January 11, 2012

Office hours:    Mon. and Wed. 3:30 p.m. – 4:30 p.m.; Fri. 11:00 a.m. – 12:00 noon
                        Other times by appointment or by chance

All exams are required. Make-ups will be given, no later than 5 pm on May 14, only if you notify me before the exam starts or as soon as possible and supply a reason. Unavoidable conflicts may be resolved by taking parallel exams before the scheduled date, if prior notice of one week is given to the instructor. Assignments submitted late are subject to a 10% penalty if submitted before any students receive their corrected solutions back in class; late assignments receive no credit if submitted after any students receive their corrected solutions back in class.

It is your responsibility to attend as many classes as possible, to do all homework problems assigned in class, and to complete all course requirements. You are responsible for any topics or problems covered in class (whether or not you are present). Attendance will be spot-checked on at least three occasions.

In the event of a university declared critical emergency, Salem State University reserves the right to alter this course plan. Students should refer to salemstate.edu for further information and updates. The course attendance policy stays in effect until there is a university declared critical emergency. In the event of an emergency, please refer to the alternative educational plans for this course located at http://www.salemstate.edu/~arosenthal/ma734. Students should review the plans and gather all required materials before an emergency is declared.

Global Goals:

1. To increase the student's understanding of the basic concepts of linear algebra.
2. To make the student aware of the crucial importance of linear algebra to many fields in engineering, science, probability, statistics, computer science and economics.
3. To enhance the student's ability to reason mathematically.
4. To enable the student to appreciate the beauty of linear algebra and its value.
5. To be able to solve simple problems involving linear algebra by hand and more complicated problems involving linear algebra using a computer.
6. To realize why many mathematicians consider the software package MATLAB (available on the computers in the Math Lab) to be the outstanding software for linear algebra, and be able to use it to solve some problems.

Learning Objectives: The student will be able to:

1. Perform numerical linear algebra with the help of the software program  MATLAB, OCTAVE or SCILAB
2. Find eigenvalues and eigenvectors of a matrix and use them to diagonalize a diagonalizable matrix.
3. Understand how the matrices of a linear transformation with respect to different bases are similar.
4. Be able to find the limit of powers of a matrix using its diagonalization and apply this to analyzing the long term evolution of discrete dynamical systems.
5. Discuss the properties of inner product spaces.
6. Find an orthonormal basis for a subspace in an inner product space using the Gram-Schmidt orthonormalization process.
7. Test whether a matrix is positive definite and understand the significance of whether it is.
8. Find the matrix of a quadratic form and the principal axes of the quadratic form.
9. Classify quadratic forms as positive definite, negative definite or indefinite.
10. Analyze quadratic forms geometrically and determine whether they correspond to ellipsoids.
11. Count the number of operations which a computer would require to perform certain numerical linear algebra algorithms.
12. Find the singular value decomposition of a matrix and be able to apply it to such applications as image processing and finding the pseudoinverse of a matrix.
13. Understand computational issues that arise when solving Ax = b and how the condition number of A affects the sensitivity of the solution to small changes in the data.
14. Use computationally efficient algorithms such as the QR algorithm to find eigenvalues of a matrix.
15. Find the sum, cartesian product and tensor product of two vector spaces.
16. Find the Jordan canonical form of a matrix.
17. Find the minimal polynomial of a linear transformation.

Course Requirements for Assessment:

Assessment	Scheduled Dates	Learning Objectives	Percent of Grade
Homework	Due Dates on most Mondays	1-17	20%
Test 1	Feb. 24	1-6	20%
Test 2	April 2	7-12	20%
Cumulative Final Exam	May 7	1-17	40%

Tentative Schedule of Course Topics:

