Linear Algebra Done Right

Sheldon Axler

Digital versions PDF
Source available No
Exercises Yes
Solutions available No
Instructor materials Slides of lecture notes
Additional resources Extensive collection of videos
Creative Commons BY-NC license
  • For a second linear algebra course for advanced students
  • Nine chapters, 390 pages
  • Originally published by Springer, now available as an open access book
  • Emphasizes abstraction and mathematical rigor
  • Key concepts are illustrated with examples
  • Numerous exercises at the end of each section
  • For more information and to download

This book is meant to support a second linear algebra course for advanced undergraduate or beginning graduate students. Though the text could be used for a one-semester course, there is likely more material than could reasonably fit into one semester; some guidance on the choice of topics is provided in the preface. The exposition presents a rigorous treatment of linear algebra with an emphasis on abstract vector spaces and the structure of linear maps. The “Done Right” in the title refers to the author’s decision to develop the theory of eigenvalues and eigenvectors without using determinants. While the language is clear and precise, the tone is also colloquial and inviting. There are very few applications, and some readers may wish for more motivation. The fourth edition, which recently became available, includes many more exercises and examples.


  1. Vector Spaces
  2. Finite-Dimensional Vector Spaces
  3. Linear Maps
  4. Polynomials
  5. Eigenvalues and Eigenvectors
  6. Inner Product Spaces
  7. Operators on Inner Product Spaces
  8. Operators on Complex Vector Spaces
  9. Multilinear Algebra and Determinants