Florian Pop: Teaching

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

E-mail: `pop AT math.upenn.edu`
Office/Phone/Fax: DRL 4E7A / 215-898-5971 / 215-573-4063
Office hours: Mo, Fr 1:00-2:30 PM. Please make an appointment in advance (by email).
TA: Kamron Vachiraprasith, Office/Phone/E-mail: DRL 3W1 / 215-746-3201 / kamronv@math.upenn.edu

### General Information

• Class: Mo 10:30-12:00 (noon), Fr 10:00-12:00 (noon) in DRL 4C8. First class on Fr, Sept 1, 2017.
• Lab: Tu, Th at 6:30-8:30 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.
• 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).
• Required background/prerequisites: Basic facts about sets and maps, and some linear algebra (e.g., Penn course Math 240), familiarity with proofs (e.g., proofs by induction).
• Resources: There are several books you can use (and some course notes might/will be provided).
• Linear Algebra Done Wrong by Sergey Treil; free available online at: LADW
• 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 midterms and a final exam (30%+35%) and everything else (35%). "Everything else" consists of regular homework, participation/performance in class, etc.
• Exam dates (tentatively): Oct 13, Nov 17, Dec 15, 2017.
• 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. The homework assignment of each week is tentatively due on Wednesday of the next week.
• Your works 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 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).