# 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.

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.
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.