Axiom 1. Mathematical potential is distributed equally among different groups, irrespective of geographic, demographic, and economic boundaries.
Axiom 2. Everyone can have joyful, meaningful, and empowering mathematical experiences.
Axiom 3. Mathematics is a powerful, malleable tool that can be shaped and used differently by various communities to serve their needs.
Axiom 4. Every student deserves to be treated with dignity and respect.
Math 540 is an introductory one-semester course on linear algebra, which is intended to be a first course in the subject. (I don't expect that you've seen any linear algebra before; the only real prerequisite is an interest in math and a willingness to dive into the subject.) This course is intended to be a gateway into the mathematics concentration and should prepare you to take MATH1530 (Abstract Algebra) and MATH1060 (Differential Geometry), both of which are offered next fall! We'll build up the theory of vector spaces and linear transformations from the foundations. Time permitting, the course will roughly cover Chapters 1-7 of Axler. Linear algebra is an excellent subject through which to begin learning to construct proofs, so along the way you will develop your mathematical proof writing skills. That being said, this is not a dedicated proof-writing course, but such a course does exist at Brown! I encourage you to shop APMA1001 (The Art of Writing Mathematics) if this type of course interests you!
I anticipate that MATH 540 will be challenging, but the TA and I are here to support you. In the end, I hope that you find it equally rewarding. Expect to spend 10-12 hours on the material outside of class every week. This includes one hour of recitation (see below).
MATH 540 versus MATH 520. The main difference between MATH 540 and MATH 520 is in emphasis. MATH 540 is a rigorous treatment of vector spaces and linear transformations emphasizing proofs and underlying structure over computational aspects and applications. The emphasis in MATH 520 is on those explicit computations and applications. If your interest in linear algebra is mainly as a tool (and it is an incredibly applicable tool!) then MATH 520 might be a better fit. Feel free to contact me to discuss your situation and/or shop both courses to find out which is right for you this semester.
Textbooks:
The only required text is Axler's Linear Algebra Done Right, fourth edition, Springer, 2024.
Download the free PDF here
Note that Axler has also made a number of videos
Links to an external site. that you may or may not find helpful. Follow the link to see if they're a good tool for you!
We will supplement that book with the following freely available resources:
Hammack, Book of Proof, third edition
Treil, Linear Algebra Done Wrong
Readings for the week will be posted in advance. I expect you to keep up with the reading, working through the examples given in the book to supplement the work we are doing in class.
Recitation sessions are an important aspect of the course. There will be two or three (identically structured) one hour sessions offered each week at a time chosen toward the end of shopping period. You are responsible for all material covered and discovered during these sessions.
Your TA will be present and a number of different things might happen in recitation. The object of these sessions is to help you deepen your understanding of the course content and to learn how to work with other budding mathematicians. You might be asked to:
solve challenging problems using your course notes, your text, and your peers.
correct a provided incorrect proof of a topic we discussed in class.
prove a proposition.
swap proofs from a problem set and provide feedback to your peers.
Your TA is there to act as a guide but will (hopefully) not be presenting solutions to the class in a lecture style. Occasionally your TA will lecture on content we didn't cover in class.
There will be one problem set per week, due (almost) every Thursday starting on February 1st at 11:59pm on Gradescope.
The schedule of assignments in this course is designed to provide you with regular feedback, allow time to address confusion, and ensure that you do not fall behind on new material. For that reason, no late problem sets will be accepted. That being said, I understand that life often gets in the way of careful planning! I will therefore drop your two lowest problem set scores.
Your TA will be providing the entire class with feedback and helpful reminders after the grading of each problem set. Look for the feedback in the Announcements.
You are encouraged to discuss and collaborate on problem set solutions with your fellow MATH 540 students. However, you must write up your own solutions to hand in. Please do not seek help on the problem sets from anyone outside of MATH 540.
Exams
Midterm 1, expected date February 29
Midterm 2, expected date April 11
Final, May 14 at 2 pm, location TBD
There will be no makeup midterm exams. If you have a good and documented reason for missing a midterm, and let me know in advance, the other midterm will count as 35% of your grade, and the final as 45% of your grade. If you are having trouble keeping up with the class for some reason, please make an appointment with me to make a plan for how to stay on track.
