Syllabus -- Linear Algebra, section 1, Spring 2007

Holy Cross Mathematics and Computer Science

Mathematics 244, section 1 -- Linear Algebra

Syllabus, Spring 2007

Professor: John Little
Office: Swords 335
Office Phone: 793-2274
email: little@mathcs.holycross.edu (preferred) or jlittle@holycross.edu
Office Hours: MWF 10-12, TR 1-3, and by appointment
Course homepage: http://mathcs.holycross.edu/~little/LA07/LA.html

What is Linear Algebra About?

Linear Algebra is the study of an extremely important class of algebraic structures known as vector spaces. 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.

Important Notes: 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 another course textbook that I can copy for you if you wish. You will need to master this basic material to understand the topics we talk about in this course. In addition, the experience gained in Algebraic Structures in developing and writing proofs of mathematical assertions will definitely be relevant in this course too and we will work on extending your fluency in this area.

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 for the course will be Linear Algebra and Its Applications, 3rd edition, by David C. Lay (publisher: Pearson-Addison Wesley). We will cover most of the material in Chapters 1 - 7 (the non-online portion) over the course of the semester.

Material We Will Study

A rough outline of the material for the course is as follows:

Unit I: Linear equations and an introduction to linear algebra via Rⁿ (7 days)
Unit II: Matrix algebra and applications (6 days)
Unit III: Determinants (4 days)
Unit IV: General vector spaces and linear mappings (8 days)
Unit V: Eigenvalues, eigenvectors, diagonalization, (6 days)
Unit VI: Inner product spaces and the spectral theorem for symmetric matrices (6 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 16.
- Friday, March 23.
- Friday, April 27.
A three-hour final examination, worth 25% of the course grade. The final exam will be given at 8:30 a.m. on Thursday, May 10.
Individual problem sets, posted each week on the course homepage. The problem sets 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 10-minute ``definitions quiz'' will be given on the Fridays when we do not have an in-class exam scheduled, starting on January 26. Of these 9 quizzes, the best 5 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.