What is the definition of the field a vector space is defined over and how does this field translate into a sub-vector space of this space?? I'm quite new to the topic of linear algebra and feel like have confused myself with the definitions of a field, vector space and sub vector space:
I'm familiar with the conditions a set of numbers must satisfy to be considered a field as well as well as the conditions a set of objects must satisfy to be considered a vector space. 
I know that a vector space is closed under scalar multiplication by members of its field. However, it seems to me that such a statement allows for a vector space to  be defined over, possibly, multiple fields (for example, if a vector space is defined over C, it should also be defined over R)
Considering the definition of a sub vector space as a subset of the vector space, V,  such that it's closed under addition and scalar multiplication by members of V's field, it follows that depending on what the field for V is taken to be, we can end up with sub vector spaces of the same space defined strictly over different fields (for example, given that V is defined over C, then we can say it's also defined over R. Using R as the field over which we define a sub vector space we can no longer say that the sub vector space has the same field as V). Doesn't this go against one of the conditions for a sub vector space that states, it must be defined over the same field as V?
 A: This is a thoughtful question. Ordinarily, we deal with a vector space $V$ as a v.s. over a particular field $K$, and the fact that $K$ may have subfields $k$, over which $V$ is also a vector space, is acknowledged, but not usually made use of.
When we speak of a sub-vector space $W$ of such a $V$ as above, we ought most correctly mention over which subfield $k$ it is that $W$ is a vector space. But almost always, what we have in mind is for $W$ to be a vector space over the $K$ that $V$ was a v.s. over.
Here’s an example: The Cartesian plane $V=\Bbb R^2$ is a two-dimensional vector space over the real field $\Bbb R$. Since I haven’t said anything about subfields of $\Bbb R$ such as the rational field or any of the infinitely many others, when I say, “Let $W$ be a proper subspace of $V$”, it would be willfully overprecise to ask me over which subfield of $\Bbb R$ I was taking as the scalar field of $W$, since it goes almost without saying that I meant for $W$ to be an $\Bbb R$-subspace of $V$.
If you want to take subspaces over other subfields of the original scalar field, you ought to specify this with wording such as: “Let $W$ be a $\Bbb Q$-subspace of $V$, now considering $V$ as a $\Bbb Q$-space.”
A: If $\varphi:K\to L$ is any homomorphism (embedding) of fields, then - as you observed - any vector space $V$ over $L$ also determines a vector space structure on the same underlying set $V$ over $K$, simply by defining
$$\kappa\cdot v:=\varphi(\kappa)\cdot v,\quad\kappa\in K\,.$$
Note that this is not a subspace of $V$, but rather a reduct (when multiplication by each scalar is regarded as a single unary operation -- we just forget the multiplications by $\lambda\in L\setminus K$).
