Basic Info

Linear Algebra: Official Course Description

Linear algebra is a foundational area of mathematics, finding widespread application in statistics, machine learning, economics, physics, and across the sciences. The starting point for this course is to consider basic properties of matrices and techniques for solving systems of linear equations. Abstracting and formalizing the process of solving linear equations leads us to the notion of a vector space and related ideas, such as linear independence, dimension, and basis. The course then turns to further properties of matrices and vector spaces, including determinants, eigenvalues and eigenvectors, and linear transformations. As time permits, we will study various applications of linear algebra, such as image compression, dynamical systems (with a focus on ecological applications), Markov chains, and Google's PageRank algorithm. Students who successfully complete this course will gain a solid introduction to the calculational techniques and key constructions and ideas of linear algebra that will prepare them for further work in the sciences and mathematics. Additionally, students will gain experience working at a level of generality and abstraction above that encountered in a typical introductory calculus sequence. Evaluation will be based on weekly problem sets. Students who enroll in this course should have successfully completed a high-school-level algebra class and be motivated to explore a powerful and broadly-used branch of mathematics that for most has a very different feel than the functions-precalculus-calculus sequence. Calculus is not a prerequisite for this class. Level: Introductory/Intermediate. Prerequisites: Highschool level algebra class. Class limit: None. Lab fee: None. Meets the following degree requirements: QR.

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Who/when/where

• Instructor: Dave Feldman
• Pronouns: he/him/his
• Meeting Times: Tuesday, Wednesday, Friday, 2:30 - 4:00
• Location: Room 103, Center for Human Ecology
• Help Sessions: Help Session Schedule (updated frequently)
• Individual Meetings: By appointment
• Tutors: None

Axioms

In mathematics, axioms are propositions that are assumed to be true. The mathematician Federico Ardila-Mantilla has written four axioms that guide the work he does in education and outreach. Federico's axioms resonate strongly with me. They are:

1. Mathematical potential is distributed equally among different groups, irrespective of geographic, demographic, and economic boundaries.
2. Everyone can have joyful, meaningful, and empowering math experiences.
3. Math is a powerful, malleable tool that can be shaped and used differently by various communities to serve their needs.
4. Every student deserves to be treated with dignity and respect.

Community Agreement

Taking the above axioms as a starting point, let's think about what type of community we want to create this term. Here is a community agreement based on one written by Federico Ardila-Mantilla.

This course aims to offer a joyful, meaningful, and empowering experience to every participant; we will build that rich experience together by devoting our strongest available effort to the class. You will be challenged and supported. Please be prepared to take an active, critical, patient, creative, and generous role in your own learning and that of your classmates.

Acknowledgments

This course is closely based on the introductory linear algebra class (MTH 204) offered by Grand Valley State University. I am grateful to Professor Matt Boelkins, who shared his syllabus and assignments for MTH204. These teaching materials were developed collaboratively by the team of faculty at Grand Valley who rotate through teaching Linear Algebra.

Huge thanks to David Austin who has made his excellent textbook, Understanding Linear Algebra freely available online. David is a professor at Grand Valley State. (Btw, Matt has also written an open textbook series, Active Calculus, that looks great. I will consider using it next fall when I teach Calculus.)

Course Information and Advice

Course Structure

1. We will closely follow Understanding Linear Algebra by David Austin. My plan is to cover Chapter 1-4.
2. There will be three sorts of work you will do for this course:
   • Daily Prep Work. For almost all classes there will be a preview assignment designed to get you warmed up for that day's class. This will be evaluated entirely on effort and completeness. If you don't complete a few of the prep assignments, that's ok.
   • Weekly Homework Assignments, described below.
   • Labs. Roughly every other week there will be a "lab" that you'll start in class and then finish on your own. You'll do this work in pairs. The labs are applications or mini case studies.
3. All work can be resubmitted without penalty up to (roughly) a few weeks after I return it to you.
4. Your final evaluation will be roughly weighted as follows: Daily Prep (20%), Weekly Homework (50%), Labs (30%).

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What is Linear Algebra?

