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
    3. The Conformal Model
  4. Appendix
    1. Prerequisites

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.

Conformal Model. As of the fourth printing there is a new Chapter 10 on the magical conformal model. The latest version of the chapter is also available here. Last update: 11/6/18.


I have created a six video YouTube playlist Geometric Algebra, about 72 minutes in all, taken from the book. Some knowledge of linear algebra is a prerequisite for the videos, but no knowledge of geometric algebra is assumed. The book assumes no previous knowledge of linear algebra.

The book promises suggestions for instructors (one page).

Current Printing

This is the printing of August 2019. It has several minor improvements and corrects all errors known to me.

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 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, founder and chief theoretician of modern geometric algebra and calculus.

Linear and Geometric Algebra is available at Amazon. See the strong reviews there.

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. It is available here. GAlgebra is cross-platform (Linux, PC, Mac), with all components freely available on the web. The software is no longer available at this page.

GAlgebra is written in Python 2. One of its components (Sympy), will stop supporting Python 2, probably by the end of 2019. Alan Bromborsky is converting GAlgebra to Python 3.

Notebooks. GAlgebra works in Jupyter (formerly IPython) and Jupyter Lab notebooks. Output is typeset in beautiful LaTeX. A Jupyter notebook to accompany the new Chapter 10 is available here. (If the file opens in your browser, save it under the name cm3.ipynb".)

GAlgebraPrimer.pdf contains instructions for installing and using GAlgebra. The primer also downloads with GAlgebra. Updated 1/24/18.

Other free geometric algebra software. I have not used them, so I am announcing, not recommending, the software. I am interested in feedback about them.
    ●  Stephen Abbott has written a symbolic geometric algebra program GAwxM, based on Maxima. It runs on both Windows and Ubuntu Linux. GitHub.
    ●  Kurt Nalty has written symbolic geometric algebra software based on GiNaC, a C++ library. It has been tested on Linux. Its chief virtue is that it is "much faster" than GAlgebra. Its output is not typeset by LaTeX. It does not run within Jupyter. GitHub.
    ●  Paul Leopardi has written GluCat, a library of C++ template classes for numerical Clifford algebra calculations. They are used as numeric classes with other template libraries. PyClical makes it easy to use Python to perform computations in Clifford algebras. GitHub.

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.
Adv. Appl. Cliff. Alg. (2016). (Improved.) From the proceedings of the conference Applied Geometric Algebra in Computer Science and Engineering, Barcelona, Spain, July 29-31, 2015.


The latest errata file is dated 8/8/19.

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

An improved version of Section 5.2 is available. This version is in books bought after 1/22/19.