This is the official syllabus for the class. I will have a pdf version of this syllabus made soon and linked here, once some details are settled. You can find links to assignments, materials, and everything else (as they become available) on the Advanced Linear Algebra course webpage.
Instructor
Andrew McIntyre. Most reliable contact method is the Bennington Math Slack; you can also email me at amcintyre, the “at” symbol, bennington.edu. At the moment, for Spring 2021, I am fully remote, and not in the office. I do not have set office hours for this term. However, you can contact me any time with questions, and I am happy to answer over Slack or to set up a Zoom meeting. During the term, I aim to always answer within 24 hours; please feel free to pester me if I don’t.
Class times and location
The class is Tuesdays and Fridays, 2:10–4:00pm. It is currently planned to be held remotely and synchronously, over Zoom. (I have sent you calendar invites for the class, please let me know if you have any difficulty.) The first class is Tuesday, February 16, 2021, and the last class is Tuesday, May 25, 2021.
Note on recording
I’m planning to record the synchronous classes, and make the recordings available on an unlisted YouTube channel. I will not record group work or in-depth class discussions, but I am planning to include student questions. If you have concerns about being recorded, please contact me.
Textbooks
There is no required textbook. We will be working mainly from problem sets; when I ask you to read a selection from a text or other source, I will provide a pdf or a link. None of the many linear algebra texts on the market satisfactorily covers all the topics I would like for this class; however, there are many excellent books from different perspectives, and during the term I will make suggestions for those who want to do further reading.
Calculators and computers
You will of course need a computer and reliable internet to participate in the synchronous classes. Please contact me and IT if you anticipate any problems, and we will do our best to sort it out.
For computations in the class, the requirements depend on whether you choose the “theoretical” or “computational” track for the assignments (see below for details). If you choose the “theoretical” option, then an ordinary scientific calculator (or app on your phone/computer) will be quite sufficient. If you choose the “computational” option, then you will need a computer algebra system/language capable of managing linear algebra seriously (and a computer capable of running it). The most popular options are Python/NumPy, MATLAB, C++, Maple, Octave, or Sage. You are welcome to use whatever you are familiar with (or want to learn!); if you have no preferences, I suggest Sage. I’ll talk more about the options and provide some links to get started in class.
Communications
There will be three main routes of communication: the synchronous classes, the Slack channel, and the webpage. The Slack channel is the main place to look for announcements: any new problem sets, written notes, or other class communications will be posted there. Please either activate notifications for Slack on your computer or phone, or check it regularly. Anything that I announce in the synchronous classes, I will repeat on the Slack. The classes will be recorded and posted on an unlisted YouTube channel (see the Slack for the link). I will collect all links to problem sets, written notes, and other resources on the webpage.
What this class covers
This is a second course on linear algebra. One focus will be on matrix decompositions (especially spectral, singular value, and QR decompositions), related concepts (e.g. Moore-Penrose psuedoinverse), and their applications. Applications will include least squares, principal component analysis, google search, data compression, and discrete and fast fourier transforms. The other focus will be on tensors, exterior algebra, Galois theory, and elementary Lie groups, with applications to geometry and physics. The class should be of interest for both theoretical and applied reasons. Students will have a choice on assignments, whether to pursue a theoretical or an applied emphasis (or some mix of both).
Prerequisites
MAT 2482 Linear Algebra or the equivalent. If you choose the “computational” option (see below), some programming fluency will also be needed.
Objectives
Every class aims at both learning and evaluation; however, these goals are often in tension. My primary concern is that this class is an effective learning environment; evaluation, while sometimes necessary, is secondary.
Specifically, I am hoping that everyone will get more fluency with linear algebra, a familiarity with more advanced tools and ideas, and a more sophisticated conceptual perspective on the subject. Depending on your priorities, I would like you to either develop more theoretical knowledge of the subject and skills with proofs, or I would like you to get a deeper understanding of numerical issues, computational tools, and their applications.
Work expected
The primary work of the class will be completing problem sets, which will be assigned at a rate of approximately one per week. I will ask you to submit written work on the Slack, and to discuss it with me in Progress Meetings (see below). I will also expect you to do some work during classes (usually collaborative work on problems). In addition, I encourage you to spend time helping your classmates; I will attempt to explicitly credit you for this if you do so. There are no tests, exams, or final projects (although you may choose to expand one of the problem set questions into a larger project).
