**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

-----------------------------

**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.

-----------------------------

**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