Linear Algebra, MTH 513
Syllabus Fall 2000
Mon. Wed. 4:30 - 5:45, Rm 106, Tyler Hall
Professor: N. Eaton
Homework due dates
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
Matrix Analysis by Roger A. Horn and Charles
Applied Linear Algebra, Third Edition by Ben
Noble and James W. Daniel.
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,
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
We will cover the following sections from Matrix
||All Eigenvalues and Eigenvectors
||Jordan Canonical Form
||The minimal polynomial
||The Reyleigh-Ritz Theorem and the Courant-Fischer
||Single Value Decomposition
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
(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,
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:
Your paper will be graded based on the following
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.
State the problem.
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.
Show how the math is used in solving the general
Illustrate with an example using maple.
Give a list of references.
The minimum length of the paper should be about four
interest of the application.
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
Due dates for homework assignments.
||2, 6, 10
|Project - Application
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