Formalities

Here is the course description, stolen from the Rutgers Summer School Catalog

LINEAR ALGEBRA AND APPLICATIONS. (CR.3.)
16:642:550: SEC. E6:90680
N.B. EVE. JUNE 23-JULY 31
MTTH 6:15-8:45
BUMBY
HILL CENTER 525

PREREQUISITE: Consent of instructor or graduate director.

Vector spaces, bases, and dimension. Linear operators, quadratic forms, and their matrix representations. Eigenvalues, eigenvectors, diagonalizability, Jordan and other canonical forms. Applications to systems of linear differential equations.

General information.

The main source of course material in this department is the World Wide Web.  The department home page is
http://www.math.rutgers.edu
That page has links to Course materials, which shows a page with links to individual courses.  Many courses, including this one, have pointers to several years of course archives.  Here is a link to last summer's page. This course has been stable for several years.  However, you can expect minor revisions in the supplements, and some new homework exercises.

I also have a personal home page that can also be reached by following the Faculty link on the department page.

The pace of the course will be enforced by a short (of duration no more than 45 minutes) exam each week.  These exams will be given at the start of the period on Monday or Tuesday, to be determined after consultation with the class.

Homework will be collected two class meetings after it is assigned. The homework will be graded.  The grade will mostly serve to identify additional work that should be done prior to the exam. 

Friday, July 4 is a holiday.  The University will be closed and no class will meet on that day, but that should not affect this course.

There will be a three hour final exam in the last class meeting on Thursday, July 31.  On that day, class will start at 6 PM and end at 9 PM.

Prerequisites.

Prior exposure to Linear Algebra at the Undergraduate level is expected, allowing the course to begin with Section 3.6 of the text.  This section gives a quick review of the topics in such a course.

Homework

A separate page contains a table showing lecture topics and homework assigned.

Supplements

These follow the supplements from Summer 2002 fairly closely, although there are minor improvements in editing. There is now an attempt to have a uniform appearance. The previous supplements 7 and 8 are now S5 and S6, so references to the supplements will have different meaning this year than they did last year. The other supplements will appear later.

• S1: Prerequisites, including Intersection of Subspaces. (alternate form of 3Y, p. 198, with an exercise)
• S2: Sign of a Permutation. (extending discussion on pp. 237-238)
• S3: The Cauchy-Binet Formula.(generalization of property 9, p. 217)
• S4: Finding eigenvectors. (more on examples from sections 5.1 - 5.3, with two exercises) [A typographical error has been noted by a student: in the last matrix on page 3, a correct pivot step leads to 38 instead of 48 in the last column. The eigenvector is correct.]
• S5: A robust method for finding characteristic polynomials (with three exercises).
• S6: The Perron-Frobenius Theorem (with four exercises).
• S7: Matrix exponentials (with three exercises). There is also an earlier version prepared for our first undergraduate course in differential equations.)
• S8: Schur's Unitary Triangularization Theorem (and other topics from section 5.6, with five exercises).
• S9: Numerical methods in Linear Algebra. A sketch of considerations involved in jumping into chapter 7 at this point of the course (with two exercises).

Grades.

The graph shows a comparison of the two components of class work (homework and class exams). In addition, there is a trend line (the best least squares fit of a linear function to the data) and lines showing the position of totals of 90%, 80% and 70%. The grades of two individuals were omitted from both the plot and the computation of the trend, because the available grades are incomplete. Students with grades below 80% should make a special effort to demonstrate mastery of the course on the final exam.

The graph shows a comparison of class work (homework and class exams) and the score on the final exam. A view of clusters of grades gives a better picture of work in the course than a simple average. Twenty grades of A and 6 grades of B+ were assigned, along with one grade of "incomplete". The graph shows everyone completing the course. The graph also includes a trend line and the location of a total of 250 points (although the individual whose scores are below that line, but include a grade of 131 on the final exam, was given an A because the final exam showed proficiency in material that had gotten an unusually low grade during the term). If the proposed grade of A- were available, a score of 275 would have been used to split the grade of A into these two grades.