Math 204: Linear Algebra

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 presumed to have a serious and enduring interest in mathematics. Most students who take this course continue 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 in your study 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, vector spaces, 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 other real 2- and 3-dimensional spaces, for example, the set of complex numbers or the set of quadratic polynomials, respectively. Further there are also infinite-dimensional vector spaces such as the set of all differentiable functions.

What allows us to call each of these objects a vector space is a shared underlying general structure and it is precisely this common structure which is the focus of Math 204. But 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 satisfies the properties of a vector space, lots of other structure automatically follows.

The other key notion in the course is idea of a linear transformation, which is a special type of 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 the plane in a line. Other examples include 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.

Vector spaces and their transformations may be 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. As time allows, we will investigate various applications.

Maple Software

During the term I will expect you to do some of your work using Maple. Maple is a computer algebra system that can handle a wide variety of mathematical calculations, not just linear algebra. You will be able to use Maple to check some of your homework answers, though you will still need to show me intermediate hand calculations. More importantly, Maple will make it possible for you to do more realistic problems that involve many variables and more complex calculations. I may do a few demonstrations in class, but I hope that you will pick up much of the syntax on your own from some of the examples that I post online.

The Colleges have a site license for Maple, but it the software can only be accessed from a computer on the Colleges' network (e.g., in Gulick, the Library, Stern). For a sample of the type of problem that is hard to do by hand but becomes easy with Maple, read about interpolating polynomials on page 26 in your text, and then read problem 34 on page 27.

Expectations and Assessment

Homework reading and practice exercises will be assigned at the beginning of each class. It is extremely important to do the practice problems and I encourage working in small groups on them. This can be very helpful in understanding the material. Come by individually or in small groups for help when you need it. Once or twice a week, there will be an assignment consisting of selected problems to hand in for grading. Unless otherwise stated graded assignments are to be your own work without collaboration. Assignments will be due at the beginning of class. No late assignments, please; they will be marked down. Once in awhile we may have a five-minute quiz on definitions. Any such quiz would be announced in advance.

In addition to the graded homework, there will be three hour tests and a final exam on the dates listed below. The final exam may include a project component using Maple that would be assigned late in the term and would be due at the final exam. 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.

As prospective mathematicians it is important to participate in a wide variety of mathematical activities. Hence, an additional requirement is to attend one Math/CS Department Colloquium and write a half to one-page review which may include a summary of the talk and what you took away from the presentation, what you liked, etc.

Your course grade will be determined as follows. Note the exam days and times that extend beyond the usual class time:

Exam 1. Monday, September 28:             16% 
Exam 2. Monday, October 26:               16%
Exam 3. Monday, November 23:              16%
Homework (and Quizzes, if any):           24%
Math Colloquium Attendance:                4%
Final Exam Friday, December 18:           24%

Because of the nature of this course, its assignments, and its assessment, your attendance and participation are 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 is required. Unexcused absences may adversely affect your grade; certainly more than three absences will lower your grade. If you must miss a class for some reason beyond your control, talk to me about it in advance.

Finally, common courtesy demands that you be on time for class and that you do not leave the room during class (unless you are ill). This will help you, your classmates, and me to give our full attention to the course.

Tips for Success

My best advice is to take good, complete notes during class. Even if you don't understand every detail during a lecture, with some patience you should be able to review each day's lecture and understand everything we did. If you don't, then you should come to see me. Here are a few simple things that you can do to be more successful in the course:

Come to all classes.
Did I mention that you should take good, complete notes? If I think it is important enough to write down, so should you!
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. (Come to see me about any material that still confuses you.) In fact, I suggest recopying the notes! This makes studying for a test easy.
Come to see me for help whenever you need it, preferably before you get too far behind.
Ask questions in class. Feel free to interrupt me when you are confused!
Visit the course website http://math.hws.edu/~mitchell/Math204F09/index.html which will have additional information, notes, and answers to previously assigned work.

Office Hours

My office is Lansing 305. My extension is 3619. My e-mail address is mitchell@hws.edu. Scheduled office hours are listed at the top of this sheet. I am often in my office much of the day; drop in to get hints or help with course assignments or just to chat.

Outline of Weekly Readings

The outline below is fairly ambitious and will be adjusted as necessary during the term.

Dates	Reading	Topic
Weeks 1-4	Sections 1.1-1.9	Systems of Equations
September 28 (Monday)	Test 1	Chapter 1
Weeks 5-6	Sections 2.1-2.3, 2.6?	Matrix Algebra
Weeks 7-8	Sections 3.1-3.2	Determinants
October 26 (Monday)	Test 2	Material from Chapters 2 and 3
Weeks 9--12	Sections 4.1-4.6	Vector Spaces
November 23 (Monday)	Test 3	Chapter 4
Weeks 13-14	Sections 4.9?, 5.1--5.3	Markov Chains?, Eigenvalues and Eigenvenctors, Diagonalization
December 18 (Friday, 1:30-4:30)	Final Exam	Cumulative

Additional Resources and Reserve Materials

If you purchased a new edition of the text, then there is a Study Guide on a CD that accompanies the text. You may find it useful to supplement the lectures. I encourage you to look at.

There is a course website where I will be posting assignments, some "lecture notes" that supplement what we do in class, and the answers to homework assignments (after they are due). Please bookmark this site: http://math.hws.edu/~mitchell/Math204F09/index.html. Several students have found this site extremely helpful in the past.

As noted earlier, I hope that you will learn to use Maple this term to be able to do more complicated applications or projects using this software. Here are some resources which can be accessed easily from links in the online version of this syllabus or at the course website. To use Maple you will need to be logged on to one of the Colleges' networked computers (e.g., in Gulick, the Library, Stern, Lansing 310).

Using Maple for Linear Algebra part 1: Matrix entry. YouTube Video. This is an excellent video targeted for new users. It is very helpful. Have the Maple software open when you start the video so you can follow along, pausing the video as needed.
Using Maple for Linear Algebra screencast 2. YouTube Video. The second part of the video above.
Maple Quick Start Video. 40 Minutes long, but you might want to watch the beginning. You should have Maple open so you can try commands as they are described in the video. The video does not cover much linear algebra, but you will see lots of things you can apply to calculus.
Maple Quick Start PDF. 18 pages. A quick intro to Maple that goes along with the Quick Start Video above.

The following texts have been placed on overnight reserve at the library. You may wish to consult them for a different point of view on the material we cover.

Elementary linear algebra / Howard Anton
Elementary linear algebra / Stanley I. Grossman
Elementary linear algebra, with applications / W. Keith Nicholson
Introduction to linear algebra / Roger C. McCann

A Note about the Center for Teaching and Learning (CTL)

Hobart and William Smith Colleges encourages students to seek the academic collaboration and resources that will enable them to do their best work. Students who would like to enhance their study skills, writing skills, or other academic skills may visit the CTL web site at: http://www.hws.edu/academics/ctl/index.aspx or contact the CTL at x3351.

Disability Accommodations: If you are a student with a disability for which you may need accommodations, you should self-identify and register for services with the Coordinator of Disability Services at the Center for Teaching and Learning (CTL), and provide documentation of your disability. Disability related accommodations and services generally will not be provided until the registration and documentation process is complete. The guidelines for documenting disabilities can be found at the following website: http://www.hws.edu/disabilities

Please direct questions about this process or Disability Services at HWS to David Silver, Coordinator of Disability Services, at silver@hws.edu or x3351.