Linear operators and vector spaces form the backbone of the operator formulation of quantum mechanics. I want to ask are there operators in classical physics too? Are these operators defined on some vector space? Examples can also be helpful.
One can also make the distinction between linear operators and their representations on vector spaces. You do not necessarily need to find a basis to work with linear operators. For the most part, one can assume that there is some representation with vector spaces possible.
Space and time are vector spaces, space-time of special relativity is a vector space. However, since the latter has a non-trivial metric tensor, it is different from $\mathbb R^4$. General relativity adds curvature to this space, so one would rather call it a manifold. All the tangent spaces, however, are still usual vector spaces.
You have vector spaces as tangent spaces wherever you have some manifold. For some trivial manifolds, they are a vector space themselves. And there are manifolds everywhere in physics:
- space and time, as well as space-time
- gauge groups like U(1) for electromagnetism and then for all the quantum field theoretical forces like electroweak (SU(2)) and strong (SU(3)) force; string theory, super-symmetry and grand unified theories have even more complex groups
- configuration space in Lagrangian mechanics
- phase space in Hamiltonian mechanics
You can also make it more general with differential geometry where there are not only vectors but also co-vectors (differential forms). And classical mechanics and classical electromagnetism can be described beautifully using differential forms.
So in short: Vectors and their generalizations are everywhere in physics.