Math 215 - Linear Algebra

We will follow a draft of a Linear Algebra book that I am writing: Current version.

For an excellent overview of the core ideas of Linear Algebra (with high-quality visualizations), I strongly recommend the video series Essence of Linear Algebra by 3Blue1Brown.

In addition, I encourage you to consult other books and online resources to help learn the material, as these sources provide different perspectives on the subject. For linear algebra books, I recommend the following:

• Linear Algebra by Jim Hefferon. Available online.
• A Course in Linear Algebra by David B. Damiano and John B. Little.

We will also spend a significant portion of time learning how to write mathematical proofs. For additional references on mathematical writing and notation, I recommend the following:

• Book of Proof by Richard Hammack. Available online.
• Mathematical Reasoning: Writing and Proof by Ted Sundstrom. Available online.
• How to Prove It by Daniel Velleman.

For general advice on making the transition from a computational perspective of mathematics to a more conceptual understanding (including how to think logically and how to write mathematics), consider reading the following:

• How to Think Like a Mathematician by Kevin Houston.
• How to Study as a Mathematics Major by Lara Alcock.

• Develop the ability to apply conceptual ideas in order to solve problems, and appreciate the importance of how the relevant theoretical ideas can simplify (or allow one to completely avoid) elaborate computations.
• Learn how to interpret and use mathematical language properly, with an emphasis on quantifiers (there exists, for all), connectives (and, or, not, implies), and the role of definitions.
• Learn how to interpret and use common mathematical notation involving sets and functions.
• Learn how to read mathematical exposition with the goal of deep understanding, rather than with the aim of solving a narrow class of rigid problems.
• Understand the logic of mathematical arguments, different ways to prove a statement, and how to construct your own proofs.
• Learn some of the fundamental concepts of linear algebra: spans, linear transformations, matrices, solving linear systems, vector spaces, subspaces, dimension, linear independence, determinants, and eigenvalues/eigenvectors.
• Develop an appreciation for the power and elegance of mathematical structures (such as vector spaces).

There will be two different types of homework assignments:

• Problem Sets will be due at 5:00pm on most Mondays and Fridays. Your solutions should be uploaded to Gradescope. These assignments contain a mixture of computational exercises, explanations, and short proofs. Your lowest two Problem Set scores will be dropped.
• Writing Assignments will be due at the beginning of class on most Wednesdays. You should turn in a paper version of your solutions in class. The problems on these assignments are more conceptual or theoretical, and will require significant explanation. Your solutions should consist of careful arguments written in complete sentences (augmented by mathematical symbolism where appropriate). A major goal of these problems is to teach the fundamentals of mathematical language and mathematical inferences, along with proper use of terminology and notation. As a result, they will be graded at a high standard involving much more than getting the "correct answer". Take the time to write and revise these as you would in a paper in other courses. Your lowest Writing Assignment score will be dropped.

There will be three in-class exams and a scheduled three-hour final exam.

In-class exams dates: September 21, October 26, and November 21.

Final exam date:
Section 1 (MWF 9:00 - 9:50): Tuesday, December 13 at 9:00.
Section 2 (MWF 1:00 - 1:50): Wednesday, December 14 at 2:00pm.

If you want to learn how to present your work professionally, as well as keep digital records, I recommend learning how to typeset your solutions. LaTeX is a wonderful free typesetting system which produces high-quality documents at the cost of only a small amount of additional effort (beyond the nontrivial start-up cost of learning the fundamentals). If you plan to do any kind of mathematical or scientific writing in the future, you will likely use LaTeX, so it is worth your time to familiarize yourself with it. See Jim Hefferon's LaTeX for Undergraduates and his LaTeX Cheat Sheet for the basics. Also, feel free to ask me questions about how to use LaTeX, and/or to send you the LaTeX file for homework assignments.

Consult the general Grinnell College policy on Academic Honesty and the associated booklet for general information.

Homework: If you enjoy working in groups, I strongly encourage you to work with others in the class to solve the homework problems. If you do collaborative work or receive help form somebody in the course, you must acknowledge this on the corresponding problem(s). Writing "I worked with Sam on this problem" or "Mary helped me with this problem" suffices. You may ask students outside the course for help, but you need to make sure they understand the academic honesty policies for the course and you need to cite their assistance as well. Failure to acknowledge such collaboration or assistance is a violation of academic honesty.

If you work with others, your homework must be written up independently in your own words. You cannot write a communal solution and all copy it down. You cannot read one person's solution and alter it slightly in notation/exposition. Discussing ideas and/or writing parts of computations together on whiteboards or scratch paper is perfectly fine, but you need to take those ideas and write the problem up on your own. Under no circumstances should you look at another student's completed written work.

I encourage you to look to other books or online sources for additional help in understanding concepts and ideas, but you must cite other books or online sources if they provide you with an idea that helps you solve a problem. However, you may not specifically look for solutions to homework problems, and you may not solicit help for homework problems from online forums.

Exams and Final: You may neither give nor receive help. Books, written notes, computers, phones, and calculators are not permitted at any time during a testing period.

I encourage students with documented disabilities to discuss appropriate accommodations with me. You will also need to have a conversation with, and provide documentation of your disability to, the Coordinator for Disability Resources, located on the ground level floor of Steiner Hall (641-269-3124).

I encourage students who plan to observe holy days that coincide with class meetings or assignment due dates to consult with me as soon as possible so that we may reach a mutual understanding of how you can meet the terms of your religious observance and also the requirements for this course.

• Actively read the course notes. Read them once before class in order to become familiar with the core concepts. Read them again, both slowly and deliberately, after class in order to build fluency. Work to understand both the computations and the theoretical discussions. Set aside time to simply think about the material and how it fits together, in addition to the time you give yourself to work on the homework. Spend your time trying to internalize rather than memorize.
• Compared to Calculus courses, a much smaller part of each class will be spent on computational examples and a much larger part of each class will be spent on the conceptual and theoretical aspects of the material. You will be expected to understand the derivations and proofs at a level far beyond the ability to parrot them back to me.
• When graded homework is returned, spend time reading the comments and reflecting on how you can improve your writing. Engaging with direct feedback on your work is one of the fastest ways to make progress. Also, read the posted solutions and compare them to your own. Examine and learn from how the solutions differ from yours in ideas, language, and organization.
• Much of your learning will happen outside of class. Although the amount of time necessary to understand the material varies, most students should anticipate spending at least 12 hours a week devoted to the course. In other words, you should schedule at least 9 hours outside of class for homework and independent reading/thinking. Learning math requires practice, patience, and endurance.
• I really enjoy interacting with students. Please come to my office hours when you want assistance! To use our time together most effectively, it helps if you have grappled with the ideas and you bring some of your scratch work and attempts. One of the most difficult parts of your mathematical education is learning how to transition from having no idea, to obtaining vague hunches, to seizing on key ideas, to writing correct proofs. If you bring your ideas and scratch work, we can focus on how to help you manage these transitions.