# matthias beck

professor of mathematics
san francisco state university
 MATH 725 Advanced Linear Algebra Fall 2021

 Lecture Mondays & Wednesdays 9:30-10:45 TH 404 Prerequisites MATH 335 (Modern Algebra) with a grade of C or better or consent of the instructor Instructor Dr. Matthias Beck Office Thornton Hall 933 & zoom Office hours (TH 933 & zoom) Mondays 11-12, Wednesdays 1-2 & by appointment

While our class is designed to meet in person, it will also be live streamed (zoom link on iLearn), so that students who cannot/should not come to campus on a given day (e.g., because they have cold symptoms) can keep up.

Course objectives. This is a second course in linear algebra in which students make the transition from Euclidean spaces and matrices to abstract vector spaces, inner product spaces and linear transformations. The emphasis is on axiomatic development, proof and conceptual understanding rather than calculation. Students will gain experience working abstractly without coordinates or determinants. This course should pave the way for further study in abstract algebra and advanced analysis. Upon successful completion, students will have a thorough understanding of the proofs of the finite dimensional versions of the Riesz Representation Theorem, the Spectral Theorem for normal operators, polar decomposition, singular value decomposition, and the Jordan canonical form.

Syllabus. Vector spaces, linear independence, bases, dimension, linear maps, null space, range, matrix representations, invertibility, invariant subspaces, upper triangular matrix representations, eigenvectors and eigenvalues, inner products, norms, orthonormal bases, Gram Schmidt process, orthogonal projection and best approximation, linear functionals and adjoints, Riesz representation, self-adjoint and normal operators, the spectral theorem, positive operators, isometries, polar and singular-value decompositions, generalized eigenvalues, characteristic and minimal polynomials, nilpotent operators, Jordan canonical form, trace, determinant.

Textbook. Sheldon Axler, Linear Algebra Done Right (3rd edition), Springer 2015. [errata, videos]

Homework. I will assign homework problems as we go through the material; the problems assigned in any given week are due at 9 a.m. of the following Friday. If you type your solutions, you are welcome to submit your solutions over email as a pdf attachment. We can discuss the homework problems at any time during class. You may hand them in early to be able to correct your mistakes. Although you may (and should) work together with your class mates, the solutions you hand in have to be your own.

 Homework 50% Midterm (October 20 ±2 days) 25% Final Exam (December 15 ±2 days) 25%

I want to ensure that each of you accomplishes the goals of this course as comfortably and successfully as possible. At any time you feel overwhelmed or lost, please come and talk with me.

The math. The way to learn math is through doing math. It is vital and expected that you attend every lecture. You will get a good feel for the math from there, but it is even more crucial that you do the homework. Working in groups is not only allowed but strongly recommended. The iLearn system allows you to send emails to anybody in your class. While I strongly encourage you to work together, the work you hand in has to be your own.

The Corona situation. We are living through a pandemic. The health and safety of you and others comes first. While in the times before Corona, attendance was highly correlated with student success in this course, I trust each of you to use your best judgement to keep you and those around you safe, and to attend our class when it makes sense to do so.

This syllabus is subject to change. All assignments, as well as other announcements on tests, policies, etc., are given in class. If you miss a class, it is your responsibility to find out what's going on. I will try to keep this course web page as updated as possible, however, the most recent information will always be given in class. Always ask lots of questions in class; my courses are interactive. You are always encouraged to see me in my office (TH 933 & zoom).

department of mathematics
san francisco state university
1600 holloway ave
san francisco, ca 94132

 becksfsu @ gmail.com