Linear Algebra

Overview

Lectures: MWF 3-4pm, Room 155 Donner Lab

Instructor: Jan Reimann

Office: 705 Evans Hall
Office hours: Mo 10:30 - 12, We 4-5:30, and by appointment

Email:
Personal Website: http://math.berkeley.edu/~reimann/

Course webpage: This site is mainly for documetary purposes. The course material (homework, sample exams, etc.) will be posted on bSpace. The course site will also have a discussion tool and a chat room. I encourage you to make use of it. Go to http://bspace.berkeley.edu and log on with your Calnet ID. Then select the accordant tab.

Course Description

Linear Algebra is essentially the study of linear transformations between (finite dimensional) vector spaces. While this is a very abstract, and therefore widely applicable, setting, we will see that, via coordinates, this is deeply connected to the theory of matrices. For example, the composition of linear transformations corresponds to matrix multiplication. This way we obtain a nice algebraic setting for studying linear transformations.

We will then develop the basic notions and deduce the fundamental results of linear algebra: rank, multiplication, and inverse, solving systems of linear equations, determinants, etc. We will go on to study of canonical forms of linear transformations. Topics will include eigenvalues and eigenvectors, diagonalization, and if time permits, the Jordan normal form.

Finally, we enrich the structure of a vector space by an inner product. This allows for the introduction of geometrical notions like orthogonality. We will see how the inner product interacts with linear transformations.

One of the goals of this course, besides learning the basics of linear algebra, is to encounter a rigorous development of a mathematical theory, in our case, that of vector spaces. Two main characteristics of mathematics are abstraction and rigorous proof. We will deal with both of them. As said above, vector spaces are very abstract objects. However, the relation with coordinate vectors and matrices greatly facilitates dealing with these abstract objects. Also, most proofs are not very hard. This is why a course in linear algebra is a very suitable introduction to higher level mathematics. You will learn how rigorously prove mathematical statements in an abstract setting.

Ideally, you are already familiar with vectors and matrices (and the accordant algebraic operations) over the real numbers. If not, the book will provide plenty of examples, but you will have to learn two things at the same time, algebraic manipulation and abstraction.

Literature

The textbook for the course is S. Friedberg, A. Insel, L. Spence, Linear Algebra. We will follow it more or less closely. We will cover Chapters 1-4 and select portions from Chapters 5-8.

Exams

There will be two midterms in class: the first on Monday, September 24, the second on Friday, November 2. Note that the first midterm is before the five-week drop-out deadline.

The final exam is held on Saturday, December 15, 12:30-3:30 pm.

All exams will be closed book exams. No cheat sheets! Bring your blue books.

The final grade will be determined as follows: 20% homework, 20% each midterm, 40% final exam.

Homework

Homework will be assigned on Wednesday and will be due on the following Wednesday in class. Homework will be graded and the lowest score will be dropped. Late homework will not be accepted.

A note on academic honesty: Collaboration among students to solve homework assignments is welcome. This is a good way to learn mathematics. So is the consultation of other sources such as other textbooks. However, every student should hand in an own set of solutions, and if you use other people's work or ideas you should indicate the source in your solutions.
(In any case, complete and correct homework receives full credit.)