Relationship between vector space and its defining field 
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*Is there a vector space $V$ over field $\mathbb F$ where the components of the vectors in $V$ are not in $\mathbb F$? I think yes, because you can think of $\mathbb C$ as an $\mathbb R$-vector space.


*Is there an example where the field $\mathbb F$ is not a subset of the "value set" of the components of $V$? I think yes, as you can think of $\mathbb R$ as a $\mathbb C$-vector space. Does an example exist using usual addition and multiplication?
Hence, can you say that there isn't a relationship which is easy to describe (wihtout the axioms defining vector space) between $\mathbb F$ and $V$?
 A: Vectors in a vector space do not have "components" until you introduce a basis (a coordinate system).
The set of points in the ordinary Euclidean plane with a fixed point specified as the origin and a fixed unit length is a vector space over the real numbers: addition is the parallelogram law and scalar multiplication is stretching lengths. Once you choose two points that form a nondegenerate triangle with the origin to specify two coordinate axes you can represent any point as a pair of real numbers.
When you think of the complex numbers as a two dimensional real vector space, each element  is represented as a pair of real numbers when you use $1$ and $i$ to determine the real and imaginary axes.
When you think of the complex numbers as a one dimensional vector space over itself, each complex number is represented by itself.
The set of real valued functions with domain the unit interval is a vector space over the real field: it's straightforward to define addition and scalar multiplication. Representing those functions with "coordinates" is a subtle problem you may encounter in more advanced courses.
