Mathematics 21b, Spring 2006

Linear Algebra and Differential Equations

course head: Clifford Taubes
office: science center 504
email: chtaubes
office hours: Tue 10-11:30, Fri 1-2

Course information

Course Head: Clifford Taubes, Science Center 504, email chtaubes@math. Drop in office hours on Tuesdays 10-11:30 and Fridays 1-2.

Prerequisites: Math 1b with a satisfactory grade, or AB-BC score of at least 4, or scores of at least 20, 8, 4 on the respective three Harvard University Math Placement Tests. Moreover, knowledge of multi-variable calculus as taught in Math 19 or Math 21a is recommended.

The sections: This course is taught in relatively small, separately meeting sections rather than in a big and rather impersonal room all together. All sections are deemed 'regular' sections except for one specially named section, the 'bio/statistics' section. Some information about how to choose a section is given below.

Textbooks: All sections require the third edition of Linear Algebra with Applications by Otto Bretscher. The publisher is Prentice Hall. Note that it is crucial that you get the third edition. Those in the bio/statistic section will be given a set of lecture notes to cover the probability and statistics material in the course. These folks must also obtain the book Probability Theory, a Concise Course, by Y. Rozanov. This book is published by Dover. (Those who took the BioChem section of this past fall's Math 21a should already have this book.)

Class meeting on February 1: The first class meeting, which everyone should attend, is on Wednesday, February 1 at 8:30 am in Science Center lecture hall D. This is going to be a roughly 1/2 hour meeting. As short as it is, don't miss it. Except for this one meeting and for the course wide exams, you meet in your assigned section.

Class meetings and problem sessions: The section meets for a total of three hours per week, either one hour each on Mondays, Wednesdays and Fridays, or for one and one half hours each on Tuesdays and Thursdays.
Each student is also assigned to a 1-hour math problem session, conducted weekly by a course assistant. The meeting time for the problem session will be arranged in your section during the first week of classes. You may attend more than one problem session per week; and the schedule of all problem sessions will be posted on the Math 21b website and on the Calculus Office bulletin board outside of Science Center 308.
The first meeting of MWF sections is on Monday, February 6 at the posted time. The first meeting for the TTh sections is on Tuesday, February 7.

Homework: Problems are assigned each meeting of your section. These are due in the first scheduled section meeting of the subsequent week except if stated explicitly otherwise. However, you are strongly urged to work the problems that are assigned in any given section meeting before the next section meeting, rather than collecting the bunch for the week and working them all at once. Here is the reason: The problems that are assigned at a given section are designed to help you digest the material so as to be ready for its applications in the subsequent section. In particular, you will more easily and efficiently follow the lecture in the subsequent section if you have already done the previous section's homework. In any event, you are very much encouraged to discuss the homework with your fellow students and to form study groups to work these assignments. However, you must write up the solutions by yourself, and you must note the names of your coworkers somewhere on the homework. (This last point is simply a matter of professional ethics.) The lowest homework score will be disregarded when your average homework grade is computed. The homework assignments will be posted on the Math 21b web site. The answers to the homework assignments will appear after the due date on the web site as well. Moreover, selected problems from the homework will be discussed in the problem sessions. Homework assignments that are submitted after their assigned due date will be accepted only at my discretion. In any event, no more than two late homework assignments will be accepted per student over the course of the semester. In addition to the weekly homework assignment, various problems of a more routine sort will be suggested for the subsequent class meeting. These are not to be turned in as their answers are either in the text book or will be provided otherwise. However, you are strongly urged to work them on your own or with others in the class because their purpose is to supply practice with the techniques and ideas that are presented in the lectures. By the way, you are also encouraged to try on your own other problems from the text to hone your ability with the concepts and techniques. Don't feel that you should limit yourself to the suggested or assigned problems. In this regard, note that the linear algebra text has answers to most of its odd numbered problems. Meanwhile, the probability book supplies answers to many of its problems.

Exams: There will be single course-wide midterm exam and a final exam. The midterm will take place on Wednesday, March 22 from 7-9pm in Science Center lecture halls A and B. Be sure to mark this date on your calendar now, as no make-up will be given. The final exam is scheduled by the University for a date in May. According to the Course Catalogue, the preliminary schedule has the final on Friday May 19 (Exam Group 3). The University will confirm this date later in the semester.

Grading: Your final grade will be based on your performance on the homework (30%), the midterm (30%), and the final (40%). A small upward adjustment in the grade is possible when the final is dramatically better than the average of the midterms and the homework. The Bio/statistics section will also have an optional extra credit project that can be used to boost the grade.

Computers and calculators: This course is teaching various concepts whose applications may or may not be facilitated by a computer. Even so, without a strong understanding of the underlying concepts, the computer will be of minimal use to you in your future scientific career. The point here is that there are now many widely available computer programs that will solve linear algebra problems in the blink of an eye. Yet, none of us will live to see the day when a computer can turn processed pig feed into gold. Thus, you should avoid at all costs using computer programs to help with your homework assignments. In any event, the use of computers and other electronic aids will not be permitted during exams. (Bring only your brain and some pencils.)

