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

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.