Florian Pop: Teaching

## Florian Pop: Math 314/514 (Advanced Linear Algebra)

E-mail: `pop AT math.upenn.edu`
Office/Phone/Fax: DRL 4E7A / 215-898-5971 / 215-573-4063
Office hours: We 4:30-5:30 PM, Fr 10:30-11:30 AM (on Zoom). Please make an appointment in advance (by email).
TAs:
• Avik Chakravarty, Office/Email: DRL 3W3, `avikchak AT sas.upenn.edu`
• Marc Muhleissen, Office/Email: DRL 4C5, `mmuhleis AT sas.upenn.edu`

### General Information

• Class: Mo, We 10:15-11:45 AM in DRL A6. First class on We, Jan 12, 2022.
• Lab: Tu, Th at 7:00-9:00 PM
• This is a rigorous proof based course in Linear Algebra. For a passing grade, the students are expected to:
• Understand and know the material and be able to solve related problems.
• Understand rigorous mathematical argumentation/proofs and to be able to write coherent mathematical proofs.
• Prerequisites: One should be familiar/used to: working with sets (including maps, order/equivalence relations); logical deduction (including usage of quantifiers, logical negation, etc.); types of proofs (proofs by induction, proofs by contradiction, direct proofs); the basics about natural numbers, integer numbers, rational numbers, real numbers. As a test for prerequisites, see Prereq HW. For some examples and applications, one needs the basics of calculus, e.g. continuity, differentiability, integration, and the elementary functions (polynomial functions, exp, log, sin, cos, etc.) and their properties.
• Syllabus: Basic algebraic structures (monoids, groups, rings, fields, modules, vector spaces). Basics of linear algebra (systems of linear equations, Gauss-Jordan elimination, matrices and their basic properties). Modules/submodules and vector spaces/subspaces, span and linear combinations, free modules, bases. Linear transformations and their relation to matrices, dual modules/vector spaces and the transpose, base change and similar matrices, dimension formulas. Bilinear and multilinear maps, (symmetric, alternating), Tensors (symmetric and skew-symmetric tensors, exterior algebra), Dimension formulas. Determinants and their properties, Cramer's rule, characteristic polynomial, Cayley-Hamilton. Eigenvectors/eigenvalues, diagonalization. Canonical forms of matrices (e.g., Jordan canonical form). Real/complex vector spaces with inner product, orthogonal bases, Gram-Schmidt. Orthogonal/unitary, symmetric, normal, hermitian/anti-hermitian transformations/matrices. Decomposition theorems, basics of spectral theory. Bilinear forms (revisited) and Quadratic forms. Applications (time permitting).
• Resources: There are several books you can use, and some (partially detailed) course notes will be provided.
• Linear Algebra Done Wrong by Sergey Treil; free available online.
• Linear Algebra by Kenneth Hoffmann - Ray Kunze, 2nd edition. Prentice Hall, Inc (standard text with complete detailed proofs/explanations).
• Serge Lang, Algebra, Part III. Springer Verlag (standard reference text which contains a very large amount of material).
• To large extent, course notes will be available online.

### Basic Rules:

• The final grade is based on two midterms (10+15 %), a final exam (25%), and everything else (50%). "Everything else" consists of regular homework, participation in class/lab; performance in lab, etc.
• Exam dates (tentatively): Feb 23, April 4, May 2, 2022.
• Miscellania: Announcements, homework assignment, notes, etc., will all be posted on web. No hard copies will be distributed. Please check this page frequently for the most updated information. Remember to use to RELOAD button of your browser.
• Homework:
• Homework will be assigned each week, and in oder to see the homework please follow the links under Homework Math 314/514. The homework assignment of each week is tentatively due on Friday of the next week.
• Your work should contain complete solutions, and rigorous and logically correct proofs for theoretical problems. (Note: such a proof must be written in grammatically correct language.)
• You are encouraged to work in groups and discuss/communicate with each other as much as possible. But the work you hand in must be your own handwritten write-up.
• Late work will not be accepted.

• Grading Note: At the end of the semester, everyone who has not withdrawn from the class will get a grade. The grade "I" (Incomplete) will not be given to avoid the grade "F" (Fail).