Linear Algebra, MTH 513 
Syllabus Fall 2000 
Mon. Wed. 4:30 - 5:45, Rm 106, Tyler Hall

Professor: N. Eaton 
Office: Tyler 222 
phone: 874-4439, Office Hours: Tues 3:30-4:00, Wed 2:00-3:00, Thurs 1:00-2:00 
or by appointment 
eaton@math.uri.edu

Homework due dates

Text

Required Text

Matrix Analysis by Roger A. Horn and Charles R. Johnson.

Recommended Text

Applied Linear Algebra, Third Edition by Ben Noble and James W. Daniel.

Course Content

This is an introductory course in Linear Algebra. We will cover concepts of linear algebra that will be useful in a wide variety other courses and applications. The required text for the course, Matrix Analysis, contains theory while the recommended book, Applied Linear Algebra contains both theory and applications. We will concentrate on the theory, using the applications mainly as motivation. Topics covered: eigenvalues, eigenvectors, eigensystems, unitary matrices and transformations, Shur decomposition, the QR decomposition, the singular-value decomposition, the Jordan form, and definite quadratic forms.

A basic knowlege of linear algebra such as is covered in an undergraduate course is required as background for this course. You must be familiar with such concepts as matrix algebra, solving systems of equations, finding inverses of matrices, vectors, vector spaces, linear transformations. This material is included in both texts and can be used for reference.

We will cover the following sections from Matrix Analysis.
Chapter Topics
1 All Eigenvalues and Eigenvectors 
2 Unitary Matrices 
Normal Matrices
Shur's Theorem
Cayley-Hamilton Theorem
QR-factorization 
3 Jordan Canonical Form 
The minimal polynomial
4 Hermitian Matrices 
The Reyleigh-Ritz Theorem and the Courant-Fischer Theorem
7 Single Value Decomposition

Motivation

Many problems in applied mathematics involve the study of transformations, that is, the way in which certain input data is transformed into output data. In many mathematical models of such complex situations, the transformations involved turn out to be linear in the sense that the sum of two inputs is transformed into the sum of their individual outputs and a multiple of an input is transformed into that multiple of the original output. The linear transformation might, for example, describe the evolution of some complicated system from one point in time to the next. The state of the system at any time might be described by the variables, xi in some vector space V, while the linear transformation A transforms the state xi in V into the state A(xi) in V. Since V is often of quite high dimension, it becomes difficult to understand how the system works. We seek much smaller subsystems. Say the subspace V0 of V is such that A(xi) is in V0 whenever xi is in V0. The subsapce V0 may actually be one-dimensional. We may find a nonzero vector x (an eigenvector) and a scalar l (an eigenvalue) for which A(x) = lx.

Determining the structure of the eigensystems of a matrix A is very important and motivates much of what is covered in the course. We see that it is equivalent to questions concerning decompositions of A and changes of basis. When the matrix A is normal, we may use unitary matices in such decompositions. We will study several such decompositions, which are not only useful in obtaining knowlege of of the structure of the eigensystems but for other applications as well. Other techniques apply to general matrices. The Jordan form is useful in analyzing defective matrices, those which are not diagonalizable.

Finally, we study quadratic forms which arise in diverse areas of applications and are useful in studying matrices.

Projects (25% of your grade)

You will have a project due after Thanksgiving break, Nov 27. Eigensystems provide useful information about matrices. Find a real world problem from behavioral, natural, physical, or social sciences, engineering, business, or computer science for which matrices serve as models and eigensystems aid in the solution of the problem. The recommended text, Applied Linear Algebra, contains many examples and references. You must use at least one source of reference other than that text. Also, run an example of your application using maple.

Each paper should contain the following elements:

  1. Give a brief overview of the area of application, enough so that the problem can be understood by a reader who is unfamiliar with the area.
  2. State the problem.
  3. Give the mathematics that are needed in solving the problem. State all theorems, even if we covered them in class. You don't have to provide the proofs of the theorems.
  4. Show how the math is used in solving the general problem.
  5. Illustrate with an example using maple.
  6. Give a list of references.
Your paper will be graded based on the following criteria:
  1. clarity,
  2. thoroughness,
  3. accuracy, and
  4. interest of the application.
The minimum length of the paper should be about four single-spaced pages.

Homework Assignments (25% of your grade)

Homework will be given from each section that we cover. The homework is to give you practice applying the theorems that we cover in class to new problems. The techniques used in your proofs will usually have the same flavor as those used to prove the theorems themselves, but often you will need to be creative in this process and put facts together in your own unique way to come up with a proof. This is the most challenging aspect of the course. The exercises assigned should be from a variety of levels so that you can work up to the hardest ones. I encourage you to work together under the following circumstances. Each person tries every problem before talking it over with someone else. Each problem that is written up and handed in should be essentially your own work. 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 in groups. It is another important part of doing mathematics to be able to communicate your ideas to someone else. Also, you may learn new approaches and techniques that you will be able apply to other problems.

Tests (Midterm 25%, Final 25%)

There will be a Midterm Exam and a Final. Your Midterm Exam will be on Oct 25 and your final will be Fri, Dec 15. Tests differ from homework in that I will test you on your understanding of the material we covered 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 that I would expect you to be able to prove.

Due dates for homework assignments.

Assignment  Problem Numbers  Due Date 
Handout  1,2,3  Sept 27
Section 1.1  1,5  Sept 27
Section 1.2 3,4 Oct  4
Section 1.3 2, 6, 10 Oct 11
Section 1.4 4,10 Oct 23
Section 2.1 1,2,3,12 Nov 8
Section 2.2 none
Section 2.3 6,8 Nov 8
Section 2.4 1,9 Nov 15
Project - Application Nov 27
Section 2.5 2,3,15 Dec 11
Section 2.6 5 Dec 11
Section 3.3 5,6 Dec 11


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