Understanding Linear Algebra

An open-access linear algebra textbook

January 2024 updates: Thanks to our friends at 619 Wreath, I'm excited to announce that there are now low-cost print versions of the textbook and activity workbook.

The newest version of the book, created in December 2023, fixed some small errors and layout issues. Nothing substantial has changed since August 2022.

Book cover

Understanding Linear Algebra is a freely available linear algebra textbook suitable for use in a first undergraduate linear algebra course. The text aims to support readers as they develop their ability to reason mathematically, their computational fluency, and their understanding of the role that linear algebra plays in shaping our society. It is also designed to support an active learning classroom environment.

Understanding Linear Algebra is written in PreTeXt and has been endorsed by the Open Textbook Initiative at the American Institute of Mathematics.


Activities workbook


  • Understanding Linear Algebra can support a two-course introductory sequence to linear algebra. While it is not meant to serve as an introduction to proofs, activities and exercises support students as they develop their ability to reason mathematically and communicate their understanding. At my university, our first two linear algebra courses do not require calculus as a prerequisite, and we find that many students thrive in it.

  • Understanding Linear Algebra is intended to support active learning. Each section begins with a preview activity for students to complete before class and subsequently contains several activities for students to complete in class, perhaps working collaboratively in small groups. A solutions manual for the activities is available upon request.

  • Each section contains numerous exercises from routine review to more in-depth, conceptual questions. Solutions are also available upon request.

  • The text contains numerous embedded Sage cells to facilitate linear algebraic computations. Relevant Sage commands are introduced as needed and summarized in an appendix.

  • Many applications are introduced and developed in depth to build an appreciation of how linear algebra plays a vital role in our society. These applications include topics such as computer animation, image compression, Google PageRank, least squares, principal component analysis, and singular value decompositions.

  • Interactive diagrams embedded in the text help students develop an intuitive understanding of key linear algebra concepts.

For instructors

First, I hope Understanding Linear Algebra is a valuable resource for you and your students in whatever way you find most appropriate to use it. I would very much like to hear from you at the email address given below. Since the book is freely available, it's difficult to know how many people are using it and what value they are finding in it.

The book is in use at over 30 institutions, from R1s to community colleges and high schools. There is a community of instructors using ULA, and we share resources such as online homework sets, syllabi, learning objectives, and more. There are also solution manuals for both the activities and exercises available upon request. Please reach out to me to connect with this community.


David Austin is a professor of mathematics at Grand Valley State University. Questions, feedback, suggestions, and anything else are always welcome:

David Austin, austind at gvsu.edu


This work is licensed under a Creative Commons Attribution 4.0 International License.

© David Austin 2023