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AA-EE-ME 510 Mathematical Foundations Of Systems Theory


Course Description
This course aims to provide a solid mathematical foundation for a number of disciplines in systems theory (communications, signal processing, control), optimization, machine learning, among others. Topics covered include elements of set and function theory, metric spaces, Hilbert spaces, and matrix theory, and their applications in systems sciences. Particular attention will be given to strengthening students ability to read and do formal mathematical reasoning as required for many graduate courses in systems, signal processing, communication, control, and optimization.


Syllabus


Textbook
P. Lax, Linear Algebra and Its Applications, Wiley, 2007.
A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis, Dover, 1999.


Schedule

9/24: (Lax; chapter 1): introduction, groups, vector spaces, finite vs. infinite dimensional vector spaces slides
9/26: (Lax; chapter 1): basis, dimension, cartesian sums, quotients spaces slides
HW#1 (Due 10/3): Lax/chapter 1: ex. 1, 3, 4, 7, 8, 10, 11, 13, 20
10/1: (Lax; chapter 1): review of quotient spaces, isomorphism, and linear maps (see also notes from 9/26)
10/3: (Lax; chapters 3, 4): linear maps and matrices slides
HW#2 (Due 10/10): Lax/chapter 1: ex. 16, 17, 19; Lax/chapter 3: ex. 2, 3, 4; and Lax/chapter 4: ex. 1
10/8: Lax Chapters 3 and 4 slides
10/10: Lax Chapters 4, 5 slides
HW#3 (Due 10/17): Lax/chapter 2: ex. 6, 7; Lax/chapter 3: ex. 5, 6, 10, 12, 14, 15; Lax/chapter 4: ex. 4
10/15: Lax Chapter 5 slides
10/17: Lax Chapter 6 slides
HW#4 (Due 10/24): Lax/chapter 5: ex. 4, 6, 7, 8, 9, 16; Lax/chapter 6: ex. 6, 10, 11, 13
10/22: : Lax Chapters 6, 7 slides
10/24: : Lax Chapter 7 slides
HW#5 (Due 10/31): Lax/chapter 7: ex. 1, 8, 17, 18
10/29: Midterm
10/31: Lax Chapters 7 slides
HW#6 (Due 11/7): Lax/chapter 7: ex. 2, 5, 11, 12, 13, 15, 19; Lax/chapter 8: 8, 9, 11, 12
11/5: Lax Chapter 9 slides
11/7: Lax Chapters 9,10 (see podcast) slides
HW#7 (Due 11/14): Lax/chapter 7: ex. 7, 20; Lax/chapter 8: ex. 6; Lax/chapter 9: ex. 2; Lax/chapter 10: ex. 1, 2, 4, 5
11/12: Veteran's Day
11/14: Lax Chapter 10 slides
HW#8 (Due 11/21): Lax/chapter 9: ex. 6, 7; Lax/chapter 10: ex. 7, 12, 13, 14; read Appendix 14
11/19: Lax Chapter 14, KF: Chapter 3 slides
11/21: Application of projection theorem: min norm control; SVD slides
HW#9 (Due 11/28): Lax/chapter 14: ex. 6, 7, 8; Reading from KF: pages 16-43; 71-81
11/26: Lax Chapters 14,15 slides + Reading from KF Chapter 3
11/28: KF Chapter 3 (contraction mapping), google PageRank, some review slides
HW#10 (Due 12/5): Lax/chapter 15: ex. 1, 2, 3, 4, 5, 6, 7, 8
12/3: review of some key concepts: spectrum/self-adjoint, generalized eigenvalues, hints on the homework slides
12/5: presentation by a few student projects, class review slides Δ
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Final Exam
The final exam will be given on Dec. 10, from 2:30-4:30 pm in EEB045. The final is closed book/notes except one page of notes (2-sided 8.5" x 11"). Please turn in your notes with your exam.
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Projects
The project for the class consists of a 4-5 pages of report using the style file on this page http://control.disp.uniroma2.it/cdc2012/author_info.php (either tex or word).

The project report format: I) abstract, 2) problem setup and assumptions, 3) basic results, simulations, proof, 4) applications, 5) references.

The project has to deal with a theoretical or an applied aspect of the topics covered in the course, including non-trivial properties of linear spaces, linear operators, matrices and matrix inequalities, normed and metric spaces, among others, and applications in engineering sciences. The mere fact that the paper contains "matrices" would NOT qualify the paper as appropriate for the project and NO CREDIT will be given to reports that do not have an analytic component that is consistent with the level of discussion/instruction in the course. Please consult with the instructor about your project report if you are in doubt.

The project report is due as a pdf file (no other formats will be accepted), submitted to the catalyst dropbox site https://catalyst.uw.edu/collectit/dropbox/mesbahi/24808 by 5:00 pm December 13th. Please note that this site closes at 5:00 pm on December 13.


Useful References
G. Strang-Linear Algebra Course
S. Boyd- EE263 course
MIT 6.241 old course notes on Linear Algebra/Functional Analysis for Systems

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Page last modified on December 05, 2012, at 01:01 PM