Linear and Geometric Algebra

Linear and Geometric Algebra

Alan Macdonald

Second printing, corrected and slightly revised.

This textbook for the second year undergraduate 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.

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.

The second printing has no major changes. It corrects all errors known to me in the first printing. I have added answers to selected problems having numerical answers. There are many improvements in wording. The numbering of equations, theorems, etc. is unchanged from the first printing.

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

Linear and Geometric Algebra is available at Amazon

Price: $30

I have had reports of poor quality shading on theorems, etc. Please let me know if you have seen this.

A sequel, Vector and Geometric Calculus, is available.

The computer exercises in the book use Python, a cross-platform language freely available on the web, its library sympy, and a module ga. The ga module is available here. (Updated September 2, 2013.) It works with Python 2.7. (Python 2.5 and 2.6 might also work.) The module is cross-platform: Linux, PC, and Mac. Instructions for ga are in LAGA Appendix B.pdf. (Updated August 30, 2013.) I use Geany, a free cross-platform IDE, to write my programs.

In addition to the above, the first printing of this book also used the file The ga module no longer requires, so it is no longer available. If you are using, it should continue to work.

Errata can be found here. Updated August 26, 2014.

"Suggestions for Instructors" can be found here.

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.

Papers mentioned in the book

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.


Alan Macdonald
Professor Emeritus of Mathematics
Luther College
Decorah, IA 52101
macdonal at luther dot edu
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