Harvard University,FAS

  Fall 2001

Mathematics Math21b
Fall 2001

Linear Algebra
and Differential equations

Course Head: Dale Winter

Office: SciCtr 506
Email: amanita@math
Mainpage Syllabus Sections Calendar Homework Exams Supplements Links

Final Exam

The information that the Registrar has provided to us regarding the location and time of the final exam is given below. The Registrar reserves the right to change the date, time and location of the final exam, so check back here just before the exam. We will update the web site immediately if we hear of any changes.

  • Day: Wednesday January 23
  • Time: 9:15 a.m. to 12:15 p.m.
  • Location: William James Hall, Room 1
  • Allowed Materials: You are allowed to use a calculator and one standard-sized (8.5 by 11 inch) sheet of notes

Review Materials

A collection of problems has been posted on the exams page.

UPDATE: We have posted some additional problems if you have completed all 32 of the practice problems that were posted in December.

Course-wide Review

There will be two course-wide reviews held on the Monday and Tuesday immediately before the exam. The times and places are given below.

  • Monday January 21. 7-9pm. Science Center 309.
  • Tuesday January 22. 7-9pm. Science Center 309.

Office Hours During Reading and Exam Periods

  • Dale Winter
    • Monday January 14. 10am-noon. SC 506
    • Monday January 14. 1pm-5pm. SC 506
    • Tuesday January 15. 10am-noon. SC 506
    • Tuesday January 15. 1pm-5pm. SC 506
    • Wednesday January 16. 10am-noon. SC 506
    • Wednesday January 16. 1pm-5pm. SC 506
    • Thursday January 17. 10am-noon. SC 506
    • Thursday January 17. 1pm-5pm. SC 506
    • Note: No office hours on Friday January 18
    • Monday January 21. 10am-noon. SC 506
    • Monday January 21. 1pm-5pm. SC 506
    • Tuesday January 22. 10am-noon. SC 506
    • Tuesday January 22. 1pm-5pm. SC 506

  • William Stein
    • To be announced

  • Spiro Karigiannis
    • To be announced


This course is intended to provide you with an introduction to linear algebra, dynamical systems, ordinary differential equations and partial differential equations. The bulk of the course is devoted to linear algebra, and the study of:
  • matrices,
  • vectors,
  • linear transformations,
  • vector spaces,
  • orthogonality,
  • determinants,
  • eigenvalues, and,
  • eigenvectors of matrices.
The rest of the course will concentrate on learning to use a collection of techniques for finding and analyzing solutions of differential equations. This part of the course will concentrate on learning to use:
  • techniques for finding symbolic solutions to some special classes of ordinary differential equations,
  • eigenvalues and eigenvectors of matrices to create graphical solutions of systems of ordinary differential equations,
  • the technique of "Separation of Variables" to reduce partial differential equations to ordinary differential equations and find symbolic solutions, and,
  • Fourier series to calculate symbolic solutions to initial and boundary value problems involving partial differential equations.
Uniting these two parts of the course will be a brief excursion into the study of vector spaces from a more abstract point of view. The issues that we will be most interested in will be:
  • the use of axioms to describe the salient features of a vector space,
  • a description of (by then) familiar concepts such as subspace, linear independence, inner product and orthogonailty in terms of axioms,
  • careful examinations of examples that satisfy the axiomatic definition of a vector space, but are quite different from the objects usually thought of as vectors, and,
  • a particularly close examination of vector spaces where the "vectors" are actually functions, and their connection to the technqiues used to solve differential equations.

Format of the Course

Math 21b is taught entirely in sections by teaching fellows, with additional weekly problem sessions conducted by course assistants. Small class sizes allow us to tailor the classes to your needs and to offer you more individual attention.

Course head

Name Dale Winter
Office Room 506, Science Center
Phone (617) 495-9063
e-Mail amanita@math.harvard.edu

I am here to help ensure that the class runs smoothly for you. My main responsibility is to coordinate all of the sections of the class, so that they all run uniformly. To this end, you should feel free to contact me at any point during this semester if any issues arise, such as a family emergency, which might cause you difficulty in keeping up with the class. In general, you should contact your section leader first, to let them know what is going on.

Teaching Fellows

Teaching Fellow e-Mail Address
Spiro Karigiannis spiro@math.harvard.edu
William Stein was@math.harvard.edu
Dale Winter amanita@math.harvard.edu

Course Text

Linear Algebra with Applications. (Second Edition.) by Otto Bretscher.

Available from the Coop. It is important for you to get the second edition of the text, as most of the homework problems will be drawn from the text.


Any graphing calculator will be a tremendous asset in this course. You will be allowed to use your calculators on all tests and examinations.

