[mlevental]

I got thrown and I know a lot of math. I got thrown because the field the vector space is over is not what I focus on when thinking about the structure of the vector space. indeed it took me a second to even come up with a reason why the ground set has to be a field at all - so that you can undo scalings - and while it's a good reason it still feels trivial.

edit: in fact i learned linear algebra from hoffman kunze

[gizmo686]

You can generalize vector spaces to work over arbitrary rings (in which case it is a module). I did have to do a double take on that question, because in my mind when I see someone talking about a vector space over a ring, I just silently translate it to a module (because 90% of the time, the person just misspoke, 9% of the time, even if they meant vectorspace, the immediate follow up would be "no, but does your point still stand of we consider it a module?", and maybe 1% of the time they are actually being adversarial.

[joe_the_user]

As I remember, most first courses in linear algebra don't deal with the subject as anything like abstract algebra and aren't going to be targeting students who can distinguish a ring from a field. Essentially, until you can past the calculus sequence, most math in the US is centered on calculation with only a few forays into proofs and definitions.