How do I conceptualize a vector space I have read a number of articles and posts about vector spaces I am still left unconfident that my conceptual model of a vector space is correct.
Here is an example to illustrate how I am thinking about them currently. To simplify things lets put a constraint on our vector space and only focus on a subset of the Real numbers. Imagine a 2d space with points (-5,5) (5,5) (-5,-5) (5,-5) that create a bounding area for our vector space. We can assume all the axioms of a vector space apply to this space (essentially scalar multiplication and vector addition).
Would the vector space be the set of all possible vectors that can fit in that box? Or would it be the space in which the vectors live (the 2d box)? Wikipedia describes a vector space as "a collection of objects, called vectors", which makes me think it is the former.

A vector space is not just a collection of vectors; it's a collection of vectors with an addition operation and a multiplication by a number operation defined on it.
An $n$-dimensional vector space is:


*

*a point when $n = 0$,

*a line when $n = 1$,

*a plane when $n = 2$,

*ordinary space when $n = 3$,

*analogous spaces when $n \geq 4$, though not as easy to visualize.


The square you describe is only part of the plane it lies in, not the whole plane, so it's not a vector space. 
Technically, you can prove this by saying that the vector $v = (5,5)$ belongs to the square, but $2v = (10,10)$ doesn't. (I am presuming that the usual multiplication rule is intended.) The square doesn't satisfy the axioms of a vector space, because the operation $2 \cdot v$ is undefined within it.
