From a purely mathematical perspective, the notion of scalars and vectors and their different roles makes sense to me. Vectors are elements of a given vector space $V$, and scalars are elements of the underlying field $F$ associated with the vector space $V$. A vector can then be represented with respect to a chosen basis, with its components with respect to that basis, in a heuristic sense (since a vector need not have a notion of magnitude and direction associated with it), quantifying how much of the vector "points" along each of the basis vectors. The scalars are then simply numbers that we can multiply vectors with, to "scale" a given vector (in some sense) or construct linear combinations of vectors, and since a scalar is simply a number, it is not represented with respect to a basis and does not have components, it is simply a number.
Does this concept of a vector space and associated scalars carry over to how vectors and scalars are defined in physics pretty much exactly, or are there any differences one should be aware of?