Offered: Fall 2008
Instructor: Kevin J. Mitchell
Office: Lansing 305
Phone: (315) 781-3619
Fax: (315) 781-3860
E-mail: mitchell@hws.edu
Office Hours: M & W 3:30 to 4:45, Tues 11:00 to 1:30 and Thurs 10:30 to 12:30.
I am often available at other times by appointment.
Class: M-W-F 12:20 to 1:15 in Napier 101.
Final Exam: Thursday, December 18, 2008 at 1:30 PM
Text: Linear Algebra & Its Applications 3rd edition (Updated) by David Lay
Course Website: http://math.hws.edu/~mitchell/Math204F08/index.html
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 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 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 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.
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. About 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. Your work will be due at the beginning of class. No late assignments, please; they will be marked down.
In addition to the graded homework, there will be three hour tests and a final exam on the dates listed 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.
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 one-page typed summary of the talk and what you took away from the presentation.
Your course grade will be determined as follows. Note the exam days and times that extend beyond the usual class time:
Exam 1. October 1 12:20 to 1:45: 16% Exam 2. October 29 12:20 to 1:45: 16% Exam 3. November 24 12:20 to 1:45: 16% Homework & Quizzes (if any): 24% Math Colloquium Attendance: 4% Final Exam December 18 1:30 to 4:30: 24%
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 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.
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.
The outline below is fairly ambitious and will be adjusted as necessary during the term.
| Time | Reading | Topic |
| Weeks 1-4 | Sections 1.1-1.9 | Systems of Equations |
| October 1 (Wed) | Test 1 | 12:20 to 1:45 |
| Weeks 5-6 | Sections 2.1-2.3, 2.6(?) | Matrix Algebra |
| Weeks 7-8 | Sections 3.1-3.2 | Determinants |
| October 29 (Wed) | Test 2 | 12:20 to 1:45 |
| Weeks 9--12 | Sections 4.1-4.6 | Vector Spaces |
| November 24 (Mon) | Test 3 | 12:20 to 1:45 |
| Weeks 13-14 | Sections 4.9 (?), 5.1--5.3 | Markov Chains (?), Eigenvalues and Eigenvenctors, Diagonalization |
| December 18 (Thurs) | Final Exam | 1:30-4:30 |
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.
Hobart and William Smith Colleges encourage students to seek the academic collaboration and resources that will enable them to demonstrate their best work. Students who would like to enhance their study skills, writing skills, or have other academic inquiries should contact the CTL. You may visit the CTL web site to learn more about the services and programs that are available. http://www.hws.edu/academics/ctl/index.aspx.
If you are a student with a disability for which you may need accommodations, you are required to register with the Coordinator of Disability Services at the CTL and provide documentation of the disability. Services and accommodations will not be provided until this process is complete. The web site for information pertaining to registration with the CTL and documenting disabilities is: http://www.hws.edu/studentlife/stuaffairs_disabilities.aspx.
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