Location: TOMP 203

Schedule: Tuesday, Thursday, 4:45pm-6:00pm

Instructor: Poorvi Vora, poorvi@gwu.edu, 706 Philips Hall. Office Hours: Tues, Thurs: 1:30-4:30 pm

TA: Ben Hosp, bhosp@gwu.edu. Office Hours: TBD in Academic Center 724.

Text: None. Class Notes.

Course Content: Complex Numbers, Linear Algebra, Algorithmic Number Theory, with applications to media compression, audio-video processing and computer security

Prerequisites: CSci 123 (Discrete Structures I) or equivalent discrete mathematics, Math 31 or single-variable calculus

Grading: HWs (30%), best 10 of 11 quizzes (15%), two tests (15% each), final (25%). HWs are due by 6 pm on the due date. Late HWs are not allowed.

224 and 124 will be graded separately. If you are an undergrad, please consult your adviser before choosing to take 224; graduate credit for 224 is not automatic for undergrads, but all those enrolled in 224 will be graded together.

Course Outline

Slides and notes on this website are not in final form till one week after the lecture covering the material. I often update notes and slides during and after the lecture

Planned Schedule

Module I: Modular Arithmetic. Suggestions for further reading: [1] MIT Open Courseware Notes, Number Theory [2] Thomas H. Barr. Invitation to Cryptology. Prentice Hall. [3] Kenneth Rosen. Elementary Number Theory and its Applications.
4 September 2007 Lecture 1: Divisibility and a review of Mathematical Induction. HW 1 assigned. Due 13 September, 6 pm References: [3], Section 1.5; [1], Section 1	6 September 2007 Lecture 2: Shift and Affine Ciphers, Modular Arithmetic. Notes, sections 1-3, definition 1 and first theorem. Quiz 1
11 September 2007 Lecture 3: Complete proofs, modular arithmetic. Notes References: [3], pg. 141-145 (omit complete system of residues)	13 September 2007 Lecture 4: Complete modular arithmetic. Begin Groups: Definitions and Examples. Notes HW 1 due Quiz 2
18 September 2007 Lecture 5: Groups: Definitions and Examples. Sections 1-4. HW 2 assigned. Due 27 September 2007, 6 pm	20 September 2007 Lecture 6: Multiplicative Invertibility modulo m Notes, section 5. GCD Notes, definition. HW 2 correction: Part A, problem 2, group operation is multiplication modulo p Rescheduled: Test 1 will be held on October 9 Quiz 3
Module II: Algorithms for Number Theory: I. Suggestions for further reading: [3] Kenneth Rosen. Elementary Number Theory and its Applications.
25 September 2007 Lecture 7: Euclidean Algorithm Notes GCD Practice Examples with Solutions	27 September 2007 Lecture 8: Euclidean Algorithm complete HW 2 due Quiz 4
2 October 2007 Lecture 9: Euclidean algorithm for inverse. Notes Modular Inverse Practice Examples and Solutions	4 October 2007 Lecture 10: Fast Exponentiation. Notes Fast Exponentiation Practice Examples and Solutions Quiz 5
9 October 2007 Lecture 11: Test 1: Lectures 1 through 9 inclusive. NOTE DATE CHANGE	11 October 2007 Lecture 12: Subgroups and Rings. Notes HW 3 assigned. Due 23 October 2007, 6 pm
16 October 2007 Lecture 13: Homomorphisms and Isomorphisms. Notes	18 October 2007 Lecture 14: Chinese Remainder Theorem. Quiz 6
23 October 2007 Lecture 15: Notes on correctness of several algorithms HW 3 due HW 4 assigned. Due 1 November 2007
Module III: Vector Spaces. Suggestions for further reading: Gilbert Strang. Introduction to Linear Algebra, 3rd Edition. Wellesley-Cambridge Press. (On reserve in library)
	25 October 2007 Lecture 16: Vector Spaces: Definition, Examples. Notes. Quiz 7
30 October 2007 Lecture 17: Subspaces.	1 November 2007 Lecture 18: Spans, linear combinations. HW 4 due Quiz 8
6 November 2007 Lecture 19:	8 November 2007 Lecture 20: Test 2. Covers Lectures 10, 12-18
13 November 2007 Lecture 21: Gaussian elimination.	15 November 2007 Lecture 22: Gaussian elimination. Quiz 9
20 November 2007 Lecture 23: Linear Independence. Extra Credit HW assigned, can be swapped with another HW for 6% credit. Due 6 December, 6 pm	22 November 2007 No class. Thanksgiving Break
27 November 2007 Lecture 24: Bases HW 5 assigned. Due 6 December, 6 pm	29 November 2007 Lecture 25: NO MATLAB TUTORIAL. Class will be held in regular classroom. Bases. Quiz 10 CS 224 Project assigned. Due December 20, 2007, by 6 pm. electronically on Blackboard.
4 December 2007 Lecture 26: Orthogonality and Inner Products Notes Norms and Orthonormal Bases.	6 December 2007 Lecture 27: Quiz 11 HW 5 due
Thursday, December 20, 2007 5:20-7:20pm Final Exam for CSci 124 in TOMP 203 HW 5 Solutions Extra Credit solutions Quiz 11 Solutions
The thoughts of students who took this class in Fall 2006: Ah! Discrete Structures! Oh my god! Another proof! Die a bit inside	Discrete Structures II is fun It makes your motor run Usually in circles Because the work is never done!
Come and calculate But do not get bent, Your time with Euclid Is very well spent. For so your happiness Will have no bound, When it's the GCD That you have found!