I have looked up precise definition of a vector field and a vector space but I could not understand the relationship between them.
On wikipedia
A vector field is:
Given a subset $S$ in $R^n$, a vector field is represented by a vector-valued function $V: S → R^n$ in standard Cartesian coordinates $(x_1, ..., x_n)$.
A vector space is:
$(V, F, +, *)$ satisfying vector space axioms
Field and space sounds like the same thing...but apparently quite different in terms of their definition. What is the key relationship between these concepts?
A vector space is the place where a vector field lives in.
More formally, a vector space is a set of vectors which satisfy some basic properties, such as the ability to add and multiply them.
A vector field is a specific function defined on part of all of the vector space.
For example, consider the example of $\Bbb{R^3}$ representing space with the Sun at the origin. Then imagine a vector field in $\Bbb{R^3}$ where each point has an arrow pointing toward the sun and representing the strength of the gravitational pull at that point.
It is very much analogous to the real number line versus a function defined on the real line.