# Is there a relation between these definitions of vectors?

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".)

• "vector" to physicists means "element of a tangent space on a manifold". (Or maybe an element of a cotangent space, or a field of such elements etc). Jun 21, 2020 at 18:09
• You may want to contemplate differing views of a complicated situation. The vector space article has historical background. Jun 21, 2020 at 18:12
• Wikipedia mentions specific differences between mathematics and physics vector concepts. Jun 21, 2020 at 18:22
• When you said triples, did you mean pairs? Jun 21, 2020 at 18:29
• It's worth noting that your physics definition is the definition of a contravariant or tangent vector given here Jun 21, 2020 at 18:31

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 $$R^9$$.