Linear Algebra Done Wrong (2004 version).
by Sergei Treil
From the Introduction:
title of the book sounds a bit mysterious. Why should anyone read this
book if it presents the subject in a wrong way? What is particularly
done "wrong" in the book?
answering these questions, let me first describe the target audience of
this text. This book appeared as lecture notes for the course "Honors
Linear Algebra". It supposed to be a first linear algebra course for mathematically
advanced students. It is intended for a student who, while not yet very
familiar with abstract reasoning, is willing to study more rigorous
mathematics that is presented in a "cookbook style" calculus type
course. Besides being a first course in linear algebra it is also
supposed to be a first course introducing a student to rigorous proof, formal definitions---in short, to the style of modern theoretical (abstract) mathematics.
The target audience explains the very specific blend of elementary
ideas and concrete examples, which are usually presented in
introductory linear algebra texts with more abstract definitions and
constructions typical for advanced books.
Another specific of the book is that it is not written by or for an
algebraist. So, I tried to emphasize the topics that are important for
analysis, geometry, probability, etc., and did not include some
traditional topics. For example, I am only considering vector spaces
over the fields of real or complex numbers. Linear spaces over other
fields are not considered at all, since I feel time required to
introduce and explain abstract fields would be better spent on some
more classical topics, which will be required in other disciplines. And
later, when the students study general fields in an abstract algebra
course they will understand that many of the constructions studied in
this book will also work for general fields.
Also, I treat only finite-dimensional spaces in this book and a basis
always means a finite basis. The reason is that it is impossible to say
something non-trivial about infinite-dimensional spaces without
introducing convergence, norms, completeness etc., i.e. the basics of
functional analysis. And this is definitely a subject for a separate
course (text). So, I do not consider infinite Hamel bases here: they
are not needed in most applications to analysis and geometry, and I
feel they belong in an abstract algebra course.