Wikipedia's entry (from 2013, I think) on linear algebra has a good summary:

Linear algebra is a branch of mathematics concerned with the study of vectors, with families of vectors called vector spaces or linear spaces, and with functions that input one vector and output another, according to certain rules. These functions are called linear maps (or linear transformations or linear operators) and are often represented by matrices. Linear algebra is central to modern mathematics and its applications. An elementary application of linear algebra is to the solution of a system of linear equations in several unknowns. More advanced applications are ubiquitous... . It has extensive applications in engineering, physics, natural sciences and the social sciences. [Linear algebra is also important to study because] nonlinear mathematical models can often be approximated by linear ones.

There are considerable differences among linear algebra courses and textbooks. Roughly speaking, there are three not mutually exclusive roles that linear algebra plays in the math curriculum:

• 1: Linear algebra teaches a set of techniques for solving problems. These classes often emphasize applications.
• 2A: Linear algebra serves as a bridge toward more formal or abstract mathematics.
• 2B: Sometimes this includes introducing students to methods of proof.
• 3A: Linear algebra can be an entry point for numerical mathematics. This includes using a computer to do mathematics and solve problems.
• 3B: This can also include analyzing the run-time and convergence properties of algorithms and writing original code.

This course is mostly of type 1 and 3A, with a little bit of 2A. I'm not certain of the exact balance; this will depend on the interests of the class. For more on the particular flavor this course, see the Our Goals section of our textbook.

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Stuff about Homework

1. There will be an assignment due almost every Friday. It is essential that you do these assignments, as this is how one learns math, and also this is most of what your evaluation will be based on.
2. There will be two parts to almost every homework assignment:
   • Problems to be submitted on WeBWorK
   • Problems to be submitted on "paper" (a scanned pdf) on google classroom
3. WeBWorK is an online homework system. There are three reasons why I use WeBWorK:
   • You get instant feedback while doing the work, so you can learn right away from your mistakes. You can submit solutions many times until you get everything correct.
   • Some problems are randomized so that you will all get slightly different versions of the questions. This means that collaborating with other students will be maximally effective, since you'll have to share solution methods and not just the final answer.
   • Since the problems are automatically marked, I can spend more time helping you and won't have to spend as much time grading.
4. I will make comments on your exercises and give one of two marks: Success! or Not Yet. A Not Yet mark means that you haven't quite demonstrated mastery/understanding of the learning goals for that exercise. You can resubmit any "Not Yet" assignments without penalty.
5. If you need extra time for one or two of the homework assignments, it's not a big deal. But be mindful to not fall farther behind every week.
6. I do not expect all of the homework assignments to be easy; I don't expect you to be able to sit down and do them easily the first time. Don't let yourself get frustrated. I strongly suggest working with others and seeking help if you need it.
7. You are strongly encouraged to work together on homework. You can also consult me, class tutors, other faculty, friends, and family. However, the homework you hand in should represent your own understanding.
8. As I plan on sending out homework assignments and other information via email/google classroom, it is important that you check your email/classroom regularly.
9. You will need access to a computer or tablet in order to read the textbook and do the homework.

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Some Warnings and Things to Think About

This is my seventh time teaching a version of this course, so I have a pretty good sense of how it will go. Some thoughts:

1. There are no official TAs for this course. So we'll have to work creatively to make sure everyone can get the help they need.
2. I am structuring things a little differently, modeled off of the GVSU course. These feels like a minor adjustment, and not a major shift in how I usually teach, but it's definitely a bit of an adjustment. Let's give it a shot and see how it goes.
3. Linear algebra is standard topic in the college mathematics curricula. It is usually taken by students in their sophomore year. Linear Algebra is required for math, physics, engineering, statistics, and economics majors. I suspect it's required for chemistry majors, but I'm not sure. In general, anyone who works in a mathematical field will need to know linear algebra.
4. My experience as a student was that linear algebra seemed very easy. So I didn't do much work in the class. But then it got harder quite quickly and I was lost and in a lot of trouble. As always, it's important to stay caught up, because you can get left behind pretty quickly in a course like this.
5. Also, when I took linear algebra I remember thinking that the material was kinda silly and that I would never use it. However, I ended up using linear algebra in almost every math and physics class I took, and I use it in my research all the time---much more than I use calculus. Some of the applications of linear algebra might not be apparent immediately, but please believe me when I say that it is a very useful topic to know.
6. Linear algebra potentially has some trap doors: things can suddenly shift from easy to very hard without you realizing it. Of course I'm going to try to not let this happen.

