> started with rings and fields
This seems unnecessary for a first class. I would agree with the parent here that the class should help build intuition and theoretical understanding. Presenting a rich theory with full generality is probably not the best way to do that though. There are some beautiful concepts and geometric understanding to be had with the reals. The abstraction to vector spaces defined over arbitrary rings and fields builds naturally once you have that base.
Do you remember whether that was a course specifically targeting CS students? I am asking because the linear algebra course at "my" university was a course for CS and math students and it was (sadly) not computational at all. If I remember correctly (it's quite some years ago), we started with rings and fields, then vector spaces, and matrices kind of naturally followed from that.