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Linear Algebra Done Right

Sheldon Axler
Publication Date: 
Number of Pages: 
Undergraduate Texts in Mathematics
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The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Allen Stenger
, on

Please see our review of the second edition. This third edition, appearing eighteen years after the second edition, is a further polishing of the existing approach. This book was and still is an interesting and useful text for a second course in linear algebra, concentrating on proofs after the concepts and mechanics have been covered in a first course.

One immediately-noticeable difference is that the book is formatted in what I call Modern Textbook Style, which means it is printed in full color and has lots of sidebars and incidental graphics. I have mixed feelings about Modern Textbook Style; it may be necessary to sell textbooks in the present environment, but (usually) it runs up the price of the book and may be distracting. Happily neither of these bad effects seem to have happened with the present edition, and the layout is pleasant and attractive.

In addition to the hundreds of little improvements and added exercises, the main substantive improvements are the addition of material on duality (functionals were covered in the previous edition, but not dual spaces), treatment of product and quotient spaces, and a new approach to deriving results for real spaces from those for complex spaces. The new approach is called here “complexification” and just means embedding the real space in a complex space (a product of the real space with itself, along with a definition of complex scalar multiplication). The author makes the observation several times that complex spaces are easier to deal with than real spaces, and the earlier edition concentrated on transformations on complex spaces and treated real spaces in the last chapter. The new edition does the same, but the complexification simplifies the real-case proofs a good bit because they can be done a corollaries of the complex case results.

Determinants still have the very small role they did in the earlier edition, and I still think this works well. Determinants are avoided by using alternative definitions that would be theorems in a determinant approach. For example, the characteristic polynomial of a linear transformation is defined in terms of its eigenvalues (the existence of which has already been studied, without determinants) rather than as a determinant.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis.

See the interactive table of contents on the publisher's web site.