189-251B: Algebra 2

Course blog (with assignments)

Professor: Henri Darmon

Classes: MWF 9:35-10:25 AM, in SADB -ζ(-1)

TEAM members:
Lucas Demuynck. Wednesday 11:30-1:00 PM, in BH 1120.
Diego Lopez. Friday 11:30-1:00 PM, in BH 719A.

Markers:
Odd-numbered assignments.
Diego Lopez.
Xianya Zhou.

Even-numbered assignments.
Dominic Petti.
Huangchen Zhou.

Office Hours:
Darmon M 10:35-11:35, and W 2:00-3:00 in Burnside Hall 1111.

Tutorials: There will be no formal tutorials for this course. However, in addition to the office hours of instructor and TEAM members, there is a Math Help Desk in BH911 operating Mondays to Fridays from noon to 5:00pm. This is a valuable ressource if you need extra help on the material or assignments, and you are strongly encouraged to make use of it.

Main text: I will be following the textbook Linear algebra and geometry by Kostrikin and Manin.

Optional Textbooks:

Linear algebra done right by Sheldon Axler.

Linear Algebra by Seymour Lipschutz (Shaum's Outline series).

Basic Algebra by Andrew Knapp.

In a more challenging vein, I highly recommend the textbook
Eléments d'analyse et d'algèbre (et de théorie des nombres) by Pierre Colmez.
It covers a lot more ground than we will in this course, and would be equally appropriate for the analysis courses that you might be taking concurrently.
It is very beautifully written and belongs on the bookshelf of any mathematics student who is passionate about her or his subject (and not afraid of math written in French...)

Several of you have asked for a supplement to the class notes which might contain a somewhat more detailed account of parts of the material and further exercises and problems for independent study.
Linear Algebra by Jim Hefferon is a book that I found on the web which looks very well written and contains plenty of exercises.

Syllabus:
This course will cover the basics of linear algebra. Linear algebra can be defined, somewhat circularly, as the branch of mathematics concerned with the study of vector spaces over a field, and the linear transformations between them. Vector spaces are an important instance of an abstract mathematical structure, just like the rings and groups that were studied in Math 235A. Surprisingly ubiquitous and flexible, they can model a bewildering variety of phenomena (both within mathematics, and in the ``real world" of applications.)

Key topics to be covered will include: Linear maps and their matrix representation. Determinants. Canonical forms. Duality. Bilinear and quadratic forms. Real and complex inner product spaces. Diagonalization of self-adjoint operators.

Should I register for 251, or 236?
Since Math 251 is an honors class, emphasis will be placed on rigorous proofs, and on developping mathematical maturity and problem-solving skills. The content will be abstract, and the pace, challenging, just as it was with Math 235A, only more so. The grading curve will thus be tougher. This reflects the stiffer competition arising from the fact that around a third (and, roughly, the more motivated, hard-working third) of the students who were in 235 is expected to move on to 251.

In particular, anyone who did not get a B in 235 will have to work much harder to earn a decent grade in 251, and should consider registering for Math 236 instead. If you got less than a B in 235 but are still keen on taking 251, that is possible in principle, but you should try to discuss with me how you plan to approach your coursework in 251.

Assignments:
Assignments are to be turned in on Wednesdays and will be returned, graded, the following Monday. There will be around eight assignments in all during the semester.

Grading Scheme : There will be two possible schemes, and I will take the maximum of those.

1. 20% Weekly assignments, 30% Midterm, 50% Final.
2. 20% Weekly assignments, 80% Final.

Midterm Exam:. The midterm exam will be held on Thursday, February 20 from 6:00 to 9:00 PM, in the McConnell Engineering building room 204 (ENGMC 204).

Final Exam: In compliance with University policy, the Final exam wil be a take home exam. It will be made available on Friday, April 17 at 9:00 AM and you will be asked to upload your answers in a single, pdf file, to MyCourses before Monday, April 20 at midnight.

The obligatory statements

McGill University values academic integrity. Therefore, all students must understand the meaning and consequences of cheating, plagiarism and other academic offences under the Code of Student Conduct and Disciplinary Procedures (see www.mcgill.ca/integrity for more information).

In accord with McGill University's Charter of Students' Rights, students in this course have the right to submit in English or in French any written work that is to be graded.

In the event of extraordinary circumstances beyond the University's control, the content and/or evaluation scheme in this course is subject to change.