189-251B: Algebra 2

Professor: Henri Darmon

Markers: Christopher Klassen and Michael Snarski.

Classes: MWF 8:35-9:25. in Burnside Hall 1B36

Office Hours:
Darmon MW 9:30-11:00, in Burnside Hall 1111.

Note. I will be absent in the first week of classes, and so the first three lectures will be given by Miljan Brakocevic. In this week, there will be no office hours but if you have any questions that cannot wait for the week after (particularly concerning your course choices) do not hesitate to send me an email at darmon@math.mcgill.ca. I will be checking my mail frequently and will endeavour to respond to your question promptly in that first week.

Tutorials: There will be no tutorials for this course. However, 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 excellent class notes that Eyal Goren has developped for this course in 2008.

Optional Textbooks:

Linear algebra done right by Sheldon Axler.

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

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 reading about math 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. 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".)

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 alot tougher. This reflects the stiffer competition arising from the fact that at most half (and, roughly, the more motivated half) of the students who were in 235 will be moving on to 251.

In particular, anyone who did not get an A in 235 will have to work much harder to earn a passing grade in 251, and should consider registering for Math 236 instead. If you did not get an A 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 returned, graded, the following Monday. There will be around ten 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, in class, on Wednesday, March 8 at the usual time (8:30-9:30).
It will test you on all the material that was covered before the study break.

I will hold extra office hours on
Friday, March 2, 9:30-11:00.
Monday, March 5, 9:30-2:00
Note however that I will not be in on Tuesday March 6. I expect you to get most of your studying done over the weekend, so that by Monday all the questions are clear in your mind...

Here is a sample midterm from a previous year for you to practice on.
Here are my corrections and comments on the midterm exam.

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.