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Linear Algebra

Jason Siefken

Preface 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.
The exposition is easy to find—it’s the text that starts each module and explains the big ideas of linear algebra. The practice problems immediately follow the exposition and are there so you can practice with concepts you’ve learned. Following the practice problems are the core exercises. The core exercises build up, through examples, the concepts discussed in the exposition.
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. When working on core exercises, think “it’s the journey that matters not the destination”. The answers are not the point! If you’re struggling, keep with it. The concepts you struggle with you remember well, and if you look up the answer, you’re likely to forget just a few minutes later.
So many definitions. A big part of linear algebra is learning precise and technical language
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Beyond three dimensions, things get very confusing very quickly. Having precise definitions allows us to make arguments that rely on logic instead of intuition; and logic works in all dimensions.
. 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. That is, don’t try to memorize definitions word for word. Instead memorize the idea and reconstruct the definition; go through the core exercises and identify which definitions appear where; and explain linear algebra to others using these technical terms.
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 and improvements. To contribute to/fix part of this book, make a Pull Request or open an Issue at https://github.com/siefkenj/IBLLinearAlgebra. If you contribute, you’ll get your name added to the contributor list.

For the instructor.

This book is 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.

License.

Unless otherwise mentioned, pages of this document are licensed under the Creative Commons By-Attribution Share-Alike License. That means, you are free to use, copy, and modify this document provided that you provide attribution to the previous copyright holders and you release your derivative work under the same license. Full text of the license is at http://creativecommons.org/licenses/by-sa/4.0/
If you modify this document, you may add your name to the copyright list. Also, if you think your contributions would be helpful to others, consider making a pull request, or opening an issue at https://github.com/siefkenj/IBLLinearAlgebra
Incorporated content. Content from other sources is reproduced here with permission and retains the Author’s copyright. Please see the footnote of each page to verify the copyright.
Included in this text are tasks created by the Inquiry-Oriented Linear Algebra (IOLA) project. Details about these tasks can be found on their website http://iola.math.vt.edu/. Also included are some practice problems from Beezer’s A First Course in Linear Algebra (marked with the symbol Beezer’s A First Course in Linear Algebra next to the problem), and from Hefferon’s Linear Algebra (marked with the symbol Hefferon’s Linear Algebra next to the problem).
Contributing. You can report errors in the book or contribute to the book by filing an Issue or a Pull Request on the book’s GitHub page: https://github.com/siefkenj/IBLLinearAlgebra/
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