MTH 513 - Fall 2007 Linear Algebra   Meets TR:  5:00-6:15 – 106 Tyler Hall Instructor:  Dr. Nancy Eaton E-mail me: Phone:  874-4439 Office:  Rm. 208, Tyler Hall Office hours:  Wednesday 9-12 or by appointment Visit my web page: http://www.math.uri.edu/~eaton/

Students who require accommodations and who have documentation from
Disability Services (874-2098) should make arrangements with me as soon as possible.

 Text:  Linear Algebra (4th Edition) (Hardcover) by Stephen H. Friedberg, Arnold J. Insel, Lawrence E. Spence Check Amazon.com

Course Content

Principle topics of linear algebra are covered.  Vector spaces are emphasized along with linear transformations and their relationship to matrices.  Topics are selected from the following list

1. Vector Spaces.  NotesCh1P1  NotesCh1P2

Ø                 Introduction. Vector Spaces. Subspaces. Linear Combinations and Systems of Linear Equations. Linear Dependence and Linear Independence. Bases and Dimension. Maximal Linearly Independent Subsets.

2. Linear Transformations and Matrices.  Notes

Ø                 Linear Transformations, Null Spaces, and Ranges. The Matrix Representation of a Linear Transformation. Composition of Linear Transformations and Matrix Multiplication. Invertibility and Isomorphisms. The Change of Coordinate Matrix. Dual Spaces. Homogeneous Linear Differential Equations with Constant Coefficients.

3. Elementary Matrix Operations and Systems of Linear Equations. Notes3&4

Ø                 Elementary Matrix Operations and Elementary Matrices. The Rank of a Matrix and Matrix Inverses. Systems of Linear Equations—Theoretical Aspects. Systems of Linear Equations—Computational Aspects.

4. Determinants.

Ø                 Determinants of Order 2. Determinants of Order n. Properties of Determinants. Summary—Important Facts about Determinants. A Characterization of the Determinant.

5. Diagonalization. Notes

Ø                 Eigenvalues and Eigenvectors. Diagonalizability. Matrix Limits and Markov Chains. Invariant Subspaces and the Cayley-Hamilton Theorem.

6. Inner Product Spaces. Notes

Ø                 Inner Products and Norms. The Gram-Schmidt Orthogonalization Process and Orthogonal Complements. The Adjoint of a Linear Operator. Normal and Self-Adjoint Operators. Unitary and Orthogonal Operators and Their Matrices. Orthogonal Projections and the Spectral Theorem. The Singular Value Decomposition and the Pseudoinverse. Bilinear and Quadratic Forms. Einstein's Special Theory of Relativity. Conditioning and the Rayleigh Quotient. The Geometry of Orthogonal Operators.

7. Canonical Forms.

Ø                 The Jordan Canonical Form I. The Jordan Canonical Form II. The Minimal Polynomial. Rational Canonical Form.

Homework - 30%
Presentation - 20%
Midterm exam  - 25%  Study Guide
Final exam  - 25%

I will use the following scale for your grade in this course:

A:  92pts or above

A-:  90pts or above

B+:  87pts or above

B:  82pts or above

B-:  80pts or above

C+:  77pts or above

C:  72pts or above

C- or lower is considered failing in a 500 level course

Exams

There will be a Midterm Exam and an in-class Final. Your Midterm Exam will be on Thursday, Oct 25 and your final will be Tuesday, Dec 18.  The exams will test your understanding of the material we cover in class.  You will need to recall the theorems that we covered and understand their significance, answer questions in general about implications of these theorems. Before each test, I will provide you with a list of theorems from class that I would expect you to be able to prove.

Homework Assignments:
Problem sets from the book will be assigned throughout the semester.  The book includes some exercises for working out examples and some statements to prove.   You do not need to hand in solutions to all the exercises assigned.  Only hand in the ones indicated.  For the others, check the answers in the back of the book.  For the theory questions, which require a proof, you are expected to write the solutions up carefully and hand them in by the due dates given.  This list will be updated as we go.

Assignments

 Section Exercises Hand In Due Date 1.2 1, 12, 13, 14, 15 - 1.3 1, 6, 8, 11, 18, 20 HK#1 -  6, 18, 20 Sept 13 1.4 1, 2a, 2c, 5a, 5c, 10, 12 HK#2  - 10, 12 Sept 18 1.5 1, 2, 9, 16 HK#3 -  16 Sept 20 1.6 1, 2c, 2e, 9, 12, 29 HK#4 -  12 Sept 25 2.1 3, 6, 13, 14, 29 HK #5  - 14, 29 Sept 27 2.2 2b, 2e, 4, 8 HK #6  -  4 Oct 2 2.3 3, 9 HK #7  -  9 Oct 4 2.4 4, 5, 6, 9, 10 HK #8  -  9, 10 Oct 9 2.5 2a, 2c, 5, 9, 10 HK #9  -  9, 10 Oct 11 2.6 3 3.1 7 HK #10  -  7 Oct 23 3.2 19, 21, 22 HK #11 -   22 Oct 30 4.2 23, 28, 30 HK #12  -  23, 28 Oct 30 5.1 HK #13 -   6, 12 Nov 1 5.2 HK #14  -  12 Nov 6 5.4 HK # 15  - 11 Nov 8 6.1 HK #16 -  10, 12 Nov 20 6.2 HK #17 – 10 Nov 27 6.3 “  – 13a Nov 27 6.4 HK #18 – 12 Nov 29 6.5 HK #19 – 10, 12, 13 Dec 4 6.6 7.1 HK#20 classwork Dec 6

HkPts:  Each exercise will be graded and comments will be given.    Each is worth up to 4 HkPts.  The grade on your homework will be the percent of points out of the total possible number of points – to be determined (close to 150).

Here are some tips in doing homework.

Try each problem on your own.

Ask me about the ones that you still do not completely understand.

Take notes whenever we go over a problem in class.

Carefully write a final version of each problem, as best you can, and keep them together in a binder.

Presentation:   Present one theorem to the class from the following list.  Also solve the homework problem given.  We will follow the book order of the theorems in this course, and when we get to yours, it will be your turn to present it.  Present it in chalk on the board, but provide a handout with all the details given.  Provide the solution to the exercise as well in your handout.  We may or may not have time to go over your solution to the exercise in the class.  Keep in mind that during your presentation, you must provide the proofs of all parts that are missing in the book.  For instance, the proof given in the book may refer to an exercise.  If so, you should provide the proof of that exercise.  Please refer to all theorems by their statements rather than their number.  These are to be typed.

1.     Theorem 1.6 and Corollary.  Section 1.6, Number 29  - M. Heissan

2.     Theorem 4.7.  Section 4.3, Number 15 –  A. Gilbert

3.     Theorem 5.22.  Section 5.4, Number 7 - N. Vankayalapati

4.     Corollary to Theorem 5.23.  Section 5.4, Number 5 – Q. Ding

5.     Theorem 5.25.  Section 5.4, Number 23 – J. Stockford

6.     Theorem 6.1.  Section 6.2, Number 6 – D. Hadley

7.     Theorem 6.2.  Section 6.2, Number 11 – C. Lynd

8.  Theorem 6.15.  Section 6.4, Number 6 – M. Drymonis

Guide to working with others:

There are benefits to discussing the problems with your classmates.  If you become stuck on a problem, fresh ideas from someone else might provide you with some new angles to try.  In the academic community as well as in business and industry, people often work in teams.  So, it is good to get some practice working with others.  Working with someone from class will help you to improve your math communication skills as well.

I encourage you to work together under the following circumstances.

Ø     Begin the problem on your own and do as much as you can.

Ø     Ask someone from the class to explain the basic outline of a solution.

Ø     If working with someone else, sometimes they will also ask you for your understanding of the basic outline.

Ø     Take what you learned and write out the solution on your own, using your own words.

Ø     Never copy word for word from anyone else’s paper.  In fact it is better not to look at anyone else’s completed written solution or to show yours to anyone else.

Ø     If you still do not completely understand the solution, you can ask your professor to look at what you wrote and try to clear up any parts of the solution that are not completely clear and accurate.

The following is an excerpt from the University Manual.

8.27.11 A student's name on any written exercise (theme, report, notebook, paper, examination) shall be regarded as assurance that the work is the result of the student's own thought and study, stated in the student's own words and produced without assistance, except as quotation marks, references and footnotes acknowledge the use of other sources of assistance. Occasionally, students may be authorized to work jointly, but such effort must be indicated as joint on the work submitted.  Submitting the same paper for more than one course is considered a breach of academic integrity unless prior approval is given by the instructors.

Schedule

 Tuesday Thursday September 6 Chapter 1 11 Chapter 1 13 2.1, 2.2 18 2.3 20 2.4 25 2.5 27 2.6 October 2 3.1 4 3.2 9 4.2 11 4.3 16 5.1 18 5.2 23 5.4 25 Midterm Exam 30 6.1 1 6.2 November 6 6.3 8 6.4 13 6.5 15 6.6 20 6.7 22 6.10 27 7.1 29 7.1 December 4 7.2 6 7.3 20 – 3pm Final