Course webpage for Math 21b: Linear Algebra with Differential Equations (Fall 2008)

If you find a mistake, omission, etc., please let me know by e-mail.

The REQUIRED Introductory Meeting for this class will take place Monday, September 15, at 8:30 AM [sorry -- I don't like that time either )-:], in Science Center Room A.

The sections will meet MWF 11, 12, and/or 1, depending on the number and schedules of students sectioning. (In particular, we cannot know yet which TF will be teaching which section.) The first sections will meet Monday, September 22.

Sectioning will begin Monday, Sep.15; please section by Wednesday, Sep.17 at 1PM. To section by computer:

Windows --> SecureCRT application --> Quickconnect, protocol ssh2, hostname, username section
Mac OS X --> Applications --> Utilities --> Terminal: at the prompt, type ssh and hit return
Unix/Linux: log into your FAS account; at the shell prompt, type ssh and hit return.

These sites will not close at the deadline (Sep.17 at 1PM), so you should use the same procedure to section if you've missed the deadline, and we'll try to accommodate you as best we can.

If you have another problem or question about sectioning that the sectioning webpage or program does not answer, please email Susan Milano (milano[AT]math[dot]harvard[dot]edu). Do not ask me: all I'll be able to do is direct you to her.

Here's the course syllabus, including the provisional homework assignments through the second midterm, just before Thanksgiving.
TF's, section times, and office hours:

11-12 section: David Geraghty [geraghty], Sci Ctr 222; office hours: Math common room (Sci Ctr 4th floor), Tuesday 4:15-5:45
12-1 section: Noam Elkies [elkies] (course head), Sci Ctr 110; office hours: Sci Ctr 335, Thursday 2:45-4:15 and by appointment
1-2 section: Wushi Goldring [wushi], Sci Ctr B10 (B=basement); office hours: Sci Ctr 425D, Monday 3:15-4:45 Sci Ctr 425D, Wednesday 4:15-5:45

Each "[username]" means the e-mail address username@math (from outside Harvard:

CA's and their problem session times:

11-12 section: Jessica Ranucci [jranucci], Monday, 8-9 PM in Sci Ctr 101B;
Beatrice Liem [bliem], Wednesday, 7:30-8:30 in Winthrop Dining Hall
12-1 section: John Tan [johntan], Tuesday, 9-10 PM in Sci Ctr 111
1-2 section: Jeremy Booher [jbooher], Thursday, 8-9 PM in Sci Ctr 103B
This time each "[username]" means the e-mail address (if you're e-mailing from a FAS account then username by itself should work).

The CA boxes are in the third floor of the Science Center, next to the sectioning office (room 308); the 21b boxes are the bottom three in the rightmost column.

Both midterm examinations will take place in Science Center Lecture Hall E.
NB: Hall E is located in the basement of the Science Center, at the NW corner of the east-west corridor. Please plan to arrive at 7:30 so that the exam can begin promptly at 7:45.

The first midterm will be in the same format(*), and on the same material (Sections 1.1-1.3, 2.1-2.4, and 3.1-3.3), as this exam from the Fall 2006 edition of 21b. Here are the answers to problems 1, 2, 3, 4. Some further practice problems are here, with solutions for problems (1,2, 3,4,5), thanks to the CA's for the Fall 2006 edition of Math 21b.

(*) That is, 3 multi-part problems, each worth 10 points for a complete solution (including showing all work), and 10 True/False questions, worth one point each (with no requirement for justifying answers).

Here's another way to see that the row rank of any matrix A equals its usual (column) rank [Fact 5.3.9(c) in the textbook]; along the way we also obtain Fact 5.4.1: the orthogonal complement of im(A) is the kernel of AT.
Here's what you have to know about determinants from Chapter 6, sections 1 and 2.
Q: What the **** is a Kyle number (mentioned in line four of the Solution on page 322 of the textbook)?
A: See page 128, specifically the footnote to the first word on the page -- thanks to D.Geraghty for remembering and locating this. Naturally it is not standard terminology in linear algebra; it does not even appear in the textbook's own index. We shall not use this term (and hope you don't either), and you're not responsible for it on the midterm and final examinations.
Some practice problems from the second midterm: this exam from the Fall 2006 edition of 21b, and more practice problems [now the correct link -- sorry it pointed to the 1st-midterm problems before!]. Here are the answers to problems 1, 2, 3, 4, 5 of the 2006 midterm, and solutions to practice roblem 1 and the T/F questions and practice problems 2,3, by 2006 TF's L.Kang and N.Sun respectively. As with the first midterm, there are plenty of other problems in the textbook that you can use for further practice. This year's midterm will have only three multi-part problems plus the ten True/False questions, that is, the same format as our first midterm, not 4+T/F as in 2006.

