We recently added Sheldon Axler’s Linear Algebra Done Right to our list of approved textbooks. While Springer publishes a print edition, Axler recently made a PDF freely available online. The fourth edition of the book, released last month, contains many new examples and exercises.
This text, written to support a second course in linear algebra, provides a rigorous treatment of its subject with an emphasis on abstract vector spaces and linear maps. The “done right” in the title refers to the author’s perspective that the theory of eigenvalues and eigenvectors is best developed without relying on determinants. While the language is clear and precise, the tone is also colloquial and inviting.
In addition to the text, Axler’s website provides an extensive collection of videos, roughly one for each section of the book, and accompanying slides of lecture notes.