Algebra

In this chapter, a detailed introduction to the algebraic properties of geometric algebra is given. The aim is to not only give an axiomatic derivation but also to discuss the calculation rules that are needed to deal effectively with geometric-algebra equations. The chapter starts with an axiomatic discussion of geometric algebra in terms of the elements of a canonical vector space basis. Since, in this text, only geometric algebras over vector spaces are used and, for any vector space, a basis can be found, this approach does not constitute a loss of generality.

After the fundamental properties of basis blades are introduced in Sect. 3.1, these are extended to general blades in Sect. 3.2. The properties derived here are those that are most often used in later chapters. While blades represent linear subspaces and thus geometric entities, Sect. 3.3 discusses versors, which represent transformation operations, such as re ection and rotation. In this context, the Clifford group and its subgroups the pin and the spin group are discussed. A more general type of transformation of multivectors is presented in Sect. 3.4, where general linear functions are considered. These are directly related to matrix algebra, and certain properties of determinants are easily derived in this context. Section 3.5 introduces the concept of reciprocal bases, which play an important role in a number of applications, for example the representation of pinhole cameras in geometric algebra (see Sect. 7.1). They are also directly related to linear functions, since they can be used to generate basis transformation matrices.

After the discussion of various transformations of multivectors, a brief introduction to multivector differentiation in Sect. 3.6 completes the picture. In this section, differentiation and integration are also discussed with respect to the tensor representation of geometric-algebra operations. This representation, which is described in detail in Sect. 5.1, basically expresses geometric-algebra operations as bilinear functions, for which differentiation and integration are well dened.