Vector spaces over a Field in linear algebra From "Linear algebra done right" by S.  Axler:

"a vector space over $\mathbb R $ is called a real vector space  and a vector space over $\mathbb C $ is called  a complex vector space"

Does this imply that the Field (by which I mean the type of the scalar used for multiplication) is systematically also the type of the coordinates of the vector ?
Do we ever study the case where the scalar used for multiplication is in  $\mathbb R $ but the vector coordinates are in $\mathbb C $ , or vice - versa ?
 A: We speak of a finite-dimensional vector space $V$ over a given field $\mathbf F$. If $\mathcal B=(u_1,\dotsc,u_k)$ is a basis for $V$, then any vector $v\in V$ may be written as a linear combination of the basis vectors:
$$
v = c_1u_1+\dotsb+c_ku_k
$$
where $c_1,\dotsc,c_k$ are the coordinates of $v$ with respect to $\mathcal B$. The coordinates $c_1,\dotsc,c_k$ each live inside of the field $\mathbf F$.
A: I would think the answer to the question is 'Yes.' To answer, I'll assume the answer is 'no;' so, let's say scalars and coordinates are in different fields. 
Let F and G be fields and let W be a vector space over F. 
Let V be a vector in W. (So V = [v1, v2, .... , vn], where the coordinates of V are in F.) Let c be an element of G (our scalar).
Then using the definition of our vector and of scalar multiplication:
cV = c[v1, v2, ..., vn] = [cv1, cv2, .... , cvn], which is required to be an element of F by definition of vector space. 
We run into a problem because multiplying V by c could put the image of V outside of the target field. For above, let F be the reals and G be the complex plane. 
