linear algebra - Is a vector field a subset of a vector space?

This is not actually a duplicate although it seems very similar to the another question I found on stackexchange titled "vector space or vector field?" If I have a vector field isn't each element of that field in n dimensions also an element of a particular vector space of similar dimension. Even if the subset does not satisfy all the properties of a vector space would it not still be a subset? Just curious if the "object" that resides in a vector field the same object that appears in some vector space of similar dimension. Thank you . Not really sure where this question belongs since one is a topic of linear algebra and the other is more in line with vector fields in calculus, which is what prompted my question to begin with.

Answer

Yes, any element of a vector field is a vector (with a dimension $n$), so ''all'' the vectors of a vector field are a subset of some vector space $W$. But a vector field is not simply a set of vectors, it is a function that assign a vector to any point in a space, that is a function $f:V \to W$ where $W$ is a vector space and $V$ can be a set, but usually has some structure (as a manifold). Also if $V$ is a vector space, this function $f$ may be not linear. So a vector field is a lot more that a subset of a vector space.