What is Linear Algebra?

Linear Algebra is a branch of mathematics as old as mathematics
itself. The solving of the simple linear equation ax + b = 0 may be
considered as the original problem of this subject. Although this
problem presents absolutely no difficulty at all, the method which
solves it (i.e. if a ‚ 0, then the unique solution is given by x =
(-b)/a, no solution if a=0 and b ‚ 0 and infinitely many solutions if
both a and b are zero), together with the properties of the
corresponding linear function y=ax+b, are fundamental models for the
ideas and methods of all of linear algebra. For example, the basic
idea behind the solution of a system of linear equations in several
unknowns is that of replacing such a system by a chain of these
simple equations (by "eliminating" the variables).

The study of systems of linear equations acquired a new significance
after the creation of analytic geometry (by Réné Descartes); it was
possible to reduce all the fundamental questions about the
arrangements of lines and planes in space to the investigation of
such algebraic systems. Here we see a fantastic idea which is the
undercurrent of most mathematical processes:

Start from a problem in geometry, translate the problem in
the language of algebra, solve the resulting algebra problem using the
algebraic tools and finally transport the solution back to geometry.

We will encounter several such examples of this process in our
course. This process of back-and-forth translation itself leads to
new discoveries in linear algebra. For example, the search in the
18th century for the general solution of n equations in n unknowns
led Leibnitz and Cramer to the notion of determinants. I can go on
and on but let me stop with this observation: today we can safely
declare that there is no branch of science which does not apply
Linear Algebra. The real world applications need Linear Algebra. That
is, perhaps, one reason why we demand Linear Algebra (and her sister,
Calculus) as a pre-requisite for all programs in the Faculty of
Science (and even in Economics, Engineering and Business
Administration). Enjoy the course!

R. Padmanabhan
Instructor, Linear Algebra
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