Welcome to Linear Algebra! Don’t worry so much about unicorns right now. Let us first focus on the Linear Algebra part. Unicorns will, of course,
appear later on naturally.

To All Readers

This text was originally written for the Linear Algebra course offered at Butler University, but we have worked to make it appropriate for use
elsewhere or for self-study. The order of the topics differs from many other Linear Algebra texts, but this was what we found worked best to help our students gain a deep understanding of the topic. We encourage you to follow the sections in order. We start with a quick, preliminary discussion of sets and functions, since these are central to the course. Then Chapters 1 and 2 focus on vector spaces. In Chapter 3, we study linear transformations, beginning with the definition and building to the connection with matrices. Finally, Chapters 4 and 5 are devoted to matrix theory.

If you would prefer to start the course with a discussion of matrices, we suggest the first part of Section 3.4, followed by Section 4.1 and all but
the last subsection of Section 4.2. This covers the definition of matrices and their connections to solving systems of equations. One could then
pick up at the beginning of the text and use the techniques just learned whenever solving a system of equations.

Some of the sections can be viewed as optional. In particular, Sections 1.5, 2.6, 3.6, 4.7, and 4.8 can be omitted without causing problems when covering future sections. After completing Sections 5.1, 5.2, and 5.3, each of Sections 5.4, 5.5, and 5.6 could serve independently as a capstone
section with some additional applications in Section 5.7.

The text is written to be read. The tone is conversational (and sometimes a little silly); there is a unicorn theme, so don’t be surprised to see pictures
of them throughout. The mathematics is written thoroughly, with most results proven in either the section or the Appendix. There are many
examples and “explorations” to engage readers as they encounter the material, written specifically with the idea that the reader work through
the explorations as they read. We leave it to the instructor’s discretion whether to provide solutions or work some of these during class meetings.
We’ve also attempted to make it lighthearted and fun in places because there’s no reason it shouldn’t be. We hope you enjoy reading it, but mostly,
we hope you learn from it.

A Note about Unicorns

While you may have heard other myths about how to summon a unicorn, we’re happy to reveal what happened to work for us. To summon a unicorn, you should include one in an example while writing a mathematics textbook. Apparently, unicorns love the study of mathematics so much, they reveal themselves to any textbook authors who appear to be receptive to their input. The ones who approached us initially were Ricky and Bubbles. We decided a nice way to include these muses was to allow them to narrate our side notes. They did get a bit carried away adding side notes of their own. Hopefully, you will not find them too distracting. It turns out it is difficult to unsummon a unicorn.

A Message for Students

Here are some goals to keep in mind when using this book:

• Learn the Content. Linear Algebra is inarguably one of the most applicable areas in modern mathematics. It is foundational to advanced mathematics courses, but is also widely used in statistics, computer science, physics, chemistry, and a host of other fields. Note here that we have used the word “applicable” rather than “applied.” You are not learning an applied version of Linear Algebra. The goal of this text is to give you a strong foundation in this topic so that you can recognize the applications in your own field as you encounter them. Thus, we begin with the conceptual definitions, but we build towards the application side as the book progresses.
• Improve Independent Learning Skills. Realistically, this is the overall goal of undergraduate education. Yes, you will learn specific
  content in courses you take, but there will always be things you still need to learn as you continue on beyond coursework. The main
  goal of college is to prepare you for life after college, and a large part of that is giving you the tools to tackle challenges and master
  new concepts outside of a classroom setting. This may be a goal in all your classes, but it is not necessarily always addressed directly.
  This text helps facilitate this goal by giving you ways to interact as you read. We encourage you to rework examples and also attempt
  all the explorations as you go through the reading. Uncovering areas in which your conceptual understanding can be refined is a
  valuable step in the learning process.
• Transition to Advanced Coursework. Linear Algebra is a prerequisite for many upper level mathematics courses and also courses in
  other departments. While some of this is due to its content, part of this prerequisite is also the experience of the course itself. For many
  students, this content will push you to think about more abstract mathematics topics than you may have in your previous experiences,
  and it will help you to see connections between different areas of mathematics, particularly algebra and geometry.

This content will be both challenging and time consuming, but it will also be rewarding. We hope you each find joy in the learning of this mathematics.

Best wishes,
Drs. Kaschner and Russell