> Pedagogically, the challenge to teaching linear algebra is that you start with "here's systems of linear equations, we can put them into matrices and now here's row operations to solve them," and you end up with "now matrices are actually representations of linear operators on vector spaces, let's analyze the properties of this specific operator." Usually, this is also coupled with a reluctance to actually discuss vector spaces, since the meat of it involves abstract algebra, which usually comes after linear algebra.

There are many, many linear algebra books. There is not just one approach; it all depends on what audience the author, or author team, is writing to.