Linear Algebra
MATH-4100-01, Fall 2007

Instructor: Victor Roytburd (Amos Eaton 405, Ph. 276-6889, roytbv at rpi dot edu)

Class meets: M, Th 10:00 – 11:50AM, CA 101

Office hours: Tu, Fr 3:00 – 5:00  (or by appointment)

Web page: http://www.rpi.edu/~roytbv/Lin_Alg/la07.html

TEXTS: (1) (required) Introduction to Linear Algebra, Third Edition, by G. Strang, Wellesley-Cambridge Press ISBN: 09614088 98
(2) (recommended) Lectures on Linear Algebra, by I.M. Gel’fand, Dover Publications ISBN: 0-4866-60826

The textbook by Strang (S) will be the principal reference. There is a web page, web.mit.edu/18.06/www, maintained for the course taught from ST at MIT. Videos of Prof. Strang’s lectures are posted on the OpenCourseWare site ocw.mit.edu. A concise and more abstract view of the subject is given in the book by Gelfand (G).

Linear Algebra is probably the most useful and applicable mathematical discipline. The goal of this course is learning how to use matrices and understanding mathematical ideas behind them.  Here is the course description from the catalog:

The theory underlying vector spaces, algebra of subspaces, bases; linear transformations, dual spaces; eigenvectors, eigenvalues, minimal polynomials, canonical forms of linear transformations; inner products, adjoints, orthogonal projections and complements.

I expect to cover the material of first seven chapters of Strang’s book. Here are the key topics:

·        Vector spaces; linear independence, basis and dimension

·        Solving Ax = b for square systems by elimination (pivots, multipliers, back substitution, invertibility of A, factorization into A = LU)

·        Complete solution to Ax = b (column space containing b, rank of A, nullspace of A and special solutions to Ax = 0 from row reduced form R)

·        Basis and dimension (bases for the four fundamental subspaces)

·        Euclidean spaces; Schwarz inequality;

·        orthogonal matrices

·        Projections. Least squares solutions (selection of best approximation curves through projections)

·        Orthogonalization by Gram-Schmidt (factorization into A = QR)

·        Properties of determinants (leading to the cofactor formula and the sum over all n! permutations, applications to inverse of A and volume)

·        Linear transformations.

·        Change of basis

·        Complex matrices

·        Eigenvalues and eigenvectors (diagonalizing A, computing powers Ak and matrix exponentials, applications)

·        Differential equations; matrix exponentials

·        Symmetric matrices and positive definite matrices (real eigenvalues and orthogonal eigenvectors, tests for xTAx > 0, applications)

·        Similar matrices 

·         Singular Value Decomposition -- orthonormal bases that diagonalize A

Next is a tentative course outline.

 

Week
Topics

Sections

8/27
Vector spaces and subspaces. Independence, basis and dimension.

 S: 1.1, 3.1, 3.5; G: Sec. 1

8/30

The geometry of linear equations. Elimination with matrices. Matrix algebra.

 2.1 – 2.5.

 9/3
 Labor Day

 

9/6

LU and LDU factorization. Transposes and permutations.

 2.6 – 2.7.

9/10, 9/13

Vector Spaces and Subspaces. The null-space of A.

 Rectangular Ax = b.

 3.1 – 3.4.

9/17, 9/20

 Row reduced echelon form. Independence, basis and dimension. The magnificent four. Euclidean spaces.

3.3–3.6; G: Sec. 2

9/24

Orthogonality of the Four. Projections.

 4.1 – 4.2

9/27

Pre-exam review

 

10/1

EXAM 1: Chapters 1 to 3.6.  See topics in blue in the outline. solutions

 

10/4

Projections. Least squares

4.3 – 4.4; G: Sec. 2

10/9

 (Monday schedule). Gram-Schmidt, and A = QR.  

4.3 – 4.4

10/11

 Determinants.

5.1 – 5.2; G: Sec. 3

10/15

Applications of determinants; volume

5.2 – 5.3; Notes

10/18

Linear operators (transformations).

7.1 – 7.2 (skip “The Identity…” pp 377-8)

10/22

Change of basis. Co- and contravariant vectors

7.3, 7.4 (Subsec. Similar Matrices). G: Sec. 9, Chap. 4

10/23 (TUESDAY)

 Pre-exam review, LOW 3039  from 5-7pm. I will go over some problems from MIT test 2, for fall ’99, fall ’02, and spring ’05. You are welcome to bring your own problems you need help with.

 

10/25

 Complex matrices

 10.1-10.3

10/29

EXAM 2: Chapters 1 to 5, 7.1, with emphasis on topics in green in the outline.  

 

11/1

Eigenvalues and eigenvectors.

6.1 – 6.2

11/05, 11/08

Diagonalization.

 6.1 – 6.2

11/12, 11/15

 Differential equations. Symmetric matrices.

6.3 – 6.4

 11/19

 Positive-definite matrices. Similar matrices

6.5 – 6.6

11/21-11/25

 Thanksgiving break

 

11/26

Singular Value Decomposition (SVD)

 6.7, 7.4

11/29

 Exam review

 

12/03

EXAM 3: Chapters 1 to 5, 6.1 – 6.6, 7.1 – 7.3. The test will contain a short extra-credit problem on SVD

 

12/06

 Matrix norms, Complex Matrices, FFT

 9.2,10.2 – 10.3

12/10-11

 Review: Amos Eaton 216. Monday 12/10, 4-6 final I will bring to class

 

Thursday, 12/13, 3-6pm

Final Exam. Sage 5101

 

 

Homework problems: There will be homework assignments that will be collected and graded. As a rule, homework assignments are due on Thursday by 4:00 pm, every week except exam weeks. Late homework is not accepted.

Tests: I plan to give three midterm tests and a final. Tests will be similar to tests at MIT (see web.mit.edu/18.06/www). The use of calculators or notes is not permitted during the exams.

Grading: Homework 15%, Tests 51%, Final 34%

MATLAB: I plan to give a few MATLAB demos during the lectures. Some of the homework problems might be easier solved with MATLAB, and you are welcome to learn it. However, learning MATLAB is not mandatory or necessary.

STATEMENT OF ACADEMIC INTEGRITY: The guiding principle is that work that you present for grading as your own should in fact BE your own. With respect to exams, this means that no assistance or collaboration of any kind is permitted. With respect to homework, you are encouraged to talk to other students about the problems. But before doing this, it is in your best interest to think over the problems on your own. Of course you must write your own solutions.

HOMEWORK ASSIGNMENTS

Assignment #1A, due 6 September; Assignment #1B, due 13 September; Assignment #2, due 20 September; Assignment #3, due 27 September

No office hours on October 5; sorry for the inconvenience

Assignment #4, due 11 October Solutions

Office HOURS Wednesday October 10, 3-5

Test 1 Solutions, Test 2 Solutions, Test 3 Solutions (for the test 3 solutions are to the last year's test, which is similar)

Assignment #5, due 18 October

Assignment #6, due Thursday October 25. solutions (new)

Assignment #7, due Thursday November 8.  Solutions

Assignment #8, due Monday November 19 (note the date change). Solutions

Assignment #9 (on SVD). This is a bonus assignment (worth 50% of a regular assignment). It is due by 4 pm Friday 12/7/2007.
Problem set 6.7: #4, 7, 15, 16

solutions will be posted on 12/8.

Back to V. Roytburd's home page 

Last updated November 30, 2007