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Help Sessions

I will have a handful of help sessions every week. You are warmly invited and encouraged to attend these sessions. Help sessions are relaxed, informal, and hopefully fun. Things that happen at help sessions:

1. I am around to offer help on the homework.
2. Some students do most of the homework while at a help session. They work through problems alone or with others, and find it comforting to know that help is immediately at hand if needed.
3. Others do the problems at home and come to the help session with specific questions.
4. Some students work in groups at help sessions, others work more or less alone.
5. Help sessions are also a chance to ask general questions about the course. Conversations also sometimes wander into other areas: politics, the state of the world, what's for dinner, what classes are offered next term, and so on.
6. Help sessions are a great way to meet other students in the class.
7. Often there will be coffee/tea and/or snacks.

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Individual Meetings

I am happy to meet with students one-on-one. The best way to set up an appointment is to send an email. There are lots of reasons why you might want to meet with me:

1. You have some in-depth questions that there isn't time to explore in a help session.
2. You have a question that you think is too basic or you're uncomfortable asking in a help session. (You shouldn't be uncomfortable, since, as the saying goes, there are no dumb questions! But I understand that you might be uncomfortable nevertheless.)
3. You want to explore possibilities for energy projects on campus or in the community.
4. You want to discuss some challenges you're facing in the class.
5. You want to talk about other classes in energy or physics, or internships, senior projects, etc.

Please don't hesitate to reach out if there's anything you want to discuss. You should also feel free to reach out to the TAs.

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What your Evaluation is Based on

Your evaluation will be based on your performance on homework assignments (approx 90%) and your contributions in lab sections (approx 10%). There will be weekly homework assignments and, towards the end of the course, some mini-case studies and projects. I think there is much to be said against grades; I believe they often interfere with genuine, reflective learning. But I am happy to assign grades if you wish. I do not have any quota of A's, B's, etc.

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Diversity, Inclusion, and Belonging

It is my intent that students from all backgrounds and perspectives be well served by this course, that students' learning needs be addressed both in and out of class, and that the diversity that students bring to this class be viewed as a resource, strength, and benefit. I aim to present materials and activities that are respectful of diversity: gender, sexuality, disability, age, religion, socioeconomic status, ethnicity, race, and culture.

Learning about diverse perspectives and identities is an ongoing process. I am always looking to learn more about power and privilege and the harmful effects of racism, sexism, homophobia, classism, and other forms of discrimination and oppression. Your suggestions are encouraged and appreciated. Please let me know ways to improve the effectiveness of the course for you personally, or for other students or student groups. If something was said or done in class (by anyone, including me) that made you feel uncomfortable, please let me know. You can also reach out to Provost Ken Hill, or Associate Kourtney Collum or Jamie McKown.

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Statements about Academic Honesty and Hours of Academic Engagement

• I am required to remind you that: "By enrolling in an academic institution, a student is subscribing to common standards of academic honesty. Any cheating, plagiarism, falsifying or fabricating of data is a breach of such standards. A student must make it his or her responsibility to not use words or works of others without proper acknowledgment. Plagiarism is unacceptable and evidence of such activity is reported to the academic dean or his/her designee. Two violations of academic integrity are grounds for dismissal from the college. Students should request in-class discussions of such questions when complex issues of ethical scholarship arise."
• I am also required to say that: You should expect to spend 150 hours of academically engaged time on this course, or 15 hours per week. In addition to 4.5 hours per week in class or discussion section, in a typical week you'll spend 2 hours reading and preparing for class and 8.5 hours attending help sessions and completing assignments.

Schedule

Important Links

Week 01

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Week 02

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Week 03

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Week 04

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Week 05

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Week 06

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Week 07

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Week 08

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Week 09

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Week 10

The building in which we gather for this class, and all of College of the Atlantic, is located on traditional lands of the Wabanaki people. The four Native American tribes in Maine today are the Maliseet, Micmac, Penobscot, and Passamaquoddy, collectively referred to as the Wabanaki. I believe it is important to acknowledge that our presence on this land entangles us in the web of colonialism, past and present. The future, however, is still unwritten.