For the midterm and final exams, you will be allowed a single 8.5x11 (one sided) sheet of paper with notes. You are not allowed to consult any other references during the exams, including: anyone else (in Math 540 or otherwise), your textbook(s), the internet, or calculators that can do linear algebra.
Information about the exam structure and content will be available no later than a week before the exam date.
As stated in the Brown University academic code, Brown students are expected to practice high standards of academic and professional honesty and integrity. Among other things, this means that you are expected to adhere to the collaboration policy of this class. If you are unsure if something is academic misconduct, ask me and I will be happy to discuss any questions you have.
Every attempt will be made to anonymize grading of problem sets and exams.
Your final grade will be determined on basis of your performance on problem sets and exams as follows:
20% Problem sets
25% each Midterm 1 and Midterm 2
30% Final
The grading system is as follows: a 90% guarantees an A, 80% guarantees a B, a 70% guarantees a C.
Students are required to adhere to the safety requirements set forth by Brown University. For details on these requirements see: https://healthy.brown.edu/. I will always wear a high-quality mask in the classroom. I ask that each of you wear high-quality (not cloth!) masks in class and office hours. If you forget a mask, I will have some high-quality masks available in class.
In situations where there is, according to Brown's policy or my best judgement, a substantial risk that I could be infectious, class will be held online on Zoom instead (see class Zoom link in the Zoom tab at the left). This would include any situation where I am symptomatic or test positive for COVID-19. If such a case arises, I will announce this over email as soon as I am aware of the situation, so please check your email before coming to class. If you are symptomatic or test positive for COVID-19 do not attend class.
Your instructor and TA will each hold 2 hours of office hours per week, at a time and location to be determined.
Office hours are times that I have set aside especially for students to come and discuss math. When you come to office hours, you can arrive at any time that is convenient for your schedule (not just at the beginning). During my office hours you can:
Ask a question about something you are confused about
Go back over a discussion from class
Look for more example problems to work through
Work on your homework alone or in a group
Ask for advice on skills for homework or test-taking
Explain solutions to your classmates
Talk about something entirely different!
I'm looking forward to getting to know you all better in office hours!
Mathematics is not a spectator sport. The best way to gain fluidity with new concepts is to practice them over and over again. Try forming a study group with some classmates and explaining solutions to each other. Challenge each other to explain the reasoning behind every step.
This may seem obvious, but it bears repeating: come to class, come to office hours, and attend recitation section.
Read the reading posted in the schedule before class, and make notes about what confuses you. If I don't clear this up in class, ask specifically about your questions.
Try completing the homework first without looking at your notes or textbook or talking with your classmates. The homework assignments are there to help you gauge if you understand the material. It is easy to overestimate how much you understand if you are getting help.
Constructing proofs is difficult, especially as a beginner. Try something! Assume that your first approach won't work (this is completely normal, even among professional mathematicians), and be ready with another approach. Your biggest asset is a willingness to be wrong a lot. Don't try to do a lot in your head -- writing things out can help you stay organized and stimulate new ideas.
Have a growth mentality! I believe that intelligence and ability can be developed and improved upon with effort and support.
I encourage you to:
Engage in challenges
See hard work as a path to success
Treat feedback as an opportunity to improve
Be inspired by the success of others
Say/think:
What am I missing?
Perhaps I need a new strategy.
This is going to take some time.
How are they coming to that conclusion?
That's an interesting idea for improvement.
Read more about the growth mentality and its impact in this article
Brown University is committed to full inclusion of all students. Please inform me early in the term if you may require accommodations or modification of any of course procedures. You may speak with me after class, during office hours, or by appointment.
Here are some additional Campus Resources that may help you this term:
Students Support Services. Deans can be a helpful resource to discuss personal, family or health-related concerns, as well as a potential academic and personal plan. They are available for same-day consult and/or scheduled appointment.
Counseling and Psychological Services. (CAPS) provides a range of mental health services to the Brown community, including individual psychological treatment, mental health support, medication management, crisis stabilization, after hours assessment and urgent care, outreach programming, and groups.
This course is heavily influenced by past iterations taught by Isabel Vogt and Melody Chan, who were themselves likely influenced by others.