**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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