ABSTRACT
This chapter studies fundamental aspects of Clifford algebra, surveys its modern formulation, looks at how imaginary numbers are dealt with in real multivector algebra, and introduces useful aspects of quaternion based geometry. The first aim is to highlight several fundamental aspects of Clifford's geometric algebra, geometric algebra (GA) or Clifford algebra. Questions which often arise when first dealing with Clifford's geometric algebra, and on how familiar linear algebra concepts are dealt with, are covered. Because Clifford's geometric algebra serves later in this book as the algebraic grammar for hypercomplex integral transforms, it is important to become familiar with this essential extension of Grassmann algebra. Therefore, an overview is provided of important geometric algebra examples of the Euclidean plane, Euclidean three-dimensional space, spacetime and of conformal geometric algebra. This is followed by a study on how real multivector elements of geometric algebra replace the many roles (e.g. in Fourier transform kernels) of imaginary numbers, providing a real geometric interpretation at the same time. Finally, the role quaternions play for rotations in three and four dimensions is shown in some detail. [179 words]
