# matthias beck

professor of mathematics
san francisco state university
 MATH 325 Linear Algebra Fall 2024

 Lecture Mondays & Wednesdays 12:00-1:40 p.m. Thornton Hall 325 Prerequisites Calculus I (Math 226 or equivalent); Proof & Exploration (Math 301) recommended as a concurrent course Instructor Dr. Matthias Beck Office Creative Arts 32 Office hours Mondays 11-12, Wednesdays 10-11, Fridays 1-2, by appointment, and via zoom

Course objectives. Linear algebra is motivated by solving system of linear equations. Despite this simple starting point, linear algebra plays deep and profound roles in both applied and "pure" mathematics, as well as many other fields such as computer science, data analysis, the natural sciences, economics, any many more. Our course will start with systems of linear equations and techniques of finding their solutions based on row operations and Gaussian elimination. This will naturally lead us to matrix algebra operation and abstract vector space notions. Representation of linear transformations by matrices and change of basis will be introduced with an eye towards diagonalization, and inner products will be used as a gateway to symmetric positive (semi-)definite matrices. Eigenvalues and eigenvectors will be introduced with applications of diagonalization in mind. We will conclude with spectral decomposition/factorization of symmetric matrices and singular value decomposition and its applications.

Syllabus. Topics in this course will include:

• Solution of system of linear equations using Gaussian elimination
• Matrix algebra, special matrices (diagonal, triangular, symmetric), matrix inverse and transpose
• Abstract vector spaces and main examples (Rn, matrix and polynomial spaces, function spaces)
• Linear dependence/independence, subspaces, their bases and dimension (rank)
• Linear transformations of vector spaces,matrix representations, change of basis
• Range and kernel of linear transformations, finding their bases, rank-nullity theorem
• Real inner product spaces and positive (semi-)definite matrices,inner products on function spaces
• Cauchy-Schwarz and triangle inequalities
• Orthogonality, orthogonal bases, orthogonal projections, Gram-Schmidt orthogonalization
• Determinants
• Eigenvalues, eigenvectors, eigenspaces and their bases
• Diagonalization and spectral factorization of symmetric matrices
• Singular Value Decomposition and applications
Texts. We will follow (selected chapters from) Peter J. Olver & Chehrzad Shakiban's Applied Linear Algebra, Springer, 2018. A pdf copy of this book can be freely downloaded via the SFSU Library. Virtually any Linear Algebra book can be used as an additional source.

Participation. A nontrivial part of the material covered in this class will be worked out in small groups during class sessions. It will thus be essential that every student participates actively in every class. If you have to miss a class due to a medical or family emergency, please let me know before the class; otherwise, I expect you to be in class and actively engaged.

Homework. I will assign homework problems as we go through the material; it is absolutely crucial for your learning process that you complete each homework problem. You may (and should) work together with your class mates. We can discuss the homework problems at any time during class, and you can hand any of your solutions for feedback. We will have a homework quiz every Wednesday at the beginning of class, in which you will be asked one definition and one problem given in the previous week.

SageMath. We will use sage in class, and you may (and should) use it outside of our class sessions, too. Here is a good introduction to sage. I will add useful sage commands for our class here.

 Homework quizzes 50% Midterm Exam (October 16, in class) 25% Final Exam (December 18, 12:30 p.m.) 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 class meeting. 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. Our class is based on Federico Ardila's Axioms:

1. Mathematical potential is distributed equally among different groups, irrespective of geographic, demographic, and economic boundaries.
2. Everyone can have joyful, meaningful, and empowering mathematical experiences.
3. Mathematics is a powerful, malleable tool that can be shaped and used differently by various communities to serve their needs.
4. Every student deserves to be treated with dignity and respect.

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.

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

 becksfsu @ gmail.com