Linear Algebra by Jim Hefferon (free)

Jim Hefferon:

Standard coverage  Linear systems and Gauss’s method, vector spaces, linear maps and matrices, determinants, and eigenvectors and eigenvalues.
Free  The book is Freely available, including its source.

Developmental approach  It covers the requisite material and proves all the results, but it does not start by assuming that students are already able at abstract work. Instead, it proceeds with a great deal of motivation, many computational examples, and exercises that range from routine verifications to a few challenges. The goal is, in the context of developing the material of an undergraduate course, to raise each student’s level of mathematical maturity.

Extensive exercise sets, with worked answers to all exercises  Sometimes material described on the web as a book is really lecture notes. That’s fine but from notes to a book is a long way. That means things like figures and an index, but most importantly means exercises. Each subsection here has many, spanning a range of difficulty. In the Answers book each exercise is covered, completely, including proofs.

Popular  Downloadable for twenty years, this book has been used in hundreds of classes at many schools as well as by thousands of individuals for independent study.
Applications  Each chapter finishes with four or five short supplemental topics. These are good for reading or projects, or for small group work.

Extras  There are beamer slides for classroom presentations, and there is a lab manual using Sage.

Prerequisite  One semester of calculus.

Reviews.  Here are some: the Mathematical Association of America review, the American Institute of Mathematics, the Open Textbook Library (includes a number of reviews), one from a longstanding site for free texts.