An Inquiry-Based Introduction to Linear Algebra and Applications

A Free and Open-Source Linear Algebra Text.

Mathematics is for everyone — whether as a gateway to other fields or as background for higher level mathematics. With linear algebra gaining importance in many applications, we feel that access to or the cost of a textbook should not stand in the way of a successful experience in learning linear algebra. Therefore, we made our textbook available to everyone for free download for their own non-commercial use. We especially encourage its use in linear algebra classrooms for instructors who are looking for an inquiry-based textbook or a supplemental resource to accompany their course.If an instructor would like to make changes to any of the files to better suit your students' needs, we offer source files for the text by making a request to the authors.

This work is licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. The graphic

Motivation.

The material in this text was developed with a strong focus on linear algebra in \(\R^n\) for the first two-thirds of the course, along with an emphasis on applications. We believe that the main ideas of linear algebra are more accessible to students if they are introduced in the more familiar context of vectors in \(\R^n\) rather than in abstract vector spaces. For this reason, vector spaces do no arise until late in the text.

Goals.

An Inquiry-Based Introduction to Linear Algebra and Applications provides a novel inquiry-based learning approach to linear algebra, as well as incorporating aspects of an inverted classroom. The impetus for this book lies in our approach to teaching linear algebra. We place an emphasis on active learning and on developing students' intuition through their investigation of examples. For us, active learning involves students — they are DOING something instead of being passive learners. What students are doing when they are actively learning might include discovering, processing, discussing, applying information, writing intensive assignments, engaging in common intellectual in-class experiences or collaborative assignments and projects. Although it is difficult to capture the essence of active learning in a textbook, this book is our attempt to do just that.

Additionally, linear algebra is one of the most applicable branches of mathematics to other disciplines. In fact, the Linear Algebra Curriculum Study Group (LACSG) (formed in 1990 to “initiate substantial and sustained national interest in improving the undergraduate linear algebra curriculum” and funded by the National Science Foundation) made recommendations for the linear algebra curriculum, one of which was making linear algebra a matrix-oriented course that includes applications to give an indication of the “pervasive use” of linear algebra in other disciplines. ² Also, the most recent Committee on the Undergraduate Program in Mathematics (CUPM) report ³ to the Mathematical Association of America recommends “every Linear Algebra course should incorporate interesting applications, both to highlight the broad usefulness of linear algebra and to help students see the role of the theory in the subject as it is applied.” An important component of this text is its inclusion of significant “real-life” applications of linear algebra to help students see that linear algebra is widely applicable in many disciplines. Our goals for these materials are several.

• To carefully introduce the ideas behind the definitions and theorems to help students develop intuition and understand the logic behind them.

• To help students understand that mathematics is not done as it is often presented. We expect students to experiment through examples, make conjectures, and then refine their conjectures. We believe it is important for students to learn that definitions and theorems don't pop up completely formed in the minds of most mathematicians, but are the result of much thought and work.

• To help students develop their communication skills in mathematics. We expect our students to read and complete activities before class and come prepared with questions. While in class, students work to discover many concepts on their own through guided activities. Of course, students also individually write solutions to exercises on which they receive significant feedback. Communication skills are essential in any discipline and we place a heavy focus on their development.

• To have students actively involved in each of these items through in-class and out-of-class activities, in-class presentations (this is of course up to the instructor), and problem sets.

• To expose students to significant real-life applications of linear algebra.

Layout.

This text is formatted into sections, each of which contains preview activities, in-class activities, worked examples, and exercises. Most sections conclude with an application project — an application of the material in the section. The various types of activities serve different purposes.

• Preview activities are designed for students to complete before class to motivate the upcoming topic and prepare them with the background and information they need for the class activities and discussion.

• We generally use the regular activities to engage students during class in critical thinking experiences. These activities are used to provide motivation for the material,opportunities for students to develop course material on their own, or examples to help reinforce the meanings of definitions or theorems. The ultimate goal is to help students develop their intuition for and understanding of linear algebra concepts.

• Worked examples are included in each section. Part of the philosophy of this text is that students will develop the tools and understanding they need to work homework assignments through the preview, in-class activities, and class discussions. But some students express a desire for fully-worked examples in the text to reference during homework. In order to preserve the flow of material within a section, we include worked examples at the end of each section.

