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 standardsized (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.
Coursewide Review
There will be two coursewide reviews held on the Monday and Tuesday
immediately before the exam. The times and places are given below.
 Monday January 21. 79pm. Science Center 309.
 Tuesday January 22. 79pm. Science Center 309.
Office Hours During Reading and Exam Periods
 Dale Winter
 Monday January 14. 10amnoon. SC 506
 Monday January 14. 1pm5pm. SC 506
 Tuesday January 15. 10amnoon. SC 506
 Tuesday January 15. 1pm5pm. SC 506
 Wednesday January 16. 10amnoon. SC 506
 Wednesday January 16. 1pm5pm. SC 506
 Thursday January 17. 10amnoon. SC 506
 Thursday January 17. 1pm5pm. SC 506
 Note: No office hours on Friday January 18
 Monday January 21. 10amnoon. SC 506
 Monday January 21. 1pm5pm. SC 506
 Tuesday January 22. 10amnoon. SC 506
 Tuesday January 22. 1pm5pm. SC 506
 William Stein
 Spiro Karigiannis
Introduction
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
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
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.
Calculators
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 coursewide 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
dropin 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% 
Midterms^{1} 
35% 
Final Exam 
30% 
Total 
100% 
^{1} Your higher scoring midterm will be worth 20%,
and the other midterm 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 
90100 
A 
8089 
B 
6579 
C 
5064 
D 
049 
E 
When the course head calculates your final grade at the end of
the course, he will calculate a score on a 0100 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."
Homework
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.)
Exams
The coursewide exams will be given on:
 Tuesday, October 23. 7:00pm9:00pm. (Science Center C.)
 Tuesday, December 4. 7:00pm9:00pm. (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, lettersized (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'.
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).
