MATHEMATICS 251- LINEAR ALGEBRA I- FALL 2005
Instructor: Dr. Alex Freire (PhD 1988; at U.T.K. since 1991)
Office: Ayres 207A, 974-4313, e-mail: freire at math dot utk dot edu
Office hours: MWF, 1:35-3:35 (or make an appointment by e-mail)
Remark: it is OK to drop in
unannounced if you just have a quick
question
GOAL: first course in linear
algebra- emphasis on solution of linear systems, geometry
of linear transformations and a survey of basic applications.
Technically the only pre-requisite
is high-school algebra, but the course will only make sense if you have
more experience
(at least calc I and II, and the course will be easier if you have Calc
III and/or DEs)
COURSE
OUTLINE
COURSE LOG
WHY THIS MATHEMATICS IS
IMPORTANT AND WHO SHOULD STUDY IT
Linear algebra is the theory behind solutions of systems of
linear equations- it underlies essentially
ALL established applications of mathematics to physics,
chemistry, biology, engineering, statistics,
economics, etc. The reason is that it is conceptually simple, and
the computational implementation
is very well developed. Some examples of applications: linear circuit
theory (EE); linear differential equations
(useful in all branches of engineering); population dynamics (math
ecology); linear models in statistics;
linear optimization (economics), image processing. It can be said that
the goal of fully one-half of
"multivariable calculus" is to reduce systems of nonlinear equations to
linear systems. In numerical analysis
(or `scientific computing') we learn how to reduce linear partial
differential equations to very large
linear systems (by `discretizing' the space or time variables). Quantum
mechanics (which underlies all modern
physics and chemistry) is based on infinite-dimensional linear algebra
(in fact, Heisenberg called it
`matrix mechanics'); all forms of `spectroscopy' in chemistry
rely on `representation theory' (Math 617), which is
basically advanced linear algebra.
The subject has computational (matrix algebra), algebraic (vector
spaces, linear transformations) and
geometric aspects (projections, rotations)- part of the reason to
introduce it early is to form the habit of moving freely
among these three different modes of reasoning. Acquiring
the skill of thinking geometrically about
algebraic problems (and vice-versa) should be a major goal of students
taking the course.
Conclusion: if your major is mathematics, statistics, physics,
chemistry, economics, earth sciences, business, or any
form of engineering, this is an important course. (Incidentally, it
includes all the material in Math 200, and
then some). For the biological sciences or pre-med students (except for
those with an interest in math
ecology), the value is in learning an important mathematical method,
but the interest of the applications may not be clear.
For students in the social sciences- any quantitative method of
analysis in your field is likely to involve linear algebra (in
particular, multivariate statistical methods are based on it).
Like most introductory topics in mathematics, the course will
seem `dry' and `unmotivated' a lot of the time;
the reason is that one has to develop a language and basic computational methods before one
can analyze
anything interesting. The student who controls a natural tendency to
reject `purely abstract' material and works
hard at learning this language will be rewarded with greatly increased
powers of quantitative analysis. I will try to
reserve at least the last 1/4 of the course (say, 10 lectures) to a
survey of applications.
COURSE
POLICIES AND PROCEDURES (a long list, including
grading- please read it carefully)
Students
with disabilities: if you need special arrangements to
tske this class (including exams) due to a disability,
please contact the Office of Disability Services (2227 Dunford Hall,
974-6087 V/T, http://ods.utk.edu/ )