The time commitment for this class should be about 10 hours per week total, 4 hours in synchronous classes and 6 hours outside of class. This should be fairly constant through the term.
theoretical versus computational option
There are two options, or “tracks”, in this class. I will expect everyone to know the definitions and theorems, to do simple calculations or proofs, and to do simple computations by hand. I will identify these problems as “core” problems. Beyond that, you have a choice: you can either do the “theoretical” problems, which will be cover more theoretical concepts and require more involved proofs, or you can do the “computational” problems, which will involve more applied ideas and computer calculations on larger systems.
How to submit work
Please submit questions as individual files (one file per question), with a descriptive file name (something like “PS2-Q4”). Please submit them in a Slack direct message to me; I have found that the easiest way for me to keep track of your work. (If you prefer to have a Google Drive folder to keep things organized for yourself, you can make a link to your Google Drive files in the Slack direct message.) I may ask you to submit only selected problems from a problem set; if so, I will let you know.
If you handwrite your solutions, please use a pdf scanner app to make a pdf. There are several good free ones available; iOS Notes has one built in, and Adobe makes a free one. Please do NOT just take pictures with your phone; these are difficult to read.
If you would like to type your solutions, I suggest learning LaTeX, which is the standard typesetting program in mathematics. The easiest way to get started is on Overleaf; I will give you more instructions in class.
Due dates and late policy
I will have suggested due dates for each problem set, to help keep you on pace. However, I do not have strict due dates, and I do not make any penalty for late submissions. The important thing for me is the level of understanding you show by the end of the term; it does not matter how or when you get there. That said, it can be hard to catch up if you let things get too far behind; if you are getting behind, I may send a concern form to that effect.
Feedback
The main instrument of feedback is the “Progress Meeting”. I will be scheduling a time to meet with each student individually for 15 minutes, about once every 2-3 weeks (a total of 4-5 times during the term). The purpose of this meeting is for me to look over your work with you, to give feedback on it, and to ask you questions.
During the meeting, I will tell you if I see problems, either with the answers to specific questions, or more generally (e.g. if you need to be writing more clearly). If problems are incomplete or need correction, I will ask you to redo them and resubmit. The aim is for all the problems you do to be done completely and correctly by the end of term. I would like you to complete all the “core” problems, and the majority of either the “theoretical” or “computational” problems. With the latter problems, you have flexibility, to spend more time on the topics that are of most interest to you.
Evaluation
My primary goal is for this class to be an effective learning experience for everyone. Evaluating is a secondary concern. But we do need to have some sort of evaluation!
There will be two summary evaluation points, at the midterm and at the end of term. At those points, I will ask you to make a written self-evaluation, and I will also write an evaluation for you. If you are choosing to get a letter grade, you and I will both propose a letter grade as well. The evaluation will be based on the following criteria:
- whether you are spending sufficient time and attention on this class
- how thoroughly you are understanding the concepts
- how thoroughly and correctly you are solving the problems
- how systematic you are at correcting errors or gaps
- how clearly and carefully you are writing your reasoning
- how persistent you are with difficult problems
- how creative you are with challenging problems
- how much of the work you are completing
- how well you are contributing to collaborative work
- how well you are contributing to class discussions (in whatever form)
- how well you are helping other students in the class, if applicable
- how consistent your presence in the class is (see below)
There are no point totals; both your and my evaluations of your work will be holistic professional judgments based on the listed criteria.
I’d of course like to encourage you to try to do well in all aspects, but you do not have to do well on all the criteria in order to get a good evaluation or grade. For example, if you did not participate much in collaborative work or class discussions, but your written work was good, I might write something like “XX was somewhat quiet, but had a comprehensive understanding of the concepts, based on their clear and thorough written solutions of problems…” Or your balance of strengths might be the opposite. Everyone is different, and there are different ways to do good work!
For the final narrative evaluation and grade, I will take your evaluation and mine both into account. If your evaluation and mine are widely different, we will talk about it and try to come to a mutual understanding of your performance in the class. If we still don’t agree, I reserve the right to have the final say, but I have found that such major disagreement almost never happens in practice.
Attendance and presence policy
The expectation is that you will attend all the synchronous class sessions. This is important since the class activities are often collaborative. Being remote, I also ask the you maintain a consistent “presence” in the class, beyond just attendance: not missing progress meetings, submitting work, asking questions and contributing to class discussions on Slack when appropriate, and working with and helping other students.
However, I completely understand that this may not always be possible. If you cannot attend class for some reason, I ask that you catch up with the class recording (the link to the YouTube channel will be posted on the Slack), and that you try a little harder to maintain class presence in other ways.
