Right, and that's a perspective you pick up on in a second course in linear algebra, typically. The key insight really is that the core concept is that of a vector space, rather than vectors per se. The only thing we really ask of vectors is that it be possible to apply linear functions with coefficients from your favorite field to them. Other than that, vectors themselves aren't that interesting: it's more about functions to and from vector spaces, whether it's a linear function V -> V or a morphism V -> W between two different vector spaces.
This is actually a common theme of mathematics, that the individual objects are in some sense less interesting than maps between them. And, of course, the idea that any time you have a bunch of individual mathematical objects of the same type, mathematicians are going to group them together and call it a "space" of some kind.
In fact, my previous paragraph is pretty much the basis for category theory. One almost never looks at individual members of a category other than a few, selected special objects like initial and terminal objects. A lot of algebra works in a similar way. If I could impart one important insight from all the mathematics I've read, done, and seen, it would be this idea of relations being more important than the things themselves.
Just plain algebra is abstract math, and even the most common everyday math most overlapping common programming work.
I didn’t even know until today there was a concept called linear algebra, it was taught to me as introductory geometry alongside other geometry concepts. So that’s neat to learn!
