Around last December, I embarked on a project of writing an introductory linear algebra textbook, starting from the definition of vector spaces and leading up to the spectral theorem. Currently I have a draft that is about 150 pages long, and I plan on improving it when I have time. The book will be available for free from the following link: A rough guide to linear algebra.
The motivation for the project was a sort of academic elitism. I wanted there to be a book that would be most useful for a student with a strong mathematical background, planning to pursue a career in pure mathematics. The philosophy is very similar to that of Math 55, although I wanted to only focus on developing the theory in an abstract, morally correct manner, rather than to introduce the reader to a variety of topics.
It strikes me as peculiar that there seems to be a universal agreement among mathematicians on what the “correct abstract viewpoint of linear algebra” is, but it is difficult to find a textbook that actually develops linear algebra from this perspective. I asked myself how I learned the abstract viewpoint, but I don’t exactly remember. I guess I partly learned it from Math 55a, and later filled in the gaps by learning other mathematics such as homological algebra or vector bundles or representation theory. Well, that is not too bad a way to learn linear algebra, but I believe it is more efficient to learn linearly algebra correctly before going on to learn more advanced mathematics. For instance, it’s probably is not a good idea to learn about vector bundles before knowing why a vector space is not the same thing as its dual vector space.
One practical problem for the abstract approach is that most students taking an introductory linear algebra course probably will not have the mathematical maturity to digest the abstract formulation, or at least, will have a very hard time learning the material. Also, in such courses, there will be a lot of students not majoring in pure mathematics, and I do not find it necessary or even desirable that such people learn the abstract theory of vector spaces. Hence I believe the book will be useful for a very narrow scope of students, those who are confident that they want to be a mathematician, and who are willing to spend energy on learning the material abstractly.
The target reader I had mind were students who have had significant experience in competition mathematics, such as mathematical olympiads. In fact, I first started developing the outline when I was giving a small mini-course to the 2017 Korean IMO team. For such students, I believe it is not a bad idea to learn linear algebra before starting college.
I would really appreciate if people send me corrections, comments, or any feedback. Since the book is likely to be read by an extremely small number of people, questions or requests for clarification are also very welcome.