I am a freshmen student in mathematics at Moscow State University (in Russia) and I'm confused with placing the subject called "analytic geometry" into the system of mathematical knowledge (if you will).
We had an analytic geometry course in fall; now we are having a course in linear algebra and it seems like most of the facts from "analytic geometry" are proved in a much more systematic and easier manner (quote from wikipedia "Linear algebra has a concrete representation in analytic geometry"). Many of our progressive professors also think that analytic geometry should be eliminated from the curriculum to clear more space for a linear algebra course.
So I'm confused: 1) if analytic geometry is a "concrete representation" of linear algebra, then why is it studied along with calculus (and not along with linear algebra) in US universities? (e.g. textbooks like Simmons )
There were, however, interesting parts of the course that were not covered in linear algebra: synthetic high-school-style treatment of beautiful topics like non-Euclidian and projective geometries. Then 2) why is not there a separate course for such topics in US curricula? As I understand US freshman math majors study 2 basic subjects - real analysis and (abstract+linear) algebra (math 55 at Harvard, 18.100 and 18.700-702 at MIT). Are these geometric topics integrated into one of these courses or are not they considered worth studying for a modern math major?
PS. This question is also important for me because it helps a lot to browse through US top universities for textbooks they use and notes. Unfortunately, Russian mathematical school is now in tatters and US textbooks are often significantly better. And since in high school geometry was among my favorite subjects I am particularly concerned about our geometry sequence and want to browse through best geometry syllabi.