Advanced Linear Algebra, Fall 2008

Logistics:

• Course: MTH MAS 5107-001, ref # 80646
• Prerequisites: Linear Algebra (MAS 3105)
• Time & Place: MW 4:35 – 5:50 in PHY 108
• Instructor: Gregory McColm
• Office: PHY 338 (inside Suite PHY 342)
• Phone: (813) 974-9550
• E-mail: mccolm@cas.usf.edu
• Office Hours: MTWR 2 – 2:50
• URL: http://shell.cas.usf.edu/~mccolm/classes/LinearAlgebra.htm
• Text: Linear Algebra by Kenneth M. Hoffman and Ray Kunze

Announcements

· Change in course policy: the lowest of the three exam grades will be dropped in computing the course grade.

Handouts

· Here is the September 3 handout on Solutions of Systems of Linear Equations.

About the Course

This course is a continuation of Linear Algebra (MAS 3105), and is one of the three core courses for the Algebra Qualifying Exam. With that in mind, it is particularly recommended for mathematics graduate students in the M.A. and Ph.D. programs. It serves to help students prepare for the Qualifying Exam, although it is expected that students will undertake additional preparation for the qualifier exam.

Specifically, we will cover (from the text):

• A very brief review of elementary linear algebra, i.e., Chapters 1 & 2 & 3.1 – 4, together with a slightly more intense introduction to functionals, etc., in Sections 3.5 - 7.
• We will skim Chapter 4, which is duplicative of the ring and field theory sections of the main Algebra sequence, and we will spend some time on Chapter 5 on determinants.
• We will then spend a lot of time on canonical forms in Chapters 6 & 7.
• The rest of the course will be on inner product spaces and various forms in Chapters 8, 9, and10, depending on the time available. All these subjects are fair game in the qualifier exam, so we will cover as much of them as possible.

Notice that this is practically the entire book, which is about four hundred pages long, and heavy reading: this is a time and energy consuming course, and historically many students find it challenging.

Incidentally, this being hurricane season, and miscellaneous muddlement being a year-round possibility, this schedule is subject to change.

Methodology & Goals

The objective of this course is to impart a working knowledge of and appreciation for the subject – working knowledge meaning one can do work in it, apply it to other areas, and, ahem, take qualifier exams in it. We take the traditional view that the most effective (if tiresome) method of achieving this objective is approaching the material as an array of skills to be mastered. There are three types of skills involved.

• The ability to carry out formulaic manipulations. These can range from straightforward manipulations to those involving "tricks".
• The ability to translate an application -- perhaps in the form of a word problem -- into a formulaic situation and find a solution for that application.
• The ability to verify (prove) the correctness of a certain assertion or technique using the two skills listed above.

The course is designed to impart these skills.

The class meets twice a week, for two 2-hour sessions. A typical class session will consist of two 50-minute stretches, with a ten-minute break in the middle. Starting September, each 50-minute stretch will include:

• Lecture presentation of new material. This will only be a guide and not a substitute for reading the text: in fact, some important material in the text will not be presented in class (not enough time), so you will have to keep up in your reading.
• Performance. Students will come to the board and present their solutions to homework problems.
• Q & A. Students will have the opportunity to ask me to do submitted or unassigned problems, or to give hints for unassigned problems.

Homework will be assigned in every session, and will be due one week hence, or at the next test review, whichever comes first. Homework is the primary activity by which students learn the material, and it is expected that students will budget AT LEAST six (6) hours a week to it. Typically, a student who does not put sufficient effort into homework will not succeed in the course.

In order to keep track of how students are doing, every Wednesday I will collect an extra credit assignment: each student composes a paragraph on how they are doing in the course and estimates time spent studying during that week. Credit is given for information; students are free to be candid. This is the only extra credit available in this course.

There will be three 1+-hour midterms, tentatively scheduled for September 29, November 3, and December 8 (the last being held during the “final exam” period from 5:30 pm to 7:30 pm). They will cover their respective thirds of the course; each will start with straightforward computational problems, and then gradually get more difficult, concluding with one or more proofs. (Thus there will be no cumulative final.) Each exam counts for a quarter of the course grade (homework counting for the other quarter); even though homework officially counts only a fourth of the grade, in reality, homework is like practice for the exams, students who do not perform well at homework do not perform will in exams. Each exam will be preceded (in the previous class session) by a review session in which no new material will be presented.

Administration

Classroom complexities. A minimal level of classroom decorum is expected.

• While class is in session, students should not read newspapers, do sudoku puzzles, surf the net, etc. In particular, cell phones should be turned off or (if one anticipates a critical call) set to “vibrate.”
• While I treat material presented in class as being in the public domain, students should inform me if they intend to make audio or visual recordings. In addition, I do not give anyone permission to make recordings for sale; however, recordings may given away for free.
• The university has a number of policies for disabled students, and disabled students are advised to consult the Office of Student Disability Services. Students should discuss options with me in advance (preferably at the beginning of the semester).

Since students practice and make presentations in class, and since this is an important part of the learning process, students are strongly advised to attend class. However, students are presumed to be sufficiently mature to make the decision to attend class on their own, and attendance is not part of the grade.

Missed Work. Students should turn in homework on time -- one of the great dangers in this course is falling behind, and students are strongly urged to start doing the homework within 48 hours of assignment (some problems will require a lot of mulling over). Attempting to submit late homework is strongly discouraged. As for exams, in order to maintain security the only time a particular exam may be taken is the time it is scheduled; make-ups or other resolutions for missed exams will be handled on a case-by-case basis. Students are strongly encouraged to make every effort to take the exams when they are held.

Office Hours. I have scheduled four fifty-minute periods each week for students who wish to see me; students may come during those hours unannounced and I will be there (unless something comes up, in which case I will announce and/or post my regrets on the class web-page). Students may also schedule appointments (I may be out if you come during a non-office hour, so an appointment is strongly advised if you need to see me at some time other than my office hours). Remember: your tuition and tax dollars are paying for your professors to have office hours, so don't be shy about visiting us and asking questions. That's what we're paid for.

Academic Honesty. We academics are rather uptight about this (for an example of Victorian attitudes, see The Adventure of the Three Students in the Return of Sherlock Holmes), so for the purpose of this course:

• In order to encourage students to work together, students may collaborate while working on homework problems.
• Students may not collaborate in any way on exams.

Academic dishonesty incidents will be dealt with in a manner consistent with university policies: see the section on Academic Policies and Regulations in the catalog.

Grading complexities. Grades normally range from A(+) to F.

• Students who prefer to be graded by the S/U system must inform me by the end of the third week of classes (September 12).
• Students may drop the course prior to 5 pm, August 29, and not have the course recorded on their transcript; students may drop the course prior to November 1 with no academic penalty. (Note: the second midterm will not be taken by then.)
• Homework scores are reduced to grades at the end of the semester, while each midterm grade is itself a letter grade from A+ to F; the average of these four grades is the course grade, using the Wisconsin system (A+ = 4+1/3, A = 4, ..., F = 0).