Determining if vector space or not. I am having a lot of trouble with what a vector space is and how I would determine if something is a vector space or not.  The question I have to answer is:  

Let $S$ be the set of all vectors $\begin{bmatrix} x \\ y \end{bmatrix}$ in $\mathbb{R}^2$ is $xy>0$, with the usual vector addition and scalar multiplication, is $S$ a vector space?

From our notes, I know that it must satisfy the conditions to be a vector space:
a) $v+w=w+v$
b) $u+(v+w)=(u+v)+w$
c) $v+0=0+v=v$
d) $v+(-v)=0$
e) $a*(bv)=(ab)*v$
f) $1*v=v$
g) $(a+b)u=a*u+b*u$  
But I have no idea how to apply any of these conditions to the question at all.
Any help would be appreciated. Thanks.
EDIT:
So looking at the back of my textbook at a similar question, it says that it is not a vector space because $u+v$ is in $V$ fails. So how would you be able to show that this fails? I've been trying to figure this out for the last 4 four hours.. I'm completely lost.
 A: Based on the comments by the OP and the question itself I think this is more of a how do I do proofs that are abstract, in some sense, where abstract in this case means showing a set of things is a vector space although you don't have specific numbers to work with.
First off I want to note that if this is the case I completely understand. My first abstract math course was in linear algebra and my first confusion arose in what the heck this vector space thing is.
Now, if I am correct thus far let's talk about vector spaces; a vector space is simply some specific set of elements (an element can literally be anything), with two operations associated with it - vector addition and scalar multiplication. From here on out I'm going to call an element of the vector space a vector. Let's discuss these two operations (vector addition and scalar multiplication) - but wait, I didn't explain what the heck an operation is. Well an operation is some sort of function that takes either two vectors or a scalar and a vector to another vector (more on what a scalar is later). For consider the vector space $\mathbb{R}$ where a vector is simply any real number. Then we can consider addition or multiplication to be operations - they take two vectors and send it to another vector. Ok now let's look at vector addition and scalar multiplication in depth:


*

*Vector Addition is the operation between any two vectors that is required to give a third vector in return. In other words, if we have a vector space $V$ (which is simply a set of vectors, or a set of elements of some sort) then for any $v,w \in V$ we need to have some sort of function called plus defined to take $v$ and $w$ as arguements and give a third vector in return; in other words we need to have some plus defined so that
$$
\mathrm{plus}(v, w) \in V
$$
Now note that instead of writing plus to denote this function, we often use the infix + to stand for this vector addition (infix means you put the function between two elements instead of before them) i.e. instead of $\mathrm{plus}(v, w) \in V$ we say that
$$
v + w \in V
$$

*Now let's look at scalar multiplication; first we need to define what exactly a scalar is - in fact this has a very nice abstract definition (it's any element in the underlying scalar field of the given vector space), but for all intents and purposes for this question, and I'm guessing for the class that this is for, you can think of a scalar as the type of number that makes up each of the vectors. That is if we consider the vector space $\mathbb{R}^2$ then a scalar for this vector space would be any real number (since real numbers make up each of the components of the vectors in $\mathbb{R}^2$. With this said we now need to define some sort of multiplication between a scalar and a vector which results in another vector; in other words we need to come up with a function multi which, for any $s$ which is a scalar for your given vector space, and any $v \in V$ we have
$$
\mathrm{multi}(s, v) \in V
$$
Again, instead of writing out multi everytime we want to signify we are using scalar multiplication we use the infix * so that instead of $\mathrm{multi}(s, v) \in V$ we write
$$
s * v \in V
$$
and even sometimes, when in context it makes sense we drop out the * and simply write
$$
sv \in V
$$
(in context means that it is understood within the given problem that $s$ is a scalar and $v$ is a vector).
Now let's talk about these conditions that a set of vectors must satisfy in order to be called a vector space; the first of these conditions, which is implied, is that the above operations must be well defined. That is, you have a well defined sense of vector addition and a well defined sense of scalar multiplication. Well defined, for vector addition, means that if we take any two vectors as input, we should get a vector as output. Similarly, well defined for scalar multiplication means that given any scalar and any vector we get another vector as output. After this we need to have that any combination of vectors and scalars satisfies the conditions that you listed.
I want to give an example of proving a few of the conditions are true for $\mathbb{R}^3$. Note that
$$
\mathbb{R}^3 = \left\{ \begin{bmatrix} a \\ b \\ c \end{bmatrix} \mid a,b,c \in \mathbb{R} \right\}
$$
First note that a scalar is any real number, since our vectors are made up of real numbers. Now we need to figure out some good idea for our definition of addition, and for scalar multiplication. The traditional way to define vector addition on $\mathbb{R}^3$ is by adding each element component by component (a component is each piece that makes up any given vector, so the first element in our vector of $\mathbb{R}^3$ is a component, and similarly so is the second and the third). In other words we define + as
$$
\begin{bmatrix} a \\ b \\ c \end{bmatrix} + \begin{bmatrix} d \\ e \\ f \end{bmatrix} = \begin{bmatrix} a + d \\ b + e \\ c + f \end{bmatrix}
$$
and we define scalar multiplication similarly component wise
$$
r * \begin{bmatrix} a \\ b \\ c \end{bmatrix} = \begin{bmatrix} r a \\ r b \\ r c \end{bmatrix}
$$
Now notice here that I represent any arbitrary vector in $\mathbb{R}^3$ as
$$
\begin{bmatrix} a \\ b \\ c \end{bmatrix}
$$
or any possible letters, with the assumption that each of the letters represents an arbitrary real number. I can do this since this is exactly what an arbitrary vector in $\mathbb{R}^3$ looks like - it looks like $\begin{bmatrix} \cdot \\ \cdot \\ \cdot \end{bmatrix}$ where each dot is an arbitrary real number (i.e. we can call each number some arbitrary letter and work from there).
Now first we need to ensure that our definitions of + and * are well defined, that is we need to make sure the result of vector addition and vector multiplication are still vectors. It turns out that given $a,b,c,d,e,f,r$ real numbers we have that $a+d,b+e,c+f,r a, r b, r c$ are all still real numbers so the elements of the form
$$
\begin{bmatrix} a + d \\ b + e \\ c + f \end{bmatrix}, \begin{bmatrix} r a \\ r b \\ r c \end{bmatrix}
$$
are still elements of $\mathbb{R}^3$. Now we need to prove that every condition in the list of conditions you provided holds for any arbitrary vector. Again I will take $a,b,c,d,e,f$ arbitrary real numbers and consