Formalities

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

LINEAR ALGEBRA AND APPLICATIONS. (CR.3.)
16:642:550:SEC. E6: 80704
N.B. EVE. JUNE 24-AUGUST 1 MTTH 6:15-8:45 (lecture)
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 you can reach 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 Tuesday, July 2 and 9, and on Monday, July 15, 22, and 29.

Homework will be collected two class meetings after it is assigned. The homework will be graded.  In the case of short exercises from the textbook, the grade will mostly serve to identify additional work that should be done prior to the exam.  A few longer calculations from supplementary notes will be assigned, and these will be given more weight since they are likely to be the only graded work on that topic. The class exams will concentrate on material from the textbook exercises.

Thursday, July 4 is a holiday.  The University will be closed and no class will meet on that day.

There will be a three hour final exam in the last class meeting on Thursday, August 1.  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

• S1: 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)
• S5: Matrix exponentials. (This is the most recent revision of an article in preparation. There is also an earlier version prepared for our first undergraduate course in differential equations.)
• S6: Schur's Unitary Triangularization Theorem. (and other topics from section 5.6, with four exercises)
• S7: A robust method for finding characteristic polynomials (with three exercises).
• S8: The Perron-Frobenius Theorem (with four exercises).
• S9: Transcript of a Maple worksheet contrasting the computation of a pseudoinverse via the SVD with the value given by the normal equations when the columns are linearly independent.
• S10: Transcript of a Maple worksheet illustrating the proof that AB and BA have the same characteristic polynomial apart from a power of the variable..

Grades.

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. Ten grades of A and 6 grades of B+ were assigned, along with one grade entered as F. Although two of the B+ grades look lower than the cluster of four, the difference is almost entirely accounted for by weakness in the homework that was corrected on exams.