# Math 535, Linear Algebra and its Applications - Fall 2019

## Instructor: Jack Huizenga

 Jack Huizenga Office: 324 McAllister email: huizenga@psu.edu URL: http://www.personal.psu.edu/jwh6013/
 Meeting times: MWF 11:15-12:05, 109 Boucke First meeting: Monday, August 26 Office hours: Monday 4-5, Tuesday 4-5, or by appointment. (Subject to change based on student availability) Textbooks: Peter Petersen ``Linear Algebra'' will be useful for most of the course. Sheldon Axler's ``Linear Algebra Done Right'' might also be useful as an easier reference for some of the material at the beginning of the course. At the end of the course, we will supplement the Petersen text with additional resources and your notes.
 Brief course description: This is a graduate course in linear algebra and its applications intended to prepare math Ph.D. students for the qualifying exam in linear algebra. The emphasis of the course will be on proofs, as this is what is tested on the qualifying exam. Homework exercises will also reinforce the theoretical concepts from the course in particular examples. We will study vector spaces, linear transformations, inner products and quadratic forms, the theory of endomorphisms of a finite-dimensional vector space, orthogonal bases, the spectral theorem, and further applications. Students from other disciplines are welcome to enroll in the course, although previous experience with mathematical proofs will be expected. The undergraduate course MATH 436 covers much of the same material at a slower pace, and spends more time developing familiarity with proofs.
 Detailed course breakdown: The graduate program provides the following more detailed description of the topics to be covered, with appropriate weights. The qualifying exam will be designed to reflect this syllabus. Fields. Vector spaces. Subspaces. Spanning sets. Linearly independent sets. Bases. Dimension. (5 hours) Linear transformations. Kernel and image. Matrices. Direct sums and quotients. Correspondence between linear transformations and matrices. Matrix operations. Standard homomorphism theorems. Rank-nullity theorem. (5 hours) Inner products and quadratic forms. Bilinear functions. Sesquilinear functions. Orthogonal sets and orthonormal sets. Norms, matrix and operator norms. Schwarz' inequality and Bessel's inequality. Adjoints. Self-adjoint, normal and unitary matrices. Orthogonal projections and orthogonal complements. Quadratic forms over the real numbers. Signature. Sylvester's law. (9 hours) Theory of endomorphisms of a finite-dimensional vector space. Determinants. Cramer's rule. Multiplicative property of the dteterminant. Minimal and characteristic polynomial. Cayley-Hamilton theorem. Rational and Jordan canonical forms. (13 hours) Orthogonal bases, spectral theorem, and applications. The Gram-Schmidt orthogonalization process, Schur form. Parseval's identity. Diagonalizability of normal operators. Positive (semi)definiteness. The spectral theorem. Singular value decomposition. Functions of matrices. Min-max, Cauchy interlacing theorem, Gerhgorin theorem. Perron-Frobenius theorem and applications to graphs. (13 hours)
 Prerequisites: Enrollment in a graduate program or consent of the instructor. Familiarity with computational linear algebra and proofs will be expected. A strong performance in the undergraduate course MATH 436 or a similar course should be sufficient.
 Homework: There will be approximately 12 problem sets, due at the beginning of class each Wednesday starting in Week 2. There will be no homework in the week following an exam. Students are encouraged to work on the problems together as desired, but your final writeup must be completed independently. Homework is by far the most important part of the course; the only way to learn advanced mathematics is to do advanced mathematics. The course grader will grade a susbset of the problems each week, and solutions will be posted for selected problems. Students are expected to turn in homework on time in order to keep up with the course. To receive a grade of ``B'' or higher, a student must submit at least 11 of the 12 assignments. Your lowest score will be dropped at the end of the semester.
 Exams: Since this is a preparatory course for the Ph.D. qualifying exam in linear algebra, we will have two two-hour midterms during the semester to serve as a progress indicator. There will be no final exam, since the qualifying exam takes place at the end of the semester. The midterms will mimic the format of the qualifying exam but cover the material that has been covered to that point in the course. Midterms will tentatively take place on the following dates. Midterm 1: Week 6, Friday Oct. 4 Midterm 2: Week 14, Friday Dec. 6
 Grading and Expectations: Homework will count for 50% of your grade and each midterm will count for 25% of your grade. Your letter grade will reflect the expected performance on the qualifying exam: A: I expect you are on track to pass the qualifying exam. A-/B+: I expect that more preparation is needed, but passing the qualifying exam is possible. B: More preparation is needed. Passing the qualifying exam seems unlikely, but the student is keeping up with the course. C or lower: The student is not meeting course expectations.
 Academic Integrity Statement: All Penn State policies regarding ethics and honorable behavior apply to this course.
 Disability Statement: Penn State welcomes students with disabilities into the University's educational programs. The Office for Disability Services Web site provides contact information for every Penn State campus: http://equity.psu.edu/ods/dcl . For further information, please visit the Office for Disability Services Web site: http://equity.psu.edu/ods .
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