Week	Topics	Sections in Text
1	Eigenvalues and Eigenvectors; Diagonalizability	5.1, 5.2
2	Matrix Limits and Markov Chains	5.3
3	Invariant Subspaces and the Cayley-Hamilton Theorem	5.4
4	Inner Products and Norms; the Gram Schmidt Orthogonalization Process and Orthogonal Complements	6.1, 6.2
5	The Adjoint of Linear Operator; Normal and Self-Adjoint Operators	6.3, 6.4
6	Unitary and Orthogonal Operators and Their Matrices	6.5
7	Orthogonal Projections and the Spectral Theorem	6.6
8	The Singular Value Decomposition and the Pseudoinverse	6.7
9	Bilinear and Quadratic Forms	6.8
10	Conditioning and the Rayleigh Quotient	6.10
11	The Geometry of Orthogonal Operators	6.11
12	The Jordan Canonical Form	7.1, 7.2
13	The Minimal Polynomial	7.3
14	The Rational Canonical Form	7.4
15	Final Exam

Bibliography:

1. Anton, Howard, Elementary Linear Algebra, 10th ed., Wiley, Hoboken, New Jersey, 2010.
2. Bretscher, Otto, Linear Algebra with Applications, 4th ed., Pearson Prentice Hall, Upper Saddle River, New Jersey, 2009.
3. Bryan, Kurt and Tanya Leise, The $25,000,000 Eigenvector: The Linear Algebra behind Google, SIAM Review,, 48, 2006, pp. 569-581.
4. Cheney, Ward and David Kincaid, Linear Algebra: Theory and Applications, 2nd ed., Jones and Bartlett Learning, Sudbury, Mass., 2012.
5. Fausett, Laurene V., Applied Numerical Analysis using MATLAB, 2nd ed., Pearson Prentice Hall, Upper Saddle River, New Jersey, 2008.
6. Johnson, Lee W., R. Dean Riess and Jimmy T. Arnold, Introduction to Linear Algebra, 6th ed., Pearson Addison-Wesley, Boston, Mass., 2008.
7. Kolman, Bernard and David R. Hill, Elementary Linear Algebra with Applications, 9th ed., Pearson Prentice Hall, Upper Saddle River, New Jersey, 2008.
8. Larson, Ron and David C. Falvo, Elementary Linear Algebra, 7th ed., Cengage Learning, Boston, Mass., 2011.
9. Lay, David C., Linear Algebra and its Applications, 4th ed., Addison-Wesley, Boston, Mass., 2012.
10. Leon, Steven J., Linear Algebra with Applications, 8th ed., Pearson Prentice Hall, Upper Saddle River, New Jersey, 2010.
11. Math Archives, http://archives.math.utk.edu (Web site).
12. Math Forum, http://mathforum.org (Web site).
13. Nicholson, W. Keith, Elementary Linear Algebra, 2nd ed., McGraw Hill, New York, NY, 2004.
14. Octave repository, http://octave.sourceforge.net/ (Web page for a freeware program very similar to MATLAB).
15. Penny, John and George Lindfield, Numerical Methods Using MATLAB, 2nd ed., Prentice Hall, Upper Saddle River, New Jersey, 2000.
16. Poole, David, Linear Algebra: A Modern Introduction, 3rd ed., Cengage Brooks/ Cole, Boston, Mass., 2011.
17. Recktenwald, Gerald W., Numerical Methods with MATLAB: Implementations and Applications, Prentice Hall, Upper Saddle River, New Jersey, 2000.
18. Sadun, Lorenzo A., Applied Linear Algebra: The Decoupling Principle, 2nd ed., American Mathematical Society, Providence, R.I., 2008.
19. Scilab home page (an open source platform comparable to MATLAB), http://www.scilab.org/ (Web site).
20. Strang, Gilbert, Linear Algebra and its Applications, 4th ed., Thomson Brooks/ Cole, Belmont, California, 2006.
21. Texas Instruments calculator software, http://education.ti.com/educationportal/sites/US/sectionHome/download.html (Web site).
22. Williams, Gareth, Linear Algebra with applications, 7th ed., Jones and Bartlett Publ., Sudbury, Mass., 2011.