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A textbook for an honorslinear algebra course (updated Sept. 4, 2017):
Linear Algebra Done Wrong.
by Sergei Treil
From the Introduction:
Thetitle of the book sounds a bit mysterious. Why should anyone read thisbook if it presents the subject in a wrong way? What is particularlydone 'wrong' in the book?
Beforeanswering these questions, let me first describe the target audience ofthis text. This book appeared as lecture notes for the course 'HonorsLinear Algebra'. It supposed to be a first linearalgebra course for mathematicallyadvanced students. It is intended for a student who, while not yet veryfamiliar with abstract reasoning, is willing to study more rigorousmathematics that is presented in a 'cookbook style' calculus typecourse. Besides being a first course in linear algebra it is alsosupposed to be a first course introducing a student to rigorousproof, formal definitions---in short, to the style of moderntheoretical (abstract) mathematics.
The target audience explains the very specific blend of elementaryideas and concrete examples, which are usually presented inintroductory linear algebra texts with more abstract definitions andconstructions typical for advanced books.
Another specific of the book is that it is not written by or for analgebraist. So, I tried to emphasize the topics that are important foranalysis, geometry, probability, etc., and did not include sometraditional topics. For example, I am only considering vector spacesover the fields of real or complex numbers. Linear spaces over otherfields are not considered at all, since I feel time required tointroduce and explain abstract fields would be better spent on somemore classical topics, which will be required in other disciplines. Andlater, when the students study general fields in an abstract algebracourse they will understand that many of the constructions studied inthis book will also work for general fields.
Also, I treat only finite-dimensional spaces in this book and a basisalways means a finite basis. The reason is that it is impossible to saysomething non-trivial about infinite-dimensional spaces withoutintroducing convergence, norms, completeness etc., i.e. the basics offunctional analysis. And this is definitely a subject for a separatecourse (text). So, I do not consider infinite Hamel bases here: theyare not needed in most applications to analysis and geometry, and Ifeel they belong in an abstract algebra course.
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