• The suggestion by the Linear Algebra Curriculum Study Group (LACSG) that, “Mathematics departments should seriously consider making their first course in linear algebra a matrix-oriented course.” is followed in this text. They suggest that this approach “implies less emphasis on abstraction and more emphasis on problem solving and motivating applications.” and that “Some applications of linear algebra should be included to give an indication of the pervasive use of linear algebra in many client disciplines. Such applications necessarily will be limited by the need to minimize technical jargon and information from outside the course. Students should see the course as one of the most potentially useful mathematics courses they will take as an undergraduate.” The focus of this text for the first two semesters of linear algebra (Section 1–27) is on \(\R^n\text{,}\) and applications of linear algebra in real spaces. At our institution, these sections comprise a two-semester entry-level sequence into our major, suitable for freshmen without a calculus prerequisite. Abstract vector spaces do not appear until Section 28, which we consider to be the beginning of a third semester of linear algebra, a junior or senior level mathematics course. Also, the 2015 Committee on the Undergraduate Program in Mathematics (CUPM) report to the Mathematical Association of America recommends that “Every Linear Algebra course should incorporate interesting applications,both to highlight the broad usefulness of linear algebra and to help students see the role of the theory in the subject as it is applied. Attractive applications may also entice students majoring in other disciplines to choose a minor or additional major in mathematics.” All but two sections in this text include a substantial application. Each section begins with a short description of an application that uses the material from the section, then concludes with a project that develops the application in more detail. (The two sections that do not include projects are sections that are essentially long proofs — one section that contains formal proofs of the equivalences of the different parts of the Invertible Matrix Theorem, and the other contains algebraic proofs of the properties of the determinant.) The projects are independent of the material in the text — the text can be used without the applications. The applications are written in such a way that they could also be used with other textbooks. The projects are written following an inquiry-based style similar to the text, with important parts of the applications developed through activities. So the projects can be assigned outside of class as independent work for students. Several of the projects are accompanied by GeoGebra applets or Sage worksheets which are designed to help the students better understand the applications.

• Each investigation contains a collection of exercises. The exercises occur at a variety of levels of difficulty and most force students to extend their knowledge in different ways. While there are some standard, classic problems that are included in the exercises,many problems are open ended and expect a student to develop and then verify conjectures.

Acknowledgements.

The many beautiful images that are in this text were created by our colleague David Austin, to whom we owe a debt of gratitude. The original LaTeX version of the book was partially supported by Grand Valley State University through an internal grant and a sabbatical.

The HTML version of the text is possible because of the work of Rob Beezer and the PreTeXt team for the development of the PreTeXt platform. David Farmer at the American Institute of Mathematics provided an initial PreTeXt conversion. From this version, Ian Curtis, Editorial Assistant for the GVSU Libraries, as part of the Accelerating Open Educational Resources Initiative at Grand Valley State University⁴, invested a significant amount of time and effort to create the PreTeXt version that you see. This work was conducted with support from the University Libraries and the GVSU President's Innovation Fund.

The code used to generate this book can be found in the GVSU OER GitHub repository⁵. This includes the PreTeXt XML files, figures, customization files (XSL and CSS), and the output (HTML and PDF). Contributors are encouraged to submit an issue or a pull request with changes, especially regarding accessibility (as mentioned below in Accessibility).

Accessibility.

In creating this PreTeXt version we (the authors and the Grand Valley State University Libraries) strive to ensure our tools, devices, services, and environments are available to and usable by as many people as possible.

The web version of An Inquiry-Based Introduction to Linear Algebra and Applications incorporates the following features to support accessibility:

• All content can be navigated by use of a keyboard

• Links, headings, and tables have been designed to work with screen readers

• Mathematics in PreTeXt are rendered with MathJax so they can be understood by people using screen readers and/or other assistive devices.

We have identified some accessibility issues which were beyond the author's and GVSU Libraries' capacity to address in the course of this adaptation. These present an opportunity for future adaptation or revision by educators with expertise in accessibility issues in mathematics documents. Known accessibility issues include:

• A small number of exercises and activities in this text require students to visually interpret diagrams or figures. These activities cannot be made accessible through alt text alone, and may need to be redesigned or replaced for full accessibility.

• Some visual representations of concepts (see, for example, Section 17) are not accessible to screen readers. However, they are fully described in text.

• The matrix table represented in Section 29 is too complex to encode with the current functionality of PreTeXt and MathJax. It is presented as an image and is not accessible to screen readers. Full accessibility likely requires redesigning the project or waiting for expanded features in PreTeXt and MathJax.

We are always looking for how we can make our resources more accessible. If you are having problems accessing this resource, please contact us by email⁶ to let us know.

To the Student.

The inquiry-based format of this book means that you can be in charge of your own learning. The guidance of your instructor and support of your fellow classmates can help you develop your understanding of the topic. Your learning will benefit best if you engage in the material by completing all preview and in-class activities in order to fully develop your own intuition and understanding of the material, which can be achieved only by reflecting on the ideas and concepts and communicating your reflections with others. Don't be afraid to ask questions or to be wrong at times, for this is how people learn. Good luck! We will be happy to hear your comments about the book.