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Linear Algebra Done Wrong
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.
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.
213 pages, Unknown Binding
First published January 1, 2004
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Displaying 1 - 7 of 7 reviews
September 12, 2020
I skimmed some of this when refreshing elementary linear algebra.
Seems like a fine first linear algebra book for mathematically inclined students. I think Axler downplays matrices too much for a first course, whereas this books strikes a golden middle ground. Namely, the abstract concepts of vector spaces, subspaces and linear maps are emphasized from the beginning, together with matrices which give more concrete examples.
Seems like a fine first linear algebra book for mathematically inclined students. I think Axler downplays matrices too much for a first course, whereas this books strikes a golden middle ground. Namely, the abstract concepts of vector spaces, subspaces and linear maps are emphasized from the beginning, together with matrices which give more concrete examples.
January 9, 2025
Not my favorite. Best to stick with linear algebra done right.
September 7, 2024
not the best introduction to the subject but acts as a nice supplement to the books by Strang and Axler. My main issue was that the notation was inconsistent, the numbering of the theorems and definitions weird (unlike in Axler say), and the back and forth between the matrix and linear map point of view a bit confusing. What I did really like however, were his discussions of change of basis matrices/maps, and the various points of view he gave on matrix multiplication. All in all, I’m glad I read it, but honestly think the path via Strang’s MIT course (the free one)/his textbook and Axler's Linear Algebra Done Right is the best way through the subject. If, however, you’re in a rush, and your somewhat familiar with mathematical proofs, this is a pretty good option.
January 5, 2023
Delicious book. Helped me prepare for my linear algebra exam by consuming the night before. When you print it out there's just enough writing on each page for the ink to season it just right. If you can, try to get thicker paper. My fiancee and I enjoyed it as a wonderful aperitif with a good cabernet, though I think it would have paired much better with a pinot. It definitely helped me understand linear algebra from the inside out.
February 24, 2025
If you had to pick one between LADR and LADW then choose LADR. However after reading LADR I would still encourage you to read this, since at this point you should be able to skim through it, to get the determinant and matrix treatment.
November 23, 2025
4 stars is likely more appropriate but I'm biased towards this book and its approach. A good foray into higher level mathematics for a young undergrad.
Best read with Axler's as a companion, or as the companion to it.
Best read with Axler's as a companion, or as the companion to it.
Displaying 1 - 7 of 7 reviews