Math 204: Linear Algebra

Offered:     Winter 1997Instructor:  Kevin J. MitchellTime:        Class: MWF 2:40 to 3:50 in NP 201                          Text:        Elementary Linear Algebra, 7th Ed. by Howard Anton             Calculator:  TI-82

About Math 204

Math 204 serves as an introduction to the core of the mathematics curriculum. Unlike in a calculus course, students in this course are preumed to have a serious and enduring interest in mathematics. Most students who take this course go on to major or minor in mathematics or a related field. Correspondingly, there is a seriousness of purpose that I expect in your approach to this course. Generally, students who take Math 204 have done well in their previous mathematics courses. Consequently the pace is quicker and there are no labs. You will need to be more independent to be successful in this course, for example you will need to do more problems and create more examples on your own.

The content of Math 204 is used throughout most upper level mathematics courses, whether they are applied or theoretical. The main object of study, a vector space, is sufficiently general so that many "systems" fall under this category. From physics, you may be familiar with vectors as "arrows" in 2- or 3-dimensional space that represent forces.But there are also infinite-dimensional vecotr spaces such as the set of all differentiable functions whose domain is all real numbers. The complexnumbers are a two-dimensional real vector space

What allows us to call all of these objects vector spaces is their underlying sturcture.Math 204 is important because it focuses on general structures. What good is it to recognize that a system has the structure of a vector space? The point, of course, is that we (you) will prove theorems about all vector spaces in general which can then be applied to any particular vector space encountered. Once you know a system has the structure of a vector space, lots of other structure automatically follows.

The ohter key notion in the course is idea of a linear transformation which generalizes the notion of a function. Linear transformations tell us how we can associate the elements or "vectors" of one vector space with those of another. Examples include geometrical transformations such as rotating a plane about the origin or reflecting theplane in a line. Differentiation which takes one set of functions and "maps" them to another set, or multiplication by i which takes one complex number and produces another are also lineartransformations.

Vector spaces and their transformations are used to model a wide variety of phenomena from how a lumber company should harvest trees in a forest, to how 3-dimensional objects should be drawn on a 2-dimensional surface such as a computer screen, to predicting which team will win the World Series, and how many games it is likely to take.

Text and Calculator

The Department of Mathematics has used Anton's linear algebra text for 15 years. Students often find the material at the beginning easy or mechanical and get "lulled to sleep." The material at the end of the course is much more abstract and requires your closest attention. There are lots of problems, both easy and difficult, in the text. Try more than just the ones that I assign!

I assume that you have (or have access to) a TI-82 graphing calculator. While it is not central to this course, you may find it handy at the beginning when doing simple matrix operations.

Assessment

Homework reading and practice exercises will be assigned at the beginning of each class. I encourage working in small groups on practice problems. This can be very helpful in understanding the material. Come by individually or in small groups for help when you need it. About once a week, there will be an assignment consisting of selected problems to hand in for grading. About every other week, these homework exercises will include some non-routine discussion problems.Unless otherwise stated graded assignments are to be your own work without collaboration. Your work will be due at the beginning of the next class. No late assignments, please.

In addition to the graded homework, there will be three hour tests and a final exam. The dates are listed in the outline below. The hour tests will be cumulative but will concentrate on more recent material. It is impossible to construct fair makeup exams in mathematics. For your own protection, my policy is that there are no makeup examinations. If for some extraordinary reason you find you are unable to take an exam, let me know as soon as possible, certainly well before the exam is administered.

Your course grade will be determined as follows.

3 one-hour exams: 25% eachfinal exam:   30% homework:     20%

I also reserve the right to take class participation and attendance into account in determining final grades.

Because of the nature of this course, its assignments, and its assessment, your attendance and participation is crucial. Mathematics is learned by regular, sustained, attentive effort over an extended period. Only when such effort has been invested will the concentrated study for an exam have any benefit. Therefore attendance at class are required. Unexcused absences may adversely affect your grade; certainly more than three absences will lower your grade. If you must miss a class or lab for some reason beyond your control, talk to me about it in advance.

Office Hours

My office is located in Lansing 305. My extension is 3619. My scheduled office hours are:

    M-W-F 10:10 to 11:10     Thurs   1:00 to 2:00

I am often in my office at other times of the day (e.g., before class), and I encourage you to drop in to get hints or help with course assignments or just to chat. My e-mail address is mitchell (or mitchell@hws.edu from off-campus).

Tips for Success

Here are a few simple things that you can to be more successful in the course

Come to all classes.
Do the readings and practice problems carefully.
Keep the answers to problems and notes on the readings in a journal.
Review the notes from the last class before coming to class again. In fact, I suggest recopying the notes! This makes studying for a test easy.
Come in for help whenever you need it.
Ask questions in class. Ask questions about the homework and readings.

Outline of Weekly Readings

This assumes a fairly rapid pace through the listed materials. We will adjust this schedule based on our actual work in class.

January 6--10: Motivation and introduction, solving linear systems, matrices and basic operations. Chapter: 1.1--1.4.
January 13--17: More on matrix algebra. Chapter: 1.4--1.7.
January 20--24: Determinants. Chapter 2.
January 27--29: The geometry and algebra of vectors in 2space and 3-space. Chapter 3.1--3.2.
Friday, January 31: First Hour Test.
February 3--7: Orthogonality and its uses. Chapter 3.3--3.5.
February 10--14: Euclidean Vector Spaces. Chapter 4.
February 17--21: More on Euclidean Vector Spaces; General vector spaces and their subspaces. Chapter 5.1--5.2.
February 23--25: Linear Independence and Spanning sets. Chapter 5.3--5.4.
Friday, February 27: Second Hour Test.
March 2--6: Basis and Dimension and its applications. Chapter 5.5--5.7.
March 9--13: General Linear Transformations. Chapter 8.1--8.2.
Final Exam on 20 March from 8:30 to 11:30.