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A textbook for an honors linear algebra course (updated July. 2011):

Linear Algebra Done Wrong.

by Sergei Treil

 

From the Introduction:

The 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? 

Before 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. 


Terms of use:

Creative Commons License
This book  is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.

Additional details: You can use this  book  free of charge for non-commercial purposes, in particular for studying and/or teaching. You can  print paper copies of the book or its parts using either personal printer or professional printing services.
Instructors teaching a class (or their institutions) can provide students with  printed copies of the book and charge the fee to cover the cost of printing; however the students should have an option to use the free electronic version.


Download the book.

In the new (July 2011) I version  corrected numerous typos, and added some more detailed  explanations. 

 

  • Introduction and Table of Contents--- PDF, 167K.
  • Text of the book (July 2011)--- PDF, 1470K
  • Errata (list of typos found in the above version)
  • For the convenience of the reader I am posting the latest version of the text, with all the typos (that are found to date) corrected  here.  It will be changing as typos are found, unlike the official text above, which will remain the same (at least untill the summer)
If you want to see the older versions of the text, you can find it below:

 

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