Linear Algebra by Jim Hefferon is a Free text for an undergraduate mathematics course. It covers the standard topics: Gauss's method, vector spaces, linear maps, determinants, and eigenvalues and eigenvectors.
The PDF of Linear Algebra, and of the answer book, the beamer slides, and the lab manual, are at the book's web page, hefferon.net/linearalgebra. You can also buy hard copies for minimal cost from booksellers such as Amazon for single-person copies, or from the college bookstore source if you are an instructor; see the web page for more information and links.
What sets this book apart is its developmental approach. Linear Algebra does not assume that students are mathematically sophisticated, so it includes lots of examples, lots of motivation, and clear exposition. At the same time, it helps instructors raise the level of sophistication of their students. Consequently, everything is proved, and it nudges readers to see the advantages of a more advanced viewpoint-- in particular, vector spaces and linear maps appear as early as is practical.
In addition to the main text, every chapter ends with a few Topics that illustrate applications or extensions of the material. These are suitable for independent reading, for group work, or for a one-day presentation as a break.
There are many exercises; I have aimed for two dozen for each class day. The answer book contains completely-worked solutions for every exercise, including the proofs. The book and answer PDFs are arranged so that if you click on the question number then you are taken to the answer and if you click on the answer number then you come back to the question.
There are beamer slides for classroom use. These are keyed to the text, including that they use the same wording and numbering for theorems, definitions, etc. The slides that I use pass over some proofs to instead do illustrative examples; there is also a version with all the proofs.
I have made videos, based on the classroom slides, covering the entire semester. If you are a classroom teacher, this is an additional resource (for some students, being able to pause or replay is helpful). If you are studying on your own then these are essentially my classroom lectures put on videos.
And, there is a lab manual for computer work, using Sage. It introduces students to doing linear algebra on a computer and also introduces some important applications that are difficult by hand but easy on a machine.
Linear Algebra is the recipient of the Mathematical Association of America's 2020 Solow Award, which recognizes its "impact on undergraduate education in mathematics."
Linear Algebra is Free. You can use it under the GNU Free Documentation license or under a Creative Common Share Alike license, at your choice.
This text has been available online for over twenty five years. It has been used in many hundreds of classrooms. It can be used as the main text or as a supplement to another text. It is also widely used for self-study.
The text is written on LaTeX and associated systems. The source is at GitLab.
This is the fourth edition of the text. This is the final edition, although there will be new printings to correct bugs.
I am always glad to hear from users, but particularly with bug reports. Please email me using my contact information on hefferon.net/linearalgebra.
If you are working with the git repository then note that I do not accept requests for project write access. In addition, I find that the pull workflow that works for writing software does not work as well for writing a book. Please, email me.
I am a professor in the Department of Mathematics and Statistics at Saint Michael's College in Colchester Vermont.
More information, and the PDF's, are at https://hefferon.net/linearalgebra.