As a rule-of-thumb, I would stay away from any "linear algebra" textbook that does not even have definition of an abstract vector space in the first few chapter

As a rule-of-thumb, I would stay away from any "linear algebra" textbook that does not even have definition of an abstract vector space in the first few chapters. In a sense, it's like programming without data types. When I was in high school, I learned linear algebra from similar textbook, but it wasn't until college (and that was an engineering linear algebra course!) that I got any sort of understanding of what it's really about.