Holy Cross Mathematics and Computer Science
Mathematics 244, section 1 -- Linear Algebra
Syllabus, Spring 2004
Professor: John Little
Office: Swords 335
Office Phone: 793-2274
email: little@mathcs.holycross.edu (preferred) or jlittle@holycross.edu
Office Hours: MW 10 - 12, TR 1 - 3, F 8 - 9, and by appointment
Course homepage: http://mathcs.holycross.edu/~little/LA04/LA.html
What is Linear Algebra About?
Linear Algebra is the study of an extremely important
class of mathematical objects known as vector spaces over a field.
Vector spaces give a common framework for understanding the properties
of vector quantities from physics (those having magnitude and direction),
properties of collections of functions closed under sums and multiplication
by constants, and many other examples which have the same underlying
structure. In the theory, the mappings between vector spaces
that respect the vector space operations -- the linear mappings
or linear transformations -- play a major role. Another major theme will be
the different coordinate systems that can be used to describe elements of a
vector space and give concrete representations of linear transformations
as matrices. Almost all of higher pure mathematics:
- abstract algebra,
- analysis,
- geometry,
and most areas in applied mathematics as well:
- ordinary and partial differential equations,
- numerical analysis,
- discrete and continuous optimization,
- probability and statistics,
- cryptography and error-control coding theory, etc.
make crucial use of the concepts of linear algebra. As a result, this course
is a prerequisite for almost all of the upper-division mathematics courses
taken in the junior and senior years by majors.
Linear Algebra and Algebraic Structures
In the Mathematics major curriculum at Holy Cross,
Linear Algebra is a continuation of the Algebraic Structures course
from the fall. Algebraic Structures is formally a prerequisite
for this course and the material we will study this semester has
many logical connections to topics studied in Algebraic
Structures. I will point out these connections from
time to time but I will try to make this course
as self-contained as possible for those students
who are taking this course without the Algebraic Structures
prerequisite.
Important Exceptions. It will be necessary to assume that you
have a strong working knowledge of
the following topics that were discussed in Algebraic Structures:
- the basic language of sets and properties of
set operations such as union, intersection, complement,
- mappings (functions) and the one-to-one and onto properties,
- relations,
- binary operations,
- proof by mathematical induction.
If you have not taken Algebraic Structures,
and if many of the concepts above are unfamiliar, you
should see me immediately. There are appendices covering these topics
in the complete course textbook that I can copy for
you. You will need to master this basic material to understand
almost everything we talk about in this course.
In addition, even if you have seen most of these topics before,
you may need to make some adjustments in how you think and work
for this course (see below). In particular, the experience gained
by the students who took Algebraic Structures in developing
and writing proofs of mathematical assertions will definitely be
relevant in this course too.
How To Approach This Course
To succeed in this course, it will help to realize
from the start that
- In many problems the goal will be first to develop, and
then to explain clearly, the complete logical argument that
establishes the truth of a general statement.
This means you will probably find you need to take more care
to develop, check, and record your thinking about problems than you
may be used to. Suggestions: Never be in a hurry to
``get assignments over with''. Take your time, think things through
calmly and carefully, and always double- and triple-check your reasoning.
Start problem sets early so you are not pressed for
time at the due date. When things are not ``coming'' to you
put the work aside for a while, tend to your other classes,
take a walk or run, work out in the
wellness center, etc. If you have ``primed'' it sufficiently,
your unconscious mind will be working on
the problem even while you are outwardly doing other things! If after
your best effort, you can see some of the way to a solution but not
all the way, have the integrity to say what you are missing.
- You will have to think about the logical structure of the subject matter
and understand the definitions of concepts and the statements and proofs of
theorems to get the ``skeleton'' of the subject firmly in place in your mind.
Suggestion: Read over your notes and the text
after every class and make
sure you understand what was done that day. Making a ``clean'' copy
of the class notes, with details filled in, original examples worked out,
added comments about things that you had to work to understand, etc.
is also highly recommended.
- You will need to commit a collection of key examples to memory and
be able to reason about their properties. Many problems on the problem sets
will deal with these key examples, and the point of doing the
problem is to help you add these new items to your ``permanent mental
furniture.'' Suggestion: Your involvement with problems
and the facts that are developed through them should not be over
when you turn the problem set in. Be sure you read through
comments on your work when the problem set is returned to see
where corrections or improvements are needed. Keep
your papers so you can refer back to them later.
You will have lots of chances
to develop and practice these new skills, and I will always be willing to
give you the benefit of my experience working with this kind of mathematics!
Even if you find this difficult at first, persistence and openness
to a different way of thinking will usually pay off in time.
Text
The text book for the course is a photocopied ``course pack''
of Chapters 1 - 4 of A Course in Linear Algebra by
David Damiano and John Little (now
formally out of print). We will cover all of the material
in these chapters this semester.
Material We Will Study
During the semester we study the following topics
- Unit I: Vector spaces, subspaces, linear combinations, linear
independence, bases, and dimension (11 days)
- Unit II: Linear transformations, matrices, kernel and image, composition
of linear transformations, inverse transformations, change of basis (13 days)
- Unit III: Determinants (4 days)
- Unit IV: Eigenvalues, eigenvectors, diagonalization, the spectral
theorem for symmetric matrices (8 days)
The other three days will be devoted to in-class examinations.
There is a more detailed day-by-day schedule posted on the
course homepage.
As always, it may become necessary to add, subtract, or rearrange topics.
I will announce any changes in class and on the course homepage.
Course Format
In order for a student to get as much as possible out of this or any
course, regular active participation and engagement with the ideas
we discuss are necessary.
To get you more directly involved in the subject matter of this course,
regularly throughout the semester the class will break down into groups
of 3 or 4 students for one or more days, and each group will work individually
for (a portion of those) class periods on a group discussion exercise.
I will be responsible for designing and preparing these exercises, and
I will be available for questions
and other help during these periods. Each group will keep a written record
of their observations, results, questions, etc. which will be handed in.
The other meetings of the class will be structured as
lecture/discussions.
Grading Policy
Grading for the course will be based on
- Three in-class tests, together worth
40% of the course grade. Tentative dates:
- Friday, February 20.
- Friday, March 26.
- Friday, April 30.
- A three-hour final examination, worth 25% of the course grade. The
final exam will be given at 8:30 a.m. on Monday, May 10.
- Individual homework assignments, given out in class and posted
on the course homepage. The homework
will count as 20% of your course grade. The individual assignments
are a very important part of this course and keeping up to date will
be necessary to succeed with this material.
No credit will be given for late homework,
except in the case of an excused absence.
- Written reports from small group discussions -- one report
from each group. Information regarding the expected format
will be given out with the assignment. Together, worth 10% of the course grade.
- A weekly 5-minute ``definitions quiz'' will be given on the Fridays
when we do not have an in-class exam scheduled, starting on January 30.
Of these 9 quizzes, the best 6 scores will be used for
the final 5% of your course average. The definition of
any term or object introduced in the course before the date of the quiz is
fair game on any of these quizzes.
If you ever have a question about
the grading policy or your standing in the course, don't hesitate to ask me.
Departmental Statement on Academic Integrity
Why is academic integrity important?
All education is a cooperative enterprise between teachers and
students. This cooperation works well only when there is trust and
mutual respect between everyone involved.
One of our main aims as a department is to help students become
knowledgeable and sophisticated learners, able to think and work
both independently and in concert with their peers. Representing another
person's work as your own in any form (plagiarism or ``cheating''),
and providing or receiving unauthorized assistance on assignments (collusion)
are lapses of academic integrity because they subvert the learning process
and show a fundamental lack of respect for the educational enterprise.
How does this apply to our courses?
You will encounter a variety of types of assignments and examination
formats in mathematics and computer science courses. For instance,
many problem sets in mathematics classes and laboratory assignments
in computer science courses are individual assignments.
While some faculty members
may allow or even encourage discussion among
students during work on problem sets, it is the expectation that the
solutions submitted by each student will be that student's own work,
written up in that student's own words. When consultation with other
students or sources other than the textbook occurs, students should
identify their co-workers, and/or cite their sources as they would for
other writing assignments. Some courses also make use of collaborative
assignments; part of the evaluation in that case may be a rating of each
individual's contribution to the group effort.
Some advanced classes may use take-home
examinations, in which case the ground rules will usually allow no
collaboration or consultation.
In many computer science classes, programming projects are
strictly individual assignments; the ground rules
do not allow any collaboration or consultation here either.
What are the responsibilities of faculty?
It is the responsibility of faculty in the department to
lay out the guidelines to be followed for specific assignments in
their classes as clearly and fully as possible, and to
offer clarification and advice concerning those guidelines
as needed as students work on those assignments.
The Department of Mathematics and Computer Science upholds the
College's policy on academic honesty.
We advise all students taking mathematics or computer science courses
to read the statement in the current College catalog carefully and
to familiarize themselves with the procedures which may be
applied when infractions are determined to have occurred.
What are the responsibilities of students?
A student's main responsibility is to follow the guidelines laid down
by the instructor of the course. If there is some point about the
expectations for an assignment that is not clear, the student is responsible
for seeking clarification. If such clarification is not immediately available,
students should err on the side of caution and follow the strictest possible
interpretation of the guidelines they have been given.
It is also a student's responsibility to protect his/her
own work to prevent unauthorized use of exam papers, problem solutions,
computer accounts and files, scratch paper, and any other materials used in
carrying out an assignment. We expect students to have the integrity to say
``no'' to requests for assistance from other students when offering that
assistance would violate the guidelines for an assignment.
Specific Guidelines for this Course
In this course, all examinations and quizzes will be closed-book and given in-class.
No sharing of information with other students in any form will
be permitted during exams and quizzes. On group discussion write-ups,
close collaboration is expected. On the problem sets, discussion of the
questions with other students in the class, and with me during office
hours is allowed, even encouraged.
If you do take advantage of any of these
options, you will be required to state that fact in a "footnote"
accompanying the problem solution. Failure to follow this rule
will be treated as a violation of the College's Academic
Integrity policy.