Harvard University,FAS
Spring 2001

Mathematics Math21b
Spring 2001

Linear Algebra
and Differential equations

Course Head: Richard Taylor

Office: SciCtr 539
Email: rtaylor@math
Mainpage Syllabus Sections Calendar Homework Exams Supplements Links


Math21b: An introduction to linear algebra, including linear transformations, determinants, eigenvectors, eigenvalues, inner products and linear spaces. We will further study linear differential equations and applications. [See below for detailed topics list.]

This course is taught entirely in sections (taught by teaching fellows [TF]), with an additional weekly problem session (conducted by a Course Assistant [CA]). Sectioning must be done via computer by noon on Thursday, Feb 1. You will be notified of your assigned section on Friday, Feb 2. Classes will begin on Monday, Feb 5.

Course Head:

Richard Taylor, Office: SC 539, Tel: 495-5487, e-mail: rtaylor@math.harvard.edu
Course website: http://www.courses.harvard.edu/~math21b
Here you'll find here homework solutions, exam review problems and solutions, and course supplements.

Exams: There will be two midterm exams and a Final Exam.

Any changes to the following exam dates will be announced here and in class.

  • Exam #1: Mon, Mar 5, 7:00-8:45pm in Sci Ctr Hall C
  • Exam #2: Tues, Apr 10, 7-8:45pm in Sci Ctr Hall C
[If you think you are unable to take the exams at either of these for any forseeable reason (eg a religeous conflict) you must inform the course head by the end of the first week of classes.]

Grades: Grades: Your overall course grade will be determined according to the following weights:

  • Midterm exams: 20% each
  • Homework: 20%
  • Final Exam: 40%
Text: Linear Algebra with Applications, by Otto Bretscher (1st edition). Available at the Coop and elsewhere.

Homework: Homework will be assigned each class and due at the start of the next class. Graded homework will generally be returned the class after that. Homework solutions will be posted on the Math 21b course web site a few days after homework is due. Your Course Assistant will establish policies regarding submitting late homework. However, it is expected that you will do your homework on schedule.

Homework problems are an integral part of this course. It is impossible to understand the material and do well on the exams without working through the homework problems in a thoughtful manner. Mathematics is not a spectator sport. Don’t just crank through computations and write down answers - think about the problems posed, your strategy, the meaning of the computations you perform, and the answers you get. Nothing prevents you from trying a few more problems in a given section if you feel it may do you some good.

We encourage you to form study groups with other students in the class so you can discuss the work with each other. Your Course Assistant will, upon request, distribute a list of names and phone numbers of those in class in order to facilitate this. Although we encourage you to talk to your classmates, work must be written up independently.

Many of the problems for homework will look different from problems you discussed in class and in the text. This is not an accident. We want you to think about the material and learn to apply it in unfamiliar settings and interpret it in different ways. Only if you understand the material (as opposed to merely knowing it) will you be able to go beyond the information you are given.

Many math students seem to subscribe to the "Ten Minute Rule": If you cannot solve it in ten minutes, you cannot solve it at all. Nothing could be further from the truth, of course. You will probably learn most from those problems which keep you busy more than ten minutes, whether you can ultimately solve them or not.

Math Question Center: In addition to class, problem sessions, and office hours, the Mathematics Department operates a Question Center in Loker on Sunday, Monday, Tuesday, Wednesday, and Thursday evenings from 8pm to 10pm. The Question Center will be staffed by Course Assistants from Math 1a, 1b, 21a, and 21b and by graduate students and others. You are encouraged to use this resource as you do your homework and when questions arise. It is intended to supplement the office hours held by your Section Leader.

Use of technology: In some of the homework problems you will be asked not to use any technology (calculators or software packages). If no restriction is made, you may use the form of technology of your choice, e.g. TI-85 calculator, Matlab, Maple, Mathematica. You may want to arrange to have access to some form of technology. Calculators will not be allowed in the exams.

Ombudsperson: If something wonderful or worrisome comes up in connection with the class let your TF know. You can also contact the course head. In addition there is an Ombudsperson you can reach by e-mailing . Messages sent there will be read by a member of the Mathematics Department who is not teaching calculus this semester, and who can, when appropriate, forward or act on information without revealing its source.

Syllabus: We will cover approximately one section of the text per class (MWF schedule). Your Section Leader will highlight the key concepts introduced in each section, but there may not be enough time to cover all the topics. You will need to study the text to fill in the details. Reading the text is an integral part of the course. On the exams, you will be responsible for all the material discussed in the text and in class. Below is the approximate day-by-day syllabus for the MWF sections of the course. Some topics may be omitted if time is limited.

1: Systems of linear equations
1.1: Introduction to linear systems
1.2: Matrices and Gauss-Jordan Elimination
1.3: On the solutions of linear systems

2: Linear transformations
2.1: Introduction to linear transformations and their inverses
2.2: Linear transformations in geometry
2.3: The inverse of a linear transformation
2.4: Compositions of linear transformations; matrix products

3: Subspaces of Rn and their dimension
3.1: Image and kernel of a linear transformation
3.2: Subspaces of Rn; Bases and linear independence
3.3: The dimension of a subspace of Rn

Review and first midterm.

4: Orthogonality and least squares
4.1: Orthonormal bases and orthogonal projections
4.2: Gram-Schmidt process and QR factorization
4.3: Orthogonal transformations and orthogonal matrices
4.4: Least squares and data fitting

5: Determinants
5.1: Introduction to determinants
5.2: Properties of the determinant
5.3: Geometrical interpretations of the determinant, Cramer's rule

6: Eigenvalues and eigenvectors
6.1: Dynamical systems and eigenvectors: An introductory example
6.2: Finding the eigenvalues of a matrix
6.3: Finding the eigenvectors of a matrix
6.4: Complex eigenvalues and rotations
6.5: Stability

Review and second midterm.

7: Coordinate systems
7.1: Coordinate systems in Rn
7.3: Symmetric matrices

8: Linear systems of differential equations
8.1: An introduction to continuous dynamical systems
8.2: The complex case: Euler's formula
`8.3': Non-linear systems (supplementary notes to be provided)

9: Linear spaces
9.1: An introduction to linear spaces

Further topics in differential equations (supplementary notes to be provided)
10.1: Ordinary linear differential equations
10.2: Fourier series
10.3: Partial differential equations I - The Heat Equation
10.4: Partial differential equations II - Laplace's Equation, the Wave Equation

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