Course Summary

Math 416, Abstract Linear Algebra, is an introduction to linear algebra with a focus on proof. The course begins by investigating techniques for solving systems of linear equations, in turn paving the way for an abstraction to matrices and matrix operations. Exploring the properties of matrices will translate our initial algebraic questions into geometric ones, a dictionary which will ultimately provide most of the interesting applications in the subject. Whereas many linear algebra courses place an emphasis on computational proficiency and give short shrift to concepts, this course will attempt to strike a richer balance, one in which computations and "real" examples provide the groundwork for a more general, abstract theory.

Though we will move towards abstraction, the concepts we will focus on are borne from concrete examples. Students who hold on to a geometric understanding of the material should find the topics we cover quite natural and easy to remember. That said, our emphasis on proof in the course means that students will need to keep up with the new terminology and theorems presented in class. You will find that the course requires you to learn a new language, and you will not be able to succeed unless you spend time learning its vocabulary and syntax.

Specific topics which will be covered include, but are not necessarily limited to: solutions to systems of linear equations, Gaussian elimination, matrix algebra, vectors and their properties, linear transformations, abstract vector spaces, dimension theory, the Gram-Schmidt algorithm, least squares, eigenvalues and eigenvectors, discrete and continuous dynamical systems, the singular value decomposition, and stochastic matrices.


The course is taught by Andrew Schultz. His office is Illini Hall, room 238. You can contact him by email at acs@math.uiuc (of course, you'll want to append a '.edu'). His office hours are Mondays and Wednesdays from noon to 1.