Words of Caution and Advice: This course may well be more demanding then your previous mathematics courses at Harvard and elsewhere. In particular, any given assignment might take more time than expected, and you should plan now to set aside regular hours to wrestle with them. It is virtually impossible to do well in this course without working the homework assignments in a timely fashion. Note also that this course is fast paced, and new material builds on old. Thus, do not fall behind. If you find yourself falling behind, please contact your section's teacher immediately to discuss options for personal help. Indeed, Harvard provides many services along these lines for its students, and your section teacher can help you find them. When you are working your assignments, keep in mind that your success in this course will require more than just memorizing formulas and "plugging in values". In this regard, you will consistently be asked to battle with homework and exam problems that differ significantly from material discussed in class.

Here is the key to success:

Understand the underlying concepts and then work enough problems so that you can employ them in any example thrown at you.

Choosing a section: Each of you is assigned a section based on your request (instructions for how to make this request are given below). You should attend all meetings of your assigned section, hand in your homework to the assigned section, and collect your graded homework and graded exams from the assigned section. Most sections will be of size on the order of 25 students. As noted above, there are two sorts of sections, all but one deemed 'regular', and the other 'bio/statistics'. The various regular sections teach the same material on the same week to week schedule, give the same weekly homework assignments, and take the same version of the midterm exam and final exam. If you opt for a regular section, choose the one whose scheduled meeting time best fits with your schedule. The curriculum as planned for the bio/statistics section differs substantially from that of the regular sections starting from the very beginning. Thus, transfer to and from the bio/statistics section after the first week of lectures is very much discouraged. I teach the bio/statistics section from 11:30-1 on Tuesdays and Thursdays. What follows tersely summarizes the curricula of the two sorts of sections: The regular sections cover a great deal of linear algebra. Any one completing a regular section of this course will see all of the basics of linear algebra that might be required for higher level courses in the physical sciences, social sciences, and in mathematics, both pure and applied. In particular, the regular sections teach

methods to solve systems of linear equations,
methods to analyze and solve systems of linear differential equations,
methods to solve discrete linear dynamical systems such as Markov processes,
least square fitting of data with arbitrary function sets,
the basics of Fourier series and its use to solve partial differential equations.

The bio/statistic section will cover less linear algebra than the regular sections, providing instead basics of statistics and probability. Moreover, the linear algebra and also the differential equations will be taught using examples from statistics and probability, with many coming from applications in the life sciences and in bioinformatics. In this regard, no apriori background in the life sciences is required here; this section makes sense for those with interests in other physical sciences and in the social sciences as well. Those in the biostatistics section of this course will see the core of the linear algebra and differential equations from the regular sections, and also see enough statistics and probability to take various higher level statistics courses (such as Stat 111 and 139).

By the way, successful completion of any section of this course will provide you with a level of sophistication in mathematics that you will not get by learning linear algebra elsewhere, either at Harvard or at any other institution. This sophistication alone will serve you well in your future science courses, and in your future scientific career.

Which section? Which kind of section should you choose? Here is some advice:

Those who should take the regular section:
a) If you contemplate physics beyond Physics 11a,b, then you are better off in a regular section.
In particular, if you plan to take the Physics 15-16 sequence, or are planning to be a physics concentrator, then you should enroll in a regular section.
b) If you are planning to be a mathematics concentrator, then you should also enroll in a regular section.
c) If you are planning to concentrate in engineering, you might be better off in a regular section.
Those who should take the bio/statistics section:
a) Anyone enrolled in Life Science 1a or 1b, or who plans to take either of these courses is strongly urged to take the bio/statistics section.
b) If you are planning to concentrate in Biology or BioChemical Sciences, then you are strongly urged by those departments to enroll in the bio/statistics section.
c) You will also be better served in the bio/statistics section if you are planning to concentrate in Environmental Science and Public Policy.
Those who can take either section:
a) Chemistry concentrators who are not planning to take advanced physics courses will find the bio/statistics section more useful than the regular section.
b) If you plan to major in Economics or other social sciences, the bio/statistics section might also be your best choice.
c) If you took Math 19, the bio/statistics section for Math 21b is a perfect follow-up.
d) If you took the BioChem section of Math 21a, the bio/statistics section of this course is a great way to learn more statistics and probability.
e) If you are planning to take the Physics 11a,b sequence, either a regular section or the bio/statistic section is appropriate.

No worries mate:
No matter what, you can't go wrong in a regular section so if you aren't sure of your concentration, take a regular section. You can always learn probability and statistics in a later course.

Sectioning: Here are the instructions for making a section request: If you have an email account, log on to the Harvard computer system, and instead of typing "pine," type

ssh section@ulam.fas.harvard.edu

and follow the instructions. Sectioning starts Monday January 30 and ends at noon on Thursday, February 2. Section assignments are emailed at 5 on Friday, February 3. If you are having trouble sectioning, if you miss the sectioning period later, or if there is a problem with your section assignment, contact Susan Milano via email at milano@math.harvard.edu.

Math21b, Linear Algebra and Applications, Spring 2006, Clifford Taubes, Harvard University, Faculty of Arts and Sciences, Harvard University