Math 725: Advanced Linear Algebra

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Dorothy Allison
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1 Math 725: Advanced Linear Algebra Course Syllabus - Fall 2005 Lectures: Every TuTh 9:35-10:50am in HSS 304 Instructor: Dr. Yitwah Cheung Office: Thornton Hall, Room 950 Phone: (415) Office Hours: Tu 12:30-1:30pm, Th 8:30-9:30am, and by appt. ycheung@sfsu.edu Prerequisites: Grade C or better in Math 335 or consent of instructor. Bulletin Description: Vector spaces and linear maps on them. Inner product spaces and the finite-dimensional spectral theorem. Eigenvalues, the singular-value decomposition, the characteristic polynomial, and canonical forms. Course Objectives: This is a second course in linear algebra in which students make the transition from Euclidean spaces and matrices to abstract vector spaces, inner product spaces and linear transformations. The emphasis is on axiomatic development, proof and conceptual understanding rather than calculation. Students will gain experience working abstractly without coordinates or determinants. In addition, they will learn how ideas from three dimensional geometry can be generalised to unify a wide variety of mathematical applications such as Fourier series, orthogonal functions, and linear regression. This course should pave the way for further study in abstract algebra and advanced analysis. Upon successful completion, students will have a thorough understanding of the proofs of the Theorem for normal operators, polar decomposition, singular value decomposition, and the Jordan canonical form. They will also be able to apply the results to specific operators. Textbook: Linear Algebra Done Right, 2nd ed. by Sheldon Axler. Problem Sets: There will be 6 Problem Sets to be written up and handed in during class on the due dates indicated in the Course Schedule. Problem Sets turned in late incur a half point penalty per day. Each problem set is worth 9 points. Students are expected to complete all the reading assignments and suggested exercises associated with each lecture in a timely manner. Midterm: In HSS 304 between 9:35-10:50am on Thursday, October 13. 1

2 Final: In HSS 304 between 8:00-10:30am on Tuesday, December 13. Grade: Your total score out of 100 is the sum of the points scored on the 6 Problem Sets (6 9=54 points), the Midterm (16 points) and the Final (30 points). Your grade for the course is determined by your total score based on the scale below. A curve, if used, will only be in your favor. Total score Grade A A B B B C C C D D D 0-59 F Academic Integrity: All students are expected to adhere to the SFSU honor code. Any student caught cheating on an examination will automatically fail the course and face expulsion from the University. Each Problem Set is to be written up individually. However, students may and are in fact encouraged to discuss the homework problems with each other, including the Problem Sets. Enrollment Status: Each student is solely responsible for maintaining his/her own enrollment status. Check your class schedule for the relevant add/drop /withdrawl dates and proper procedures for maintaining your enrollment status. Students with Disabilities: The University is committed to providing reasonable academic accomodations to students with disabilities. To qualify, you must register with the Disability Programs and Resource Center (DPRC) in the Student Services Building, room 110. Phone (415) (voice/tty), Fax (415) , dprc@sfsu.edu Subject to Change: Any changes to this syllabus will be announced during class and posted the same day on the course webpage at URL address: 2

3 Math 725: Course Schedule Fall 2005 Text: Linear Algebra Done Right, 2ed. by Sheldon Axler. Note: X.Y refers to the Yth lecture on chapter X of the text. 8/23 Tu no class 10/18 Tu 6.1 8/25 Th /20 Th 6.2 8/30 Tu /25 Tu 6.3 9/1 Th /27 Th 7.1 9/6 Tu /1 Tu 7.2, PS #4 due 9/8 Th 2.2, PS #1 due 11/3 Th 7.3 9/13 Tu /8 Tu 8.1 9/15 Th /10 Th 8.2 9/20 Tu /15 Tu 8.3, PS $5 due 9/22 Th 3.3, PS #2 due 11/17 Th /27 Tu /22 Tu /29 Th /24 Th Thanksgiving 10/4 Tu /29 Tu /6 Th 5.2, PS #3 due 12/1 Th /11 Tu /6 Tu 9.2, PS #6 due 10/13 Th Midterm - in class 12/8 Th 9.3 Midterm: In room HSS 304 between 9:35-10:50am on Thursday, October 13 Final exam: In room HSS 304 between 8:00-10:30am on Tuesday, December 13 3

4 Math 725: Reading Assignments Note: The main topic for Lecture X.Y is covered in Chapter X. Lecture Main Topic 1.1 Fundamental concepts (also read Munkres 1-1) 1.2 Vector spaces, subspaces 1.3 Direct sums 2.1 Span & linear independence 2.2 Bases and coordinates 2.3 Dimension 3.1 Linear maps (also read Munkres 1-2) 3.2 Null space & range 3.3 Matrix of a linear map 3.4 Invertibility 4.1 Fundamental theorem of algebra 5.1 Invariant subspaces 5.2 Diagonalisability 5.3 Upper trianglular matrices 6.1 Inner product spaces 6.2 Orthogonal bases & projections 6.3 Linear functionals, adjoint 7.1 Self-adjoint & normal operators 7.2 Spectral theorem, positive operators 7.3 Polar & singular value decomposition 8.1 Generalised eigenvectors 8.2 Jordan decomposition 8.3 Canonical form 10.1 Trace 10.2 Determinant 10.3 Volume 9.1 Real operators with complex eigenvalues 9.2 Block upper triangular matrices 9.3 Structure theorem 4

5 Math 725: Suggested Exercises Munkres 1-1: (1), 3, 4, 7 Chapter 1: 3-7, 8-10, 13, 15 Chapter 2: 1-5, (6), 8, 9, 10, 11, 13, 14, 16 Munkres 1-2: 4, (6) Chapter 3: 1-5, (6), 7, 8, 12, -14, 16, Chapter 4: 2, 3, (4) Chapter 5: 1, 3, 4, 5, 7, 8, 9, 10, 12, 17, 21 Chapter 6: 2, 3, 4, 5, 7, 10, 15-17, 24, 27, 30 Chapter 7: 2, 4-6, 8, 9, 11, 14 Chapter 8: 10, 11, 12, 17, 23, 25, 30 Chapter 10: 3, 7, 8, 9, 10, 11, 16, 19 Chapter 9: 1-3, 6, 8, 9, 12, 14 Note: Optional exercises indicated in parentheses. 5

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