I know there are a few similar questions already asked about Vector Spaces, but maybe asking in a different way may help others who are stuck, and it may prompt a reply from a different angle.
My understanding so far is below, but can't quite put it together:
Vector space is a concise term when referring to any mathematical object (functions, column matrix/vector, etc.) that can be multiplied by a number and added.
So, you have a set of vectors, V and a Field (F) with its elements, the scalars.
Now something is a field if it is a set that allows us to do addition, subtraction, multiplication, and division AND where doing so would keep you within that set. So, if 7 and 10 are in the set, but the set included only positive numbers like 7 and 10, this is a problem as doing subtraction of those two mentioned elements, 7 - 10, would lead us to a negative number..so it does not meet the property of staying in F after these operations. Such a set could not be a field?
Then we have the two operations, vector addition and scalar multiplication, which conclude the definition of a vector space. We can add two vectors to make a third, and we can scale a vector and create a new one.
So, the question:
- Is the Field then simply referring to the set of numbers that we can use to multiply a vector in Ax=b because it meets the properties discussed above?
X1 * Vector1 X2 * Vector2 and so forth?
"Vector space is a concise term when referring to any mathematical object (functions, column matrix/vector, etc.) that can be multiplied by a number and added". What can "be multiplied by a number and added" is not the vector space, but the elements of a vector space.
"a field if it is a set that allows us to do addition, subtraction, multiplication, and division AND where doing so would keep you within that set". A set does not allow to do "addition, subtraction,...", a set is just a collection of elements. A field is a set $X$ AND two operations $+,\cdot:X \times X\rightarrow X$ satisfying certain conditions. What can happen is that you have a field $(X,+,\cdot)$ and wonder if a subset $Y\subseteq X$ is closed under $+,\cdot$ (that is, $a,b\in Y\rightarrow a+b,ab\in Y$). The same set $X$ can support different, non-equivalent operations $+,\cdot$, so being a field is not a property of $X$, but a structure we consider on it.
A field is a structure given before the vector space, that is, a vector space is a vector space over a field. You generally fix a field $F$ and then consider a vector space $V$ over that field. The field $F$ are those elements that multiply the vectors of $V$. For example, $\mathbb R^2$ is usually a vector space over $\mathbb R$ (the classic $\alpha(v_1,v_2)=(\alpha v_1,\alpha v_2)$), but it can also be a vector space over $\mathbb C$ (as $(a+ib)(v_1,v_2)=(av_1-bv_2,av_2+bv_1)$), you need to specify which field act on $\mathbb R^2$.
Now, if $V$ is a vector space over a field $F$ and you have a subfield $F'\subseteq F$, then you can make $V$ into a vector space over $F'$ (simply restrict multiplication by scalar to $F'$). So $F$ acts as a "parent" field, from which you extract a smaller field $F'$ over which you consider scalar multiplication. In the previous example, once you give the structure to $\mathbb R^2$ of a vector space over $\mathbb C$, then you can restrict to any subfield of $\mathbb C$ (for example $\mathbb R,\mathbb Q$,etc)
Edit A field $F$ is something like $\mathbb Q$, it is a set $F$ with an addition $+$ (which is a function $F\times F\rightarrow F$) and a multiplication $\cdot$. The multiplication on a field is a form of scalar multiplication (every field is a vector space over itself, like $\mathbb R$ is a real vector field, that is, a vector field over $\mathbb R$). A vector space over $F$ is, similarly, a set $V$ with an addition $+:V\times V\rightarrow V$ (different from the addition in the field $F$, take for example $\mathbb R^2$ with componentwise sum $(v,w)+(v',w')=(v+v',w+w')$) and a "scalar multiplication" $\cdot:F\times V\rightarrow V$ satisfying certain conditions (like $a(v+w)=av+aw$ for all $a\in F,v,w\in V$, etc). Usually "scalar multiplication" is reserved for vector spaces, while "multiplication" is reserved for fields. What's important is that addition and multiplications on a set $F$ are not necessarily unique, that is, the same set $F$ can have other operations on it making it into a field. So a field (and similarly a vector space, or any other algebraic structure really) is not simply a set, but a set with some fixed operations.
Finally, "closed under" usually means that a certain set contains the results of certain operations performed on its elements. In general, if I have a set $X$ and a certain operation $f:X^n\rightarrow X$ (like addition or multiplications, or something more general), is say that a set $Y\subseteq X$ is "closed under $f$" if $$f(a_1,\cdots,a_n)\in Y,\text{ for all }a_1,\cdots,a_n\in X$$ that is, when $f$ has elements of $Y$ as "input", it returns an element of $Y$. For example $\mathbb Z\subseteq \mathbb Q$ is closed under addition and multiplication ($\{0\}$ is also closed under multiplication and addition), $\{-1,1\}$ is closed under multiplication, but not addition and $\{-1\}$ is not closed under multiplication nor addition.