matthias beck

MATH 325 Linear Algebra Fall 2025

Lecture	Mondays & Wednesdays 2:00-3:40 p.m. Thornton Hall 404
Prerequisites	Calculus I (Math 226 or equivalent); Proof & Exploration (Math 301) recommended as a concurrent course
Instructor	Dr. Matthias Beck
Office	Thornton Hall 933
Office hours	Mondays 10-11, Wednesdays 11-12, 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 sections from) two books:

• 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
• David Austin's Understanding Linear Algebra, an open-source book with built-in SageMath boxes.

Virtually any Linear Algebra book can be used as an additional source, and I invite you to find one that is particularly useful for you. I will keep a brief diary here which sections of the text books we covered each week:

• Week 1: Olver-Shakiban Sections 1.1, 3 & 4, Austin Sections 1.1-2. Homework (from OS): 1.1.1 & 2, 1.3.1, 2 & 7, 1.4.3. Turn in 1.3.1(d), 1.3.7, 1.4.3
• Week 2: Olver-Shakiban Sections 1.2 & 8, Austin Sections 1.4, 2.1-2. Homework (from OS): 1.2.7, 8 & 14, 1.8.1, 4 & 7. Turn in 1.2.14, 1.8.4, 1.8.7(d).
• Week 3: Olver-Shakiban Sections 7.1, 1.5, Austin Sections 2.5-6, 3.1. Homework (from OS): 7.1.3 & 7, 1.5.7, 16, 25 & 31. Turn in 7.1.7, 1.5.7, 1.5.31(c).
• Week 4: Olver-Shakiban Sections 1.4-6, 2.1-2.
• Week 5: Olver-Shakiban Sections 2.2-3, Austin Sections 2.3, 3.5. Homework (from OS): 1.4.21, 2.1.6, 2.2.2, 9 & 27, 2.3.3 & 7. Turn in 1.4.21(c), 2.1.6, 2.2.9.
• Week 6: Olver-Shakiban Sections 2.3, Austin Sections 2.1-4. Homework (from OS): 2.3.22, 23 & 32, 2.4.3, 5 & 9. Turn in 2.3.22, 2.4.5 & 9.
• Week 7: Olver-Shakiban Sections 2.4-5, Austin Sections 3.2. Homework suggestions (from OS): 2.4.6, 8, 11 & 14.
• Week 8: Olver-Shakiban Sections 1.6 & 3.1, Austin Section 6.1. Homework (from OS): 2.5.2 & 7, 3.1.1, 5 & 8. Turn in 2.5.7(d), 3.1.1 & 8.
• Week 8: Olver-Shakiban Sections 3.1 & 2, Austin Section 6.1.

The math. The way to learn math is through doing math. One doesn't become a good piano player by listening to music, and one doesn't learn how to shoot free throws by watching basketball games. Similarly, you don't learn mathematics by watching someone else do mathematics. You will be doing a lot of the math in this course. 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.

Axioms & first principles. 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.

I would emphasize within the last axiom that dignity and respect is also something you should treat yourself to, and that every person in our course, students and professor, are expected to be honest. It is likely that some or all of us will experience challenges and disruptions this semester. Please choose to act with compassion for others and for yourself as we navigate any situations that arise.

Homework. I will assign homework problems as we go through the material. You may (and should) work together with your class mates. You may not search the internet or use AI tools for solutions to problems; we will use our creativity, course texts, and peer collaboration as our tools: we will learn math by doing math. We can discuss the homework problems at any time during class, and you can hand in (or bring to my office hours) any of your solutions for feedback. The homework problems from any given week will be due before the following Wednesday class. You can either bring your homework to class or send me a pdf copy via email (please use your first name as file name).

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

Grading system. In our course, we will use a specifications-based assessment and grading system, which has the following key features:

• There are four elements to assessment and grading: participation, homework, and exams with definition and problem sections.
• There are no points and no partial credit; all assignments will be either be given a pass or an incomplete. This grade, for each problem, will be based on a combination of (a) correctness of the solution and (b) quality of communication in your writing.
• Your class grade will be determined by the amount of course material you have passed, with the requirements to achieve each letter grade specified below. By focusing on learning and mastery instead of accruing points, we can keep our attention focused on what we know and where we need to improve.
• You can revise some of your incomplete work. My goal is for each of you to develop as mathematical thinkers, learners, and collaborators. To support this goal, you are allowed to revise and resubmit selected work during the semester.

The four elements of assessment and grading are as follows.

• Participation: I expect you to be present and engaged every class session. Every student begins the semester with a participation grade of 20 and will lose one point for each unexcused absence.
• Homework: On non-exam Wednesdays, you will submit three homework problems, assigned the previous week. You will be allowed to revise and resubmit them (once) by the following Wednesday.
• Exams: we will have four cumulative exams, on
  • 22 September
  • 15 October
  • 12 November
  • 15 December
  Each exam will consist of three linear algebra definitions and three problems similar to the homework. You will be allowed to revise and resubmit selected problems (once) within a week of when I return your work.

To earn a particular grade in the class, you need to complete all of the requirements in the column corresponding to that grade in the table below; i.e., if the entry lists x then you need to receive a pass grade on x of those items to earn the corresponding column grade.

	A	B	C	D
Participation (out of 20)	17	15	13	11
Homework (out of 33)	29	26	22	16
Definition exams (out of 12)	10	8	6	4
Problem exams (out of 12)	10	9	7	6

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.

Fine print.
SFSU academic calender
BS rule
Academic Integrity and Plagiarism
Tutoring
CR/NCR grading
Incomplete grades
Late and retroactive withdrawals
Student disclosures of sexual violence
Students with disabilities
Religious holidays

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.