Course webpage for Math 21b: Linear Algebra with Differential Equations (Fall 2006)

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

Information on the FINAL EXAM and review sessions during reading period is at the bottom of this page.
The REQUIRED Introductory Meeting for this class will take place Monday, September 18, at 8:30 AM [sorry -- I don't like that time either )-:], in Science Center Room A.

The sections will meet MWF 10 and 11, and also MWF 12 with sufficient enrollment. The first sections will meet Monday, September 25.

Sectioning will begin Monday, Sep.18; please section by Wednesday, Sep.20 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.

Here's the course syllabus, including the provisional homework assignments through Thanksgiving.

NB The homework from 7.6 is now due Wednesday Nov.22 (since Nov.20 is the midterm date). The next few homeworks are:

The topic of class on Friday, Dec.1 will be nonlinear dynamical systems, following this handout (please correct these errata in your printout). The homework for Monday, Dec.4, is exercises 1,2,3,4 [extra=5*] from pages 9-10 on the same handout.

Afterwards we'll follow a combination of some of the remaining sections of the textbook and ``chapter 10'', Prof. Taubes' notes on linear differential equations written over the last year or two for the end of Math 21b. Please also note these errata and comments in your printout.

Here are some practice exercises on this material. Please ignore problems 11, 19, 20, 21, and 24, which concern material we won't cover. Please also the following typographical errors in the solutions: in solution 29, the exponents -16t, -12t, 64t should be -16t, -36t, and -64t; in solution 34, the factor -16/Pi should be -8/Pi. 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?
[Added Jan.13 and 21: E. Beh notes that Problem 31 is incorrect as stated, and that the printed solution indicates that the intended differential equation was f'''+2f''+2f'=0, not f'''-2f''+2f'=0; that in Solution 33, the last inequality should be x>=0 rather than x>=1; and that the orthonormal basis in Solution 35b should begin with 1/sqrt(2) as usual, not 1.]

The remaining homework assignments for the semester are:

Note that (as suggested by some of the homework assignments and extras) we'll also be using Bretscher 9.3 for the 10.1-2 material, and Bretscher 5.5 for Fourier series (10.3). You might also refer to Bretscher 4.1 (and parts of later sections of chapter 4) for further background concerning the 10.1-2 material.
TF's, section times, and office hours:

10-11 section: Elizabeth Denne [denne], Sci Ctr 507; office hours: Sci Ctr 535, Monday 2-3, Tuesday 3-4, and Friday 1-2
11-12 section: Noam Elkies [elkies] (course head), Sci Ctr 309A; office hours: Sci Ctr 335, Thursday 2:45-4:15 and by appointment
12-1 section: Maksym Fedorchuk [fedorchuk], Sci Ctr 310; office hours: Sci Ctr 242B, Monday 11-12 and Thursday 5-6

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

CA's and their problem session times:

10-11 section: Heather Carmichael [carmich], Friday 3:30-5 PM, Sci Ctr 101B
11-12 section: Louis Kang [lkang], Monday 7:30-9 PM, Sci Ctr 216;
11-12 section: (William) Lin Cong [lincong], Thursday 6-7:30 PM, Sci Ctr 116;
12-1 section: Nike Sun [nsun], moved to Thursday 4-5:30 PM, Sci Ctr 103B.

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

A message concerning the Math Question Center:
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Please take a few minutes to fill out a quick survey so we can make the MQC even better!

Both midterm examinations will take place in Sever Hall, Room 113.

Some practice problems for the first midterm (taken from last year) are here. The actual exam will have 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).

Thanks to the CA's for quickly putting together answers for the practice problems (1,2, 3,4,5)!

Here are the answers to problems 1, 2, 3, 4 of the first midterm exam.

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.
Here are some practice problems for the second midterm; we now have solutions to Problem 1 and the T/F questions (L.Kang) and Problems 2,3 (N.Sun); thanks to the CA's for quickly putting those together. Also, Here are some problems on this material from last semester's final exam, including short answers. Some problems from past midterms (not covering the material in Chapters 6 and 7, but still including short answers) are here and here.

The actual exam will have 4 multi-part problems (but with roughly the same total number of parts as on the first midterm), 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 are the answers to problems 1, 2, 3, 4, 5 of the second midterm exam.

The FINAL EXAMINATION will take place Monday, Jaunary 22, at a place and during a 3-hour period to be announced by the Registrar. Update: The Registrar has announced the time and place of all exams. Math 21b is scheduled for Monday, January 22, at 9:15 AM in Sever 113 (as it happens this is the same room where both 21b midterms were held). The exam will emphasize material developed since the second midterm, but will cover earlier topics as well. All material on which homework was assigned is fair game for the final, except the ``Chapter 8.3'' handout, because not all three sections had homework assigned from it. In particular, the syllabus for the final exam includes material from sections 1 through 4 of ``Chapter 10'', but does not include sections 5 and 6 of ``Chapter 10''.

The review sessions for the final exam will take place during the second week of reading period in Science Center Room 507: Monday January 8 from 11AM to 1PM, and Friday January 12 from 10AM to 12 Noon.

Here are copies of recent final exams in Math 21b: May 2004, May 2005 (with solutions at the end(*)), and January 2006 (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.)

Brief solutions for the homework from the ``Chapter 10'' handout are here (remember that you're not responsible for 8.3 on the final); thanks to the CA's for sending these.