Linear Algebra Book & Presentation

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 

A First Course in Linear Algebra is an introductory textbook designed for university sophomores and juniors. Typically such a student will have taken calculus, but this is not a prerequisite. The book begins with systems of linear equations, then covers matrix algebra, before taking up finite-dimensional vector spaces in full generality. The final chapter covers matrix representations of linear transformations, through diagonalization, change of basis and Jordan canonical form. Along the way, determinants and eigenvalues get fair time. There is a comprehensive online edition and PDF versions are available to download for printing or on-screen viewing. Physical copies may be purchased from the print-on-demand service at Lulu.com.

It seems therefore desirable to give permanent form to the lessons of experience so that others can benefit by them and be encouraged to undertake similar work.
H.M. Cundy and A.P. Rollet, Mathematical Models, 1952

This textbook has more freedom than most (but see some exceptions). First, there is no cost to acquire this text, and you are under no obligation whatsoever to compensate or donate to the author or publisher. So in this most basic sense, it is a free textbook. Therefore you can also make as many copies as you like, ensuring that the book will never go out-of-print. You may modify copies of the book for your own use - for example, you may wish to change to a prefered notation for certain objects or add a few new sections. I have applied a copyright to the book, and subsequently licensed it with a GNU Free Documentation License (GFDL). It is this combination that allows me to give you greater freedoms in how you use the text, thus liberating it from some of the antiquated notions of copyright that apply to books in physical form. The main caveat is that if you make modifications and then distribute a modified version, you are required to again apply the GFDL license to the result so that others may benefit from your modifications.

If nature has made any one thing less susceptible than all others of exclusive property, it is the action of the thinking power called an idea, which an individual may exclusively possess as long as he keeps it to himself; but the moment it is divulged, it forces itself into the possession of every one, and the receiver cannot dispossess himself of it. Its peculiar character, too, is that no one possesses the less, because every other possesses the whole of it.
Thomas Jefferson, Letter to Isaac McPherson, August 13, 1813

It is hoped that by this arrangement, others will help improve the book through rapid correction of errors and contributions of exercises and new material. Examine an evaluation copy, and if you like what you see, consider a donation or purchase to support this experiment in providing quality textbooks at reasonable prices.

"Instructors who wish to teach a pure linear algebra course that emphasizes rigor and formal mathematics will be able to make good use of this material and feel secure in the knowledge that the book is not going to go out of print. Finally, the price is right."
SIAM Review, Book Reviews, December 2007

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