Linear Algebra

Linear Algebra is a free, open source textbook aimed at a first or second year one-semester introductory linear algebra course. The book is designed for a non-proofs-focused course, but could be used in a course that focuses on proofs.

Originally designed for MAT223 at the University of Toronto, student-facing (in-class) questions are a main feature of Linear Algebra, and the textbook doubles as a workbook with space for students to write their solutions to questions. It is designed to be used in classes taught using Inquiry Based Learning or other Active Learning methods.

Get the book

The entire book is available for free as a PDF.

LaTeX source

The source files for this book are available on GitHub. If you find a mistake or have a suggestion for the book, please file an issue on GitHub.

Instructor and student resources

There are youtube video playlists, lesson plans, and an instructor guide available. For access to the instructor guide, please contact the author.

• MAT223 Video Playlists. Video play lists to support a 12-week course taught with Linear Algebra.

If you are using the book in a class you are teaching, instructor resources are available by request. Just contact the author. You can also request WeBWorK homework sets if you have access to a WeBWorK server.

About this Book

For the student

This book is your introductory guide to linear algebra. It is divided into modules, and each module is further divided into exposition, practice problems, and core exercises.

To optimally learn from this text, you should:

• Start each module by reading through the exposition to get familiar with the main ideas and linear algebra terminology.
• Work through the core exercises to develop an understanding and intuition behind the main ideas and their subtleties.
• Re-read the exposition and identify which concepts each core exercise connects with.
• Work through the practice problems. These will serve as a check on whether you've understood the main ideas well enough to apply them.

The core exercises. Most (but not all) core exercises will be worked through during lecture time, and there is space for you to work provided after each of the core exercises. The point of the core exercises is to develop the main ideas of linear algebra by exploring examples.

So many definitions. A big part of linear algebra is learning precise and technical language. There are many terms and definitions you need to learn, and by far the best way to successfully learn these terms is to understand where they come from, why they're needed, and practice using them.

Contributing to the book. Did you find an error? Do you have a better way to explain a linear algebra concept? Please, contribute to this book! This book is open-source, and we welcome contributions.

For the instructor

This book was designed for a one-semester introductory linear algebra course course with a focus on geometry (MAT223 at the University of Toronto). It has not been designed for an “intro to proofs”-style course, but could be adapted for one.

Unlike a traditional textbook that is grouped into chapters and sections by subject, this book is grouped into modules. Each module contains exposition about a subject, practice problems (for students to work on by themselves), and core exercises (for students to work on with your guidance). Modules group related concepts, but the modules have been designed to facilitate learning linear algebra rather than to serve as a reference. For example, information about change-of-basis is spread across several non-consecutive modules; each time change-of-basis is readdressed, more detail is added.

Using the book. This book has been designed for use in large active-learning classrooms driven by a think, pair-share/small-group-discussion format. Specifically, the core exercises (these are the problems which aren’t labeled “Practice Problems” and for which space is provided to write answers) are designed for use during class time.

A typical class day looks like:

1. Student pre-reading. Before class, students will read through the relevant module.

2. Introduction by instructor. This may involve giving a definition, a broader context for the day’s topics, or answering questions.

3. Students work on problems. Students work individually or in pairs/small groups on the prescribed core exercise. During this time the instructor moves around the room addressing questions that students may have and giving one-on-one coaching.

4. Instructor intervention. When most students have successfully solved the problem, the instructor refocuses the class by providing an explanation or soliciting explanations from students. This is also time for the instructor to ensure that everyone has understood the main point of the exercise (since it is sometimes easy to miss the point!).

   If students are having trouble, the instructor can give hints and additional guidance to ensure students’ struggle is productive.

5. Repeat step 3.

Using this format, students are thinking (and happily so) most of the class. Further, after struggling with a question, students are especially primed to hear the insights of the instructor.

Conceptual lean. The core exercises are geared towards concepts instead of computation, though some core exercises focus on simple computation. They also have a geometric lean. Vectors are initially introduced with familiar coordinate notation, but eventually, coordinates are understood to be representations of vectors rather than “true” geometric vectors, and objects like the determinant are defined via oriented volumes rather than formulas involving matrix entries.

Specifically lacking are exercises focusing on the mechanical skills of row reduction and computing matrix inverses. Students must practice these skills, but they require little instructor intervention and so can be learned outside of lecture (which is why core exercises don’t focus on these skills).

How to prepare. Running an active-learning classroom is less scripted than lecturing. The largest challenges are: (i) understanding where students are at, (ii) figuring out what to do given the current understanding of the students, and (iii) timing.

To prepare for a class day, you should:

1. Strategize about learning objectives. Figure out what the point of the day’s lesson is and brain storm some examples that would illustrate that point.

2. Work through the core exercises.

3. Reflect. Reflect on how each core exercise addresses the day’s goals. Compare with the examples you brainstormed and prepare follow-up questions that you can use in class to test for understanding.

4. Schedule. Write timestamps next to each core exercise indicating at what minute you hope to start each exercise. Give more time for the exercises that you judge as foundational, and be prepared to triage. It’s appropriate to leave exercises or parts of exercises for homework, but change the order of exercises at your peril—they really do build on each other.

A typical 50 minute class is enough to get through 2–3 core exercises (depending on the difficulty), and class observations show that class time is split 50/50 between students working and instructor explanations.

Please contact the author with feedback and suggestions, or if you are decide to use the book in a course you are teaching. You can also easily submit feedback about an error or typo by creating a GitHub issue.

About the author

License

Linear Algebra by Jason Siefken is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License. You are free to download, use, print, and even sell this work as you wish to. You can also modify the text as much as you like (create a custom edition for your students, for example), as long as you attribute the parts of the text you use to the author.

If you are interested in using parts of the book combined with another text with a similar but different license (GFDL, for example), please reach out to get permission to modify the license.