If you can’t be present in class (both attendance and other ways), I ask that you keep in touch with me about it, and we can try to make arrangements together to keep you on track. I do not have any particular number of allowed absences, and I do not intend to penalize you on evaluations for absences. However, if you are not present for more than the occasional class, I will send a concern form; and, as a matter of College-wide policy, being absent for two weeks or more in total may lead to a marginal pass or failure even if your other work is good. That said, please talk with me about any gaps in your presence in class, and we should be able to work out a solution. My main concern is that it is difficult to do well if you are not regularly present and engaged. (In particular, I would caution you against the temptation to “skip this one class so I can get caught up”: this never works out well.)
Inclusivity
Unfortunately, mathematics has been and is often tied up with excluding people: both in that people are excluded from mathematics, and in that mathematics is used as a vehicle of exclusion. This is a big, complicated problem, with many facets. The aspects I want to address here are the ingrained attitudes about who is “supposed” to be “good at math”, and who is assumed to be “not good at math”. We have all grown up surrounded by these attitudes to a lesser or greater extent, and they have affected us all, both by the assumptions people make about us, and by the assumptions (perhaps unconscious) that we make about ourselves.
Let me be clear: based both on all the research people have done, and on my own years of personal experience teaching, I see no evidence of any systematic difference in “math ability” (however you measure that) between people based on gender, race, or any other grouping you might make.
Going further, I would assert (again, based both on the research and on my personal experience) that the idea of “math ability” is highly oversold anyway. Most anyone can excel in mathematics, up to whatever level they choose, given a supportive environment.
Although I strongly believe that there are no “math people” and “non-math people”, and that everyone can succeed, there ARE very strong systematic differences in how people have been treated in and around mathematics, and also consequently in how people see themselves. If you have had obstacles put in your path, and have been made to doubt yourself, please know that you are capable, you can excel, and that I will do whatever I can to make a supportive environment for you to do so. On the other hand, if you have been on the other side of these ingrained attitudes—if you have had the advantage of not being systematically minimized or doubted or excluded—then I ask that please recognize the responsibility that comes with that advantage: to not make these unfair assumptions about others, to consciously make space for others, and to help create a culture of mutual respect.
Classroom norms
Everyone is welcome in this classroom, and in mathematics. Please be kind and respectful to one another.
IF YOU ARE NERVOUS ABOUT BEING IN THIS CLASS, please know that, in my experience, everyone can do mathematics, and do it well. You may have had negative experiences with math, and you may have had assumptions made on your ability for invalid reasons. It’s normal to be nervous. You can do this, and you will be supported here.
IF YOU FEEL CONFIDENT ABOUT BEING IN THIS CLASS, (you might sometimes belong to both categories!), please be kind and respectful to your classmates. Do not talk over them, or make it seem like they should know things that they don’t. You will find, as the class goes on, that we are all struggling with math problems in our own way. We are all in this together.
IF YOU ARE MADE UNCOMFORTABLE IN THIS CLASS for any reason, please talk to me about it. (Aside from being uncomfortable about hard problems, but you can talk to me about that too!) I will do what I can to solve the issue. Please do not hesitate to be direct with me if you are having a problem, including if the problem is me: I promise not to be offended or upset. If I do not resolve the issue, or if I am the one making you uncomfortable, please talk to a college administrator (e. g., from The Office of Diversity & Inclusion, Student Life, or Academic Services) or other faculty member.
Academic Accommodations and Health Resources
I am happy to work with you on any needed accommodations. Please let me know the situation.
Here is the official statement: Bennington College provides reasonable accommodations to students with documented disabilities when such accommodations are requested and necessary to ensure equal access to College programs and facilities. If you believe you are entitled to an accommodation speak with Katy Evans, the Academic Services and Accommodations Advisor, about any disability-related needs. If approved, you will receive a memo detailing your specific accommodations; it is your responsibility to provide me with the memo and discuss the implementation of accommodations. Note that I will not be aware of your needs if you do not share this memo with me. Accommodations are not retroactive, so the sooner we meet to discuss your needs, the better. Also, students experiencing mental and/or physical health challenges that are significantly impacting their academic work are encouraged to speak with their faculty advisor and member of Academic Services (academicservices@bennington.edu or 440-4400) about the impact and to connect with resources through health and psychological services (440-4426 or 440-4451).
Basic needs
At Bennington College, we understand that basic needs (food, housing, and wellness) have a direct impact on academic performance, mental-emotional-physical health, professional development, and holistic success of our students. If you have a personal circumstance or need that will affect your learning or performance in this course, please let me or your faculty advisor know so that we can direct you to available resources to help support you during the term.
Conclusion
You can find a more general statement of my teaching principles here. Please contact me if you have any questions at all!
