I've come across two different definitions of vectors.
The first one is from linear algebra and it just defines vectors to be the elements of a vector space.
The other one is the one that we were taught in the physics class. There a vector was defined as anything whose coordinates transform in a particular way under certain transformations of space (the professor specifically gave example of the pairs of numbers $(x, y)\in\mathbb R^2$ and the transformation on $\mathbb R^2$ as rotation by angle $\theta$ so that $(x, y)\mapsto (x\cos\theta+y\sin\theta, -x\sin\theta + y\cos\theta)$).
Question: Is there a relation between the two? And can someone help making the second definition mathematically precise, and more general? (I don't know how to generally define "space" in "transformations of space".)
To oversimplify, mathematicians love abstraction and physicists are practical: they want to calculate stuff.
You are being subjected to both viewpoints. Physics often uses $R^3$, which satisfies the defining properties (axioms) of a vector space that were given in your math class. Thus, $R^3$ satisfies all the theorems about vector spaces that were proved there.
However, there is no harm in just understanding $R^n$ for now. It's much easier to understand the theory there.
BTW, the linear transformations from $R^3$ to itself (or the matrices that represent them) have a multiplication defined but, as a vector space, it's just $R^9$.