Attendance and Excellence

I understand that some of you may have already encountered some of the material of this course. I believe that you will strive for excellence in this course, and will be disappointed if your performance does not measure up to your expectations. In this class, we will not have a course-wide attendance policy for attending section. Instead, we will have an excellence policy. As long as you are able to understand the material and produce work that is acceptable to both you and your section leader, you are free to make your own decisions about class attendance. We will follow the schedule of topics on the course syllabus closely. If you are not performing up to the standards that you and your teaching fellow feel appropriate, then we will meet to discuss the situation, and make plans for improvement. I hope that you will get a lot out of class, see class as a profitable use of your time and want to come along each day. My experiences of teaching indicate that students who:
  • read the textbook and class notes on a regular basis,
  • attend class regularly,
  • complete homework assignments on time and to the best of their ability,
  • seek help when they need it,
  • study seriously for tests and exams
tend to have a much better chance of achieving excellent results in the class.

Math Question Center

During the semester, the Math Department operates and staffs a drop-in center where you can go for help with your math courses.
This Math Question Center is staffed by the course assistants from courses Xa through 21b, as well as some of the graduate students and faculty from the Math department.
The Math Question Center is normally open between 8pm and 10pm, Sunday through Thursday. (It is closed on Friday and Saturday night.) At present the Math Question Center is located in Loker Commons. It will open in the next few weeks. For the latest news on the Math Question Center, click here .

Grading, Homework, Tests and Exams

Grade Breakdown

Your semester grade is based on a weighted average of all scores accumulated from the different parts of the course. The table given below shows the weight that will be given to each of the parts of the course.

Component of Coursework Percentage Contribution to Grade
Homework 35%
Mid-terms1 35%
Final Exam 30%
Total 100%

1 Your higher scoring mid-term will be worth 20%, and the other mid-term will be worth 15%.

If your grade on the final exam is higher than the grade from your composite score, then your final grade for the course will be the same as your grade on the final exam.

The Curve

In this class we will have a simple way to convert numerical scores to letter grades. This method is:
Range of Numerical Values Letter
90-100 A
80-89 B
65-79 C
50-64 D
0-49 E

When the course head calculates your final grade at the end of the course, he will calculate a score on a 0-100 scale using the scores that you have obtained during the course and the grade breakdown given above. Your course grade will be then be obtained using this table. In the event of a fractional score, the course head will always round up to the nearest integer. The course head may modify the final letter grades with a "+" or "-" if he and your section leader believe that your performance in the class warrants this. There is only one set of circumstances under which the course head will deviate from the policy outlined above. This will be to ensure that at least 20% of the people in the class get grades of "A" or "A-" and at least 30% of people in the class get grades of "B+," "B" or "B-."


Each week, you will be assigned two homework assignments. The days on which assignments will be given out, and days on which they are due are given below.

Day homework assigned Due date
Monday Friday of the same week
Wednesday Monday of the next week

Solutions to the homework assignments will be posted to the course web site very soon after the homework is collected. In general, no late homework can be accepted. The only exceptions to the regulation will be serious personal emergencies such as hospitalization or an absence from class that is sanctioned by Harvard University. At the end of the semester, you will able to drop the lowest two homework scores. In the few situations where the due date of a homework assignment or lab report falls on a day when an exam is scheduled, the due date for the homework will be oficially postponed to the next class day following the exam, or the homework assignment may be cancelled at the discretion of the course head. Each homework assignment will consist of ten questions, mainly drawn from the text book. Over the course of the semester, the composition of these questions will average out to be something like:
  • 65%: Mathematical operations, solving equations, making calculations. Fairly straight- forward use of course content and concepts that do not involve complicated applications or modeling.
  • 25%: Mathematical modeling of phenomena, starting with a fairly precise and explicit description of the phenomena that can be readily translated into mathematical symbols, graphs, etc. (Problems like this will typically also involve calculations, solving equations, etc.)
  • 10%: Investigations into more complicated phenomena. (This will likely also involve both modeling and mathematical operations.)


The course-wide exams will be given on:
  • Tuesday, October 23. 7:00pm-9:00pm. (Science Center C.)
  • Tuesday, December 4. 7:00pm-9:00pm. (Science Center C.)
  • Wednesday, January 23, 2002. (This is subject to change. Time and place to be announced.)
If you find that you have an unavoidable conflict with either of the first two exams times, please contact the course head at the first opportunity. If you have a truly unavoidable conflict with the final exam time, then you must petition the registrar. Neither the course head nor any of the teaching fellows can change the time or day of the final exam, and are expressly forbidden from making special arrangements for individual students to take the final exam on alternate days or at alternate times. During the exams, you will be allowed to use your calculators and one standard, letter-sized (8.5 by 11 inches) page of notes. Yes, you can write on both sides of the sheet of paper (and the edges too, if you like).

To section: Use any Harvard computer to telnet to 'hilbert.math.harvard.edu'. When prompted to 'login', type 'section'. At the password prompt, press 'enter'.
telnet to hilbert.math.harvard.edu
Follow the online instructions from there. Alternatively, from any web browser go to the Math department home page and click on the sectioning link in the upper right hand corner. If there is a problem with your sectioning assignment, contact Susan Milano in office 308 in the Science Center (milano@math.harvard.edu).

Please send comments to math21b@fas.harvard.edu.