Aan ) . (an A ... Aad ~ =;=1 L (-I) k+i (al A ... ilk .. Aan)· (an A ... il; ... A ada; (aIA ... Aan)·(anA ... 8) 29 Geometric Algebra Whlle we are on the subject of frames, we may as well show how to construct a basis and its reciprocal for the complete Geometric Algebra ~(An) from a basis for ~I (An). But the mere existence of a ~sis for ~(An) is all that we shall ever appeal to in the rest of the book, so the details of the following construction may be passed over without loss.

1) Hence it is appropriate to call dn the vector space of A. Conversely, every n-dimensional vector space dn uniquely determines two unit n-blades ±I. The n-blade formed by outer multiplication of any set of n vectors alo ... 2) and the scalar A vanishes if and only if the ak are linearly dependent. Any nonzero scalar multiple of I is called a pseudoscalar of JiI" . Assignment of an orientation to JiI" is equivalent to associating a unique unit n-blade I with JiI,,; in that case, we say that I is the tangent or the direction of JiI".