Linear Algebra - Fall 2006

(last updated 11/30/2006)

 


Instructor: S. M. Mintchev

Meeting Time: TR 8:55am - 10:45am

Location: Waverly Bldg., Rm. 435

Instructor Information:

            · email address: mintchev@cims.nyu.edu
 
            · office: Warren Weaver Hall, Rm. 718
 
            · telephone: 212-998-3107
 
            · office hours: M 10:00am - 11:00am; 2:00pm - 3:00pm; T 11:00am - 12:00pm

 

General Information about the Course: This course is an introduction to linear transformations of finite-dimensional real and complex vector spaces. It begins with the study of real systems of linear equations and their solution sets.  We will first develop the necessary matrix theory tools to solve linear systems, then move toward understanding linear transformations through their matrix representations. Throughout the semester we will see real-world applications of this material, in the hope that students develop an intuition for recognizing problems that fit into this framework and solving them systematically.

Text: Linear Algebra and Its Applications, third edition. David C. Lay. Addison Wesley. 2003.

Grading Policy: There will be five, approximately-bi-weekly homework assignments, posted on this website at the beginning of the discussion period for the required material, and due at the beginning of class on the posted date (see below for the list of assignments). Usually, about 20 problems will be assigned at a time. Of these, the grader for the course will choose about 8 at random and grade them. Late homework will NOT be accepted, except with a relevant doctor's note, etc. Some of the homework assignments (see syllabus below) will be followed by a 15 min. in-class quiz consisting of two or three problems related to the homework material. In addition to these assessments, there will be two exams for the course. A midterm exam will be given approximately in the third week of October. The midterm will cover the first half of material for the course - in particular, this will be the material from the first three homework assignments. There will also be a final examination; this exam will cover the second half of material taught in the course. The values of these components towards the final grade in the course are presented below:

Component
 Value
Homework
 15 %
Quiz Total
 25 %
Midterm Exam
 30 %
Final Exam
 30 %

Students are advised to solve homework problems carefully and to write up neat presentations of their solutions. While collaboration is encouraged, students should make a strong individual (meaning: separated from each-other in space, and entirely independent) effort on a problem before attempting collaboration with peers. Such work will be key to placing the student in a position to perform well on the examinations. In addition, since there will be no examinations before the ADD/DROP deadline for most NYU programs, the first two homework assignments should be interpreted as being illustrative of the workload required for passing the course; the grades on these, as well as on any quizzes given during the first 2-4 weeks should be used to make the ADD/DROP decision if necessary.

Syllabus (please note that this schedule is tentative and will likely be adjusted as the semester progresses):

Date Sections Topic
09/05/2006 1.1, 1.2 Linear Systems; Row Reduction
09/07/2006 1.3, 1.4 Vector Equations; Matrix Equations
09/12/2006 1.5, 1.6, 1.7 Solution Sets; Applications; Linear Independence
09/14/2006 1.8, 1.9 Linear Transformations / Matrix Representation
09/19/2006 2.1, 2.2, QUIZ 1 Matrix Operations; Inverse of a Matrix
09/21/2006 2.3, 2.4 Invertible Matrices; Partitioned Matrices
09/26/2006 2.5, 2.6 Matrix Factorization; Leontief Input-Output Model
09/28/2006 2.8, 2.9 Subspaces; Dimension and Rank
10/03/2006 3.1, 3.2 Determinants and their Properties
10/05/2006 3.3, 4.1, 4.2 Cramer’s Rule, Vector Spaces, Subspaces
10/10/2006 4.3, 4.4 Linear Independence; Bases; Coordinate Systems
10/12/2006 4.5, 4.6 Dimension of a Vector Space; Rank
10/17/2006 4.7, 4.8, 4.9 Change of Basis; Difference Equations; Markov Chains
10/19/2006 5.1, 5.2 Eigenvectors, Eigenvalues; The Characteristic Equation
10/24/2006 5.3, 5.4, QUIZ 2 Diagonalization; Eigenvectors / Linear Transformations
10/26/2006 5.5, 5.6 Complex Eigenvalues; Discrete Dynamical Systems
10/31/2006 MIDTERM EXAMINATION (covers through Homework 3)
11/02/2006 6.1, 6.2 Inner Product; Length; Orthogonality / Orthogonal Sets
11/07/2006 6.3, 6.4 Orthogonal Projections; Gram-Schmidt Process
11/09/2006 6.5 Least-Squares Problems
11/14/2006 6.6 Linear Models
11/16/2006 7.1 Diagonalization of Symmetric Matrices; Quadratic Forms
11/21/2006 7.2 Quadratic Forms; Constrained Optimization
11/23/2006 7.2
11/28/2006 7.2
11/30/2006 7.2, 7.3
12/05/2006 7.4, QUIZ 3 Singular Value Decomposition
12/07/2006
Review
12/12/2006
FINAL EXAM

Homework Assignments (please see grading policy section for requirements):

Assigned
 Due
 Homework Exercises
09/05/2006
 09/19/2006
 Section 1.1: 6, 10, 28
Section 1.2: 11, 16
Section 1.3: 23, 24, 28
Section 1.4: 31
Section 1.5: 12, 25
Section 1.6: 6, 12
Section 1.7: 22, 31
Section 1.8: 20, 21, 22
09/27/2006
 10/10/2006
 Section 1.9: 23
Section 2.1: 15, 16, 33
Section 2.2: 12, 30, 31
Section 2.3: 27, 38
Section 2.4: 13, 14
Section 2.5: 13, 19
Section 2.6: 5
Section 2.8: 23, 30
Section 2.9: 12, 27
Section 3.1: 13, 42
10/09/2006
 10/24/2006
 Section 3.2: 33, 34
Section 3.3: 16, 17
Section 4.1: 19, 26, 33
Section 4.2: 34
Section 4.3: 31, 32
Section 4.4: 18, 23
Section 4.5: 26
Section 4.6: 3
Section 4.7: 6, 13
Section 4.8: 25, 35
11/09/2006
 11/21/2006
 Section 5.1: 16, 20, 33
Section 5.2: 18, 19
Section 5.3: 25, 31, 32
Section 5.4: 19, 20, 21, 22
Section 5.5: 4, 6
Section 5.6: 1, 3
Section 6.1: 17, 18, 30
Section 6.2: 23, 24, 29
Section 6.3: 9, 24
Section 6.4: 3, 19
11/30/2006
 12/05/2006
 Section 6.5: 12, 19, 21
Section 6.6: 15,16
Section 7.1: 23, 36
Section 7.2: 6, 8
Section 7.3: 13