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The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.
This book can be thought of as a very pure-math version of linear algebra, with no applications and hardly any work on matrices, determinants, or systems of linear equations. Instead it focuses on linear operators, primarily in finite-dimensional spaces but in many cases for general vector spaces.
The book is not as bold as its title indicates; “done right” refers to the very technical device of avoiding determinants. But avoidance does have the useful effect of forcing you to think directly in terms of vectors and operators and not dive into a pile of calculations. Axler has come up with some very slick proofs of things that normally require a lot of grinding away, and that in itself makes the book interesting for mathematicians. The book is also very clearly written and fairly leisurely.
The book is pitched as a second course in linear algebra, although it doesn’t require any previous knowledge of the subject. I’m not sure why it is so pitched — possibly it is so readers won’t expect to learn all those calculation techniques that it doesn’t cover, or it is to scare away students who aren’t ready for this level of abstraction.
The book resembles in many ways Halmos’s Finite-Dimensional Vector Spaces. The focuses are somewhat different; Halmos wanted to write a book that would deal with all the properties of infinite-dimensional spaces but only prove them for finite-dimensional spaces. Halmos also deals with a wider variety of structures, in particular tensor products and projections and various types of subspaces. Axler concentrates on the properties of linear operators, and doesn’t introduce other concepts unless they’re really necessary. Halmos is (as usual) very concise, while Axler is more expansive and easier to follow.
Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at MathNerds.org, a math help site that fosters inquiry learning.