Linear and Geometric Algebra

Vector and Geometric Calculus


  1. Linear Algebra
    1. Vectors
    2. Vector Spaces
    3. Matrices
    4. Inner Product Spaces
  2. Geometric Algebra
    1. 𝔾3
    2. 𝔾n
    3. Project, Rotate, Reflect
  3. Linear Transformations
    1. Linear Transformations
    2. Representations
  4. Appendices
    1. Prerequisites
    2. Software

Read a detailed table of contents, the preface, and the index.

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.

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 have created a six video YouTube playlist Geometric Algebra, about 72 minutes in all, taken from the book. Unlike the book, some knowledge of linear algebra is a prerequisite for the videos. The geometric algebra starts at the beginning.

The book promises suggestions for instructors (One page.).

Third Printing

The third printing has no major changes. It corrects all errors known to me in the first printing. There are many improvements in wording. The numbering of equations, theorems, etc. is unchanged from the first printing.

From a review of Linear and Geometric Algebra:

"I commend Alan Macdonald for his excellent book! His exposition is clean and spare. He has done a fine job of engineering a gradual tran sition 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

A sequel, Vector and Geometric Calculus, is available.

Computer Exercises

GAlgebra. The computer exercises in the book use GAlgebra, a Python module written by Alan Bromborsky. GAlgebra is cross-platform (Linux, PC, Mac), with all components freely available on the web. The software is no longer available at this page.

Notebook. GAlgebra is available in a jupyter (formerly IPython) notebook. Output is typeset in beautiful LaTeX.

GAlgebraPrimer.pdf contains instructions for installing and using GAlgebra. The file also downloads with the module. It will be updated as necessary.

Appendix B of the book also contains instructions for installing and using the module. But the information there will become dated as GAlgebra evolves. So I recommend that you use GAlgebraPrimer.

Papers mentioned in the book


The latest errata file is dated November 17, 2015.

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