This textbook for the sophomore level linear algebra course presents a unified treatment of linear algebra and geometric algebra, while covering most of the usual linear algebra topics.

Geometric algebra and its extension to geometric calculus simplify, unify, and generalize vast areas of mathematics that involve geometric ideas. Geometric algebra is an extension of linear algebra. The treatment of many linear algebra topics is enhanced by geometric algebra, for example, determinants and orthogonal transformations. And geometric algebra does much more, as it incorporates the complex, quaternion, and exterior algebras, among others.

Geometric algebra and calculus provide a unified mathematical language for many areas of physics (classical and quantum mechanics, electrodynamics, relativity), computer science (graphics, robotics, computer vision), engineering, and other fields.

The book can be used for self study by those comfortable with the theorem/proof style of a mathematics text.

The table of contents, preface, and index are available here. (Modified 28 December 2010.)

This book can be used as a linear algebra text, without geometric algebra, as outlined in the preface. Thus an instructor can include geometric algebra as time permits, or teach a two track course, with some students studying geometric algebra and some not.

I commend Alan Macdonald for his excellent book! His exposition is clean and spare. He has done a fine job of engineering a gradual transition from standard views of linear algebra to the perspective of geometric algebra. The book is sufficiently conventional to be adopted as a textbook by an adventurous teacher without getting flack from colleagues. Yet it leads to gems of geometric algebra that are likely to delight thoughtful students and surprise even the most experienced instructors.
-- David Hestenes, Distinguished Research Professor, Arizona State University

The computer exercises in the book use Python, a multiplatform language freely available on the web, and its NumPy and SymPy libraries. When the book was published, the libraries were available only for Python 2.5 and 2.6. Now they are also available for 2.7, 3.1, and 3.2. I cannot yet recommend these new versions. I use Geany, a free multiplatform environment, to write my programs.

There is a geometric algebra and calculus module GA in SymPy. For many of the exercises in the book you will also need the file laga.py.

Added March 27, 2012. There is a bug in the SymPy method eigenvects used to compute eigenvectors. Unfortunately, it affects Problem 9.4.10c. I have revised laga.py to avoid the problem. The function printeigen, described in Appendix B, should now compute eigenvalues and eigenvectors correctly.

Errata can be found here. Updated March 27, 2012.

Please send corrections, typos, or any other comments about the book to me. I will post them here as appropriate.

Alan Macdonald is Professor Emeritus of Mathematics at Luther College in Decorah, Iowa. He received a Ph.D. in mathematics from The University of Michigan in 1970. Other than geometric algebra, his research interests include the foundations of physics and generalized Riemann integration. His web page is here.

A Survey of Geometric Algebra and Geometric Calculus
An introduction to geometric algebra and geometric calculus for those with a knowledge of undergraduate mathematics. The section Further Study lists many papers available on the web.

An Elementary Construction of the Geometric Algebra
Adv. Appl. Cliff. Alg. 12, 1-6 (2002). (Improved.)
Abstract: We give a simple, elementary, direct, and motivated construction of the geometric algebra over Rⁿ.