MCS-221 Linear Algebra

Announcement

Catalog description: An introduction to the theory and applications of linear algebra. Topics include vector spaces, matrices, linear transformations, determinants, eigenvalues and eigenvectors, and inner product spaces.
Prerequisite: MCS-121 or MCS-131. MCS-220 is recommended prior to or concurrently with MCS-221.

What is linear algebra?

Linear algebra is a branch of mathematics that is remarkable for its breathtaking beauty and ubiquitous utility. It provides a compact, expressive language for the description of models arising in a wide variety of fields of study, including biology, chemistry, computer science, data transmission, ecology, economics, engineering, image processing, medicine, physics, psychology, sociology, and statistics, as well as other areas of mathematics. Its study typically is a requirement for students of engineering, science, and mathematics.

The fundamental theme of linear algebra is the solution of systems of linear equations, like

2 x + 3 y = 4 and 5 x - 6 y = 7. The equations are called "linear" because geometrically they describe lines (or planes or higher-dimensional counterparts). The central equations of linear algebra, in matrix notation, are the linear system A x = b and the eigenvalue problem A x = m x. At any given time, a significant fraction of the planet's computing resources is involved in solving such problems.

The fundamental concept of linear algebra is the vector space. This notion embraces lines, planes, and Euclidean 3-dimensional space. But vector spaces include much, much more. The student who masters this abstraction will obtain a key to understanding a wide variety of problems. Another key concept is that of a linear transformation between vector spaces. This abstract concept has concrete representations in the form of rectangular arrays of numbers called matrices.

MCS 221 students will be encouraged to understand the fundamental ideas of linear algebra the GAC way: geometrically, algebraically, and computationally. They will apply their understanding both to "problems to solve" and to "problems to prove." Through a variety of applications, students will gain an appreciation of the widespread usefulness of linear algebra. In addition, students will find themselves seduced by the beauty of the subject as they gain experience in mathematical abstraction and logical deduction.

Instructor: John Holte

Office: Olin 307
Telephone: x7465
Office hours: M 1:30-2:20; T 9:00-9:50; W 10:30-11:20, 12:-30-1:20; R 1:30-2:20
Feel free to see me at other times as well.
E-mail: holte@gac.edu
My homepage: http://www.gac.edu/~holte
Course homepage: http://www.gac.edu/~holte/courses/mcs221/fall00

Textbook

• Schaum's Outline of Theory and Problems of LINEAR ALGEBRA, second edition
  by Seymour Lipschutz
  McGraw-Hill, New York, 1991
• You will be asked to consult another text of your choice during the course.

Class meetings

• 8:00-8:50 Monday, Tuesday, Thursday, and Friday in Olin 318
• Regular attendance is expected. Class time will be used for a variety of activities--lectures, discussion, Maple labs, problem solving, and presentation of solutions. You should come to each class prepared for the day's work.
• If you have a disability that requires an accommodation, please see me as soon as possible.

Homework and collaborative learning

I encourage you to work with other students in learning the material, both in class and on homework, provided that you do so in such a way that everyone in your group learns the material. You should write your solutions neatly and show clearly how you get your answers, at least in nontrivial problems. Your write-ups should be done independently, in your own words; solutions should not be copied from someone else's paper. You must give credit to those who helped you, including fellow students, and if you get part of your solution from any text (including ours) or internet source, you must cite it so that it can be traced.

Homework rules

• Acknowledge your sources (people and texts).
• In nontrivial problems, show how you get your answers.
• Turn in neat, well-written solutions, not messy first drafts. Trim "fringes."
• Do not copy collaborative solutions; write up solutions in your own words.
• Turn in homework on time. Each class day late reduces the possible points by 25%.
• Do extra credit problems entirely on your own.

Problem assignments

Check here at http://www.gac.edu/~holte/courses/mcs221/fall00/problems.html   for assignment updates.

MCS 221 Syllabus--Fall 2000

Chapter	Class dates	Tests
1. Systems of linear equations	9/7-9/14
2. Vectors in Rn and Cn, spatial vectors Review	9/15-9/22	Monday Sept. 25
3. Matrices	9/26-9/28
4. Square matrices, elementary matrices	9/29- 10/10	Thursday Oct. 12
5. Vector spaces, I	10/13-10/19
Fall Reading Break	10/20-10/23
5. Vector spaces, II	10/24-10/30	Tuesday Oct. 31
6. Inner product spaces, orthogonality	11/2-11/10
7. Determinants	11/13-11/16
8. Eigenvalues, I	11/17-11/20	Tuesday Nov. 21
Thanksgiving Break	11/23-11/26
Eigenvalues, eigenvectors, diagonalization	11/27-11/30
9. Linear mappings	12/1-12/5
10. Matrices & linear mappings Review	12/7-12/12
Final Examination	8:00-10:00 a.m.	Tuesday Dec. 19

Academic honesty

Page 31 of the Gustavus Adolphus College Academic Bulletin states in part: "The faculty of Gustavus Adolphus College expects all students to adhere to the highest standards of academic honesty... In all academic exercises, examinations, papers, and reports, students shall submit their own work. Footnotes or some other acceptable form of citation must accompany any use of another's words or ideas." See pp. 68-69 of the Gustavus Guide for details.

Make-up policy

Make-up exams will not be given except for medical or family emergency reasons. Students who will be absent from an exam for a school-sponsored event should arrange with me in advance to have an exam sent along with the coach. Late homework will be penalized as stated under Homework rules.

Grading

Assuming regular attendance, your work will be weighted as follows:
• 60%   Tests
  • You will have four unit tests and one final examination. The best four will be counted at 15% each.
• 25%   Problem assignments
  • Regular homework problems
  • Proof portfolio
  • Maple exercises
  • Book report
• 15%  Attendance, participation, and performance
  • Presentation of problem solutions in class
  • The following factors also contribute to participation and performance.

    Positives	Negatives
    Faithful attendance This counts a lot!	Missing classes, tardiness
    Being prepared	Being unprepared
    Paying attention in class	Not paying attention, sleeping, doing something else, talking while the prof is talking
    Contributing to class discussions	Disruptive behavior
    Asking relevant questions-- I am impressed when I am asked more often about points of mathematics than points of grading.
    Answering questions in class, whether asked aloud or on a quiz	Not knowing the answers
    Curiosity, appreciation, cheerfulness	Apathy, resentment, sullenness
    Turning work in on time	Turning work in late
    Neat, well-written work	Messy work
    Working hard	Hardly working
    Improvement during the term, overcoming setbacks	Going downhill
    Attendance the day before and the day after a break	Skipping class the day before or the day after a break