Question on the definition of vector spaces. My question is perhaps useless, but I want to shed some clarity on this matter.
I'm bothered by people that say a vector space is a "bunch of vectors". Or that a vector space "consists of vectors"/"is the set of vectors".
I think it makes more sense to say that a vector space of n-dimensions is an n-dimensional space (whatever a space is exactly, IDK), that contains all possible n-dimensional vectors, and that if one operates with them via addition and scalar multiplication, you don't jump out of that space (i.e. you stay within that space).
I don't like to say that this space is made up of vectors, or consists of them. I'm just confused about what those statements even mean.
Could anyone tell me what people mean by saying that it consists of vectors or that a vector space is just a set of vectors?
Thanks.
 A: If you think about the axioms of a vector space and think about them algebraically you  can see that a vector space is a set $S$ with a binary operation of vector addition which gives $S$ the structure of an abelian group, and a binary operation of scalar multiplication by elements of a field $F$ is "compatible" with the group structure (distributive, associate, identity). This gives $S$ the structure of a module over $F$, and as $F$ is a field this is just another way of saying it is a vector space. 
In this sense we often call the elements of $S$ "vectors", so saying a vector space is made up of vectors is, as you suggest, meaningless in some sense, it isn't that the space is made up of some special things called vectors, but rather we just happen to call the elements of $S$ "vectors" to remind us that as we are considering these elements in the context of the vector space structure (and not just as elements of a set). 
Not sure if that helps, as I'm not sure whether you are familiar with groups and modules, but I hope it is a little clearer. 
If you are thinking of vectors geometrically, then $S$ is a set consisting  of elements that represent displacements of the space (e.g translations of the Euclidian plane), these displacements are given by a direction and a magnitude, and scalar multiplication on these elements alters the magnitude. In this sense then "vector" is just the same as saying "displacement".
A: Your first encounter with linear algebra can be confusing if you haven't seen a lot of vector arithmetic in physics.  Perhaps think about it this way.  We are quite comfortable with addition, subtraction, multiplication, and division in $\mathbb{R}$.  How do we generalize this arithmetic to pairs of real numbers?  That is, lets define $\mathbb{R}^2$ to be the set of all pairs of two real numbers $(a,b)$.  How should we define addition and subtraction in this case?  The natural choice is to simply add and subtract componentwise: $(a,b)+(c,d)=(a+c,b+d)$, where $(a,b)$ and $(c,d)$ are just two elements of $\mathbb{R}^2$, and indeed this definition is free of any pathologies or "weirdness".  
What about multiplication, though?  We could define $(a,b)(c,d)$ by $(ac,bd)$, in a similar way to addition, but it turns out this is a bad definition.  To see this, consider $(1,0)(0,1)=(0,0)$ by this definition, hence $(1,0)$ and $(0,1)$ wouldn't have multiplicative inverses (division is the same as multiplication by a multiplicative inverse). Since this definition of multiplication doesn't work, we choose the next best thing: multiplication by scalars.  So we can take a real number $\alpha\in\mathbb{R}$ and multiply it by a pair $(a,b)$ to get $(\alpha a,\alpha b)$.
That's what a vector space is (a more mathematically robust definition exists, sure, but I think this intuitive picture will serve you better at this point).  In the case of $\mathbb{R}^2$, its the set of all pairs of real numbers with a notion of addition and the best kind of multiplication we can manage.  By extension, $\mathbb{R}^n$ is the set of all lists of $n$ real numbers $(a_1,a_2,\dots,a_n)$ with addition defined by component-wise addition and multiplication again defined by scalar multiplication.
A: A vector space is a set (with some special properties), so it is a collection of elements. To keep the breadth of vocabulary in check, we all agree to call the elements in a vector space vectors instead of elements. So "a vector space is a collection of vectors" is just defining what a vector is. The term vector space has already been defined without referring to vectors.
You want to come to the table from that perspective, not by imagining a bundle of arrows or anything like that.
