The College of the Holy Cross

Mathematics 44, section 1 -- Linear Algebra

Syllabus, Spring 1999

Professor: John Little
Office: Swords 335
Office Phone: 793-2274
email: little@math.holycross.edu or jlittle@holycross.edu
Office Hours: MWF 10-12, TR 1-3, and by appointment.

Course Description

Linear Algebra is the continuation of the Algebraic Structures course from the fall and will draw heavily on some of the ideas introduced there. The major topics we will deal with this semester all concern the properties of an extremely important class of algebraic structures known as vector spaces over a field and the mappings between vector spaces that preserve the operations defined in the vector space. Almost all of higher pure mathematics -- abstract algebra, analysis, geometry, differential equations, etc. -- and most areas where mathematics is applied to solve real-world problems use the concepts of linear algebra in some form. 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. As was true last semester, in almost all of our work this term, the goal will be first to develop, and then to explain clearly, the complete logical argument that establishes the truth of a general statement, starting from a clearly stated set of assumptions and using results established previously. 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. You will need to commit a collection of key examples to memory and be able to reason about their properties. You will need to document much more of your thinking about problems than you probably have done before. While a good intuition may guide you to correct statements, just making correct statements will not be enough.

Text

The text book for the course is Linear Algebra by L. Smith, 3rd edition. We will cover most of the material in Chapters 1-11 and 13-16 this term.

Material We Will Study

During the semester we study the following topics

• Unit I: Vector spaces, linear combinations, linear dependence and independence, bases, dimension (about 9 days; Chapters 1-7 of Smith)
• Unit II: Linear transformations, matrix representations, change of basis, solving systems of linear equations (about 17 days; Chapters 8-11,13 of Smith)
• Unit III: Eigenvalues/eigenvectors, diagonalization of linear transformations, inner product spaces, the Spectral Theorem for self-adjoint transformations (about 11 days; Chapters 14-16 of Smith)

The remaining three class days will be devoted to in-class exams. See below for dates. A more detailed, day-by-day schedule is available for your consultation. 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

1. Three in-class tests, together worth 45% of the course grade. Tentative dates:
   • Friday, February 19.
   • Friday, March 26.
   • Friday, April 23.
2. A three-hour final exam, worth 30% of the course grade. The final examination will be given at 2:30 p.m. on Tuesday, May 11.
3. Written reports from small group discussions -- one report from each group. Information regarding the expected format will be given out with the first assignment of this kind. Together, worth 10% of the course grade.
4. Individual homework assignments, given out in class. The homework will count as 15% 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, or with my permission.

If you ever have a question about the grading policy or your standing in the course, don't hesitate to ask me.