Now available in Open Access, this best-selling textbook for a second course in linear algebra is aimed at undergraduate math majors and graduate students. The fourth edition gives an expanded treatment of the singular value decomposition and its consequences. It includes a new chapter on multilinear algebra, treating bilinear forms, quadratic forms, tensor products, and an approach to determinants via alternating multilinear forms. This new edition also increases the use of the minimal polynomial to provide cleaner proofs of multiple results. Also, over 250 new exercises have been added.
The novel approach taken here banishes determinants to the end of the book. The text focuses on the central goal of linear algebra: understanding the structure of linear operators on finite-dimensional vector spaces. The author has taken unusual care to motivate concepts and simplify proofs. A variety of interesting exercises in each chapter helps students understand and manipulate the objects of linear algebra. Beautiful formatting creates pages with an unusually student-friendly appearance in both print and electronic versions.
No prerequisites are assumed other than the usual demand for suitable mathematical maturity. The text starts by discussing vector spaces, linear independence, span, basis, and dimension. The book then deals with linear maps, eigenvalues, and eigenvectors. Inner-product spaces are introduced, leading to the finite-dimensional spectral theorem and its consequences. Generalized eigenvectors are then used to provide insight into the structure of a linear operator.
From the reviews of previous editions:
Altogether, the text is a didactic masterpiece. ― zbMATH
The determinant-free proofs are elegant and intuitive. ― American Mathematical Monthly
The most original linear algebra book to appear in years, it certainly belongs in every undergraduate library ― CHOICE
M. Abicht –
PRO:+ Very good structure, absolutely no technical ballast+ clear definitions and theorems+ The high level of abstraction right from the start helps students to get to grips with the nature of not only linear algebra, but also mathematics itself.+ Nice notes in the margins of the text about history, well-known mathematicians and interesting facts.+ Due to its brevity and the omission of unnecessary technical burden, the book also serves as a perfect reference work (I used it a lot during my undergrad and grad studies, even today while working in a consulting job).+ New Chapter on Multilinear Algebra and a lot of other new helpful additions to edition 3 of the book (see screenshot)+ Very, very, very good tasks (especially helpful to learn conceptual thinking)+ Book came in perfect binding quality (see screenshot). The third edition was not printed by springer themself, rather from a print on demand Shop in the Netherlands with lower quality, this one ist far better than the third edition.+ OPEN ACCESS Book. You can support the author and buy the book, like I did OR you can get the free PDF by Springer or from the authors website “axler.net”. Sheldon, THAN YOU for supporting the next generation of Mathematicians.+ The third edition of this book laid the foundation for me to become the mathematician I am today. When I struggled during second semester of my undergrad studies (I even failed on my first try in Linear Algebra 1), THIS book kept me on track, showed me the beauty of math and led me to a masters, graduated with distinction (compare LinkedIn, if you do not believe me) . THANK YOU Sheldon and greetings from a guy somewhere in Europe whose mathematical career YOU saved.CONTRA:- NO SOLUTIONS TO THE EXERCISES, but then again, that is intended (But … in my opinion, one should be free to decide to use solutions in general and when to use them).- The topic of Linear Systems of Equations and how to calculate with them is completely missing (Again, this is intended, because it serves as a second course in Linear Algebra – For instance, in Germany you do two courses of Linear Algebra with highest rigour right from the beginning, so it could be useful to get a second source for things like Gauss Algorithm et cetera if you are not studying in the US).- Not only is there no emphasis on calculation techniques, they are completely avoided.
Maxim Nikolaev –
The title of the book says it all. Sheldon has done it right. I guess all I need to mention is that the printing quality is very good
ausbin –
I am a graduate student who studies an area steeped in linear algebra. I have found this book invaluable, particularly the author’s hard work on making proofs elegant and readable. I had bought a copy of the third edition previously, but I bought the fourth edition because I greatly appreciate the Axlerization of the tensor product (which I live and breathe) in the last chapter.
Erika –
Came new and sealed. No complaints!
vicente –
estubo muy bien la compra y fue muy buena la entrega
MV –
For those of you who wish to really know whats behind those elementary, middle school silliness of matrices truly and understand vectors, use two things:
1. This book
2. ChatGPT
This book will give you a really solid foundation of:
1. Vector and vector spaces
2. Linear Maps and matrices
3. Eigenvalues and EigenVectors
And linear algebra in general. I wish I’d read this book in my elementary – I’d have more interest in math if I did.
Remember to really ask questions with your choice of AI – mine is ChatGPT – combining this book with a dialog and doing problems can really clarify a lot of things. Good luck!