Here are solutions for the second midterm.

Here is the schedule of homework assignments past the second midterm date.

The topic of class on Monday, Nov.24 will be nonlinear dynamical systems, following this handout (please correct these errata in your printout).

Here is the latest version of “Chapter 10” by Prof. C.H.Taubes (NB Bretscher's textbook has only 9 chapters), which we'll use together with the Bretscher text for most of December. Many thanks to Prof. Taubes for putting this text together 2.5 years ago and allowing us to post it online now. Please also note these errata and comments (updated Dec.3) in your printout.

The FINAL EXAM for Math 21b will take place on TUESDAY, JANUARY 13 in SCIENCE CENTER A, STARTING AT 9:15 AM. The date is as expected from the “exam group” of the class; the time is according to this schedule which the Registrar sent out a few days before the Thanksgiving break -- see the last line of page 4. Note also the instructions at the beginning of that document:
Students requesting to take the exam on a different date/time/location, or requesting special accommodations, should contact one of the administrative offices below:
The last regular class meeting is Monday, December 15. There will be two review sessions during reading period: Wednesday and Thursday, January 7 and 8, both from 1PM to 3PM in Room 507 of the Science Center. The first will concentrate on material covered since the second midterm, and the second on material already covered on midterms 1 and 2 (which of course you're also responsible for in the final exam).
ADDED JANUARY 2 (would have been earlier but for computer access issues during the winter break):

Here are copies of recent final exams in Math 21b: May 2004, May 2005 (with solutions at the end(*)), and January 2006 and January 2007 (problems only). Note that there are a handful of problems on material (usually from 10.5 and 10.6) that is not on the syllabus for this year's exam.
(*) Corrections: in the 2004 exam, T/F question 7b, the bottom two entries of A were evidently switched, so the problem as stated is FALSE rather than the intended TRUE [thanks to Eugene Beh]; in the 2004 solutions, Question 6, parts b and c should have factors of sin(2x) and cos(2x) instead of sin(x) and cos(x) [thanks to Eric Petersen]; in the 2005 solutions, Question 6, part b should have cos(2t)+sin(t) instead of cos(t)+sin(t) as the last basis element [thanks again to Eugene Beh]. (In question 6c, if you listed the natural basis elements of 6a in a different order, then your answer to 6c will look different though the two 4*4 matrices must be similar.)

Here are some more practice exercises on the “Chapter 10” material. You may ignore problems 11, 19, 20, 21, and 24, which concern material we didn't cover. Problem 31 is incorrect as stated; the printed solution indicates that the intended differential equation was f'''+2f''+2f'=0, not f'''-2f''+2f'=0. Please note also the following typographical errors in the solutions: in solution 29, the exponents -16t, -12t, 64t should be -16t, -36t, and -64t; in solution 33, the last inequality should be x≥0 rather than x≥1; in solution 34, the factor -16/π should be -8/π; and the orthonormal basis in solution 35b should begin with 1/sqrt(2) as usual, not 1. Finally, problem 26 is incorrect as stated: the stated problem is basically the same as problem 29, but the solution is different. Can you figure out how to change the differential equation and the initial conditions so that the given solution 26 becomes correct? [Thanks to E.Beh for the corrections to Problems 31, 33, and 35b.]


READING PERIOD OFFICE HOURS are either by appointment or at the following dedicated times:

David Geraghty: 11:00(AM)-12:30(PM), Saturday Jan.10, and 11:00(AM)-12:30(PM), Sunday Jan.11, in the math department common room (Sci Ctr 4th floor)
Noam Elkies: 3:00-4:30, Thursday Jan.8, Sci Ctr 335; 12:00-1:00, Monday Jan.12, Sci Ctr 335
Wushi Goldring: 6:30-7:30, Tuesday Jan.6, Sci Ctr 425D; 4:30-6:00, Wednesday Jan.7, Sci Ctr 425D