Professor:  Dr. Nancy Eaton

Office:  Tyler 222

Phone:  874-4439 

Office Hours:  Mon 12:30 – 1:30, Tuesday 12:30-2:30, or by appointment

E-mail:  eaton@math.uri.edu
 
 
 

Text:Matrix Analysis, by Roger A. Horn and Charles R. Johnson.

Link to Homework Assignments

Link to Point System for Homework

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 other books that you can check out from the library, such as Applied Linear Algebra, by Noble and Daniel contain both theory and applications. We will concentrate on the theory, using the applications mainly as motivation. Topics covered from: eigenvalues, eigenvectors, eigensystems, unitary matrices and transformations, Shur decomposition, the QR decomposition, the singular-value decomposition, the Jordan form, Hermitian matrices, and definite quadratic forms.
 

A basic knowledge 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, vector spaces, bases, dimension, and rank.Also solving systems of equations, row-reduced echelon form of matrices, determinants , finding inverses of matrices, vectors, vector spaces, linear transformations. This material is included in the introduction of our text.You may find it useful to have an undergraduate text such as the one used for MTH215 handy for reference.
 

We will cover the following sections from Matrix Analysis (time permitting).
 

1	Eigenvalues, eigenvectors, and similarity
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 Thm
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 vector, x, in some vector space, V, while the linear transformation, A, transforms the state x in V into the state Ax 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 Ax is in V₀ whenever x is in V₀. The subspace V₀ may actually be one-dimensional.We may find a nonzero vector x (an eigenvector) and a scalar l (an eigenvalue) for which Ax = 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 matrices in such decompositions. We will study several such decompositions, which are not only useful in obtaining knowledge 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. 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. Look in the library for books that give applications. You must use at least one source of reference other than our text. Also, run an example of your application using maple.

Each paper should contain the following elements:

1.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.Problem statement

3.The mathematics background which is needed to solve the problem. State all theorems, even if we covered them in class. You don't have to provide the proofs of the theorems that we covered in class.

4.An explanation of how the math is used to solve the general problem.

5.An example illustrated with Maple.

6.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 paper should be at least four single-spaced pages long.