Comment by kccqzy
I have taken several linear algebra courses, one from my high school and two from universities. The thing is, not all courses of linear algebra will discuss rotations the way you discuss it. One reason is that sometimes a high school linear algebra course cannot assume students have learned trigonometry. I've seen teachers teach it just to solve larger linear systems of equations. Another reason is that sometimes a course will focus just on properties of vector spaces without relating them to geometry; after all who can visualize things when the course routinely deals with 10-dimensional vectors or N-dimensional ones where N isn't a constant.
I think teaching beginner linear algebra using matrices representing systems of equations is a pedagogical mistake. It gives the wrong impression that matrices are linear algebra and makes it difficult for students to think about it in an abstract way. A better way is to start by discussing abstract linear combinations and then illustrating what can be done with this using visualizations in various coordinate systems. Once the student understands this intuitively, systems of equations and matrices can be brought up as equivalent ways to represent linear transformations on paper. It’s important to emphasize that matrices are convenient but not the only way to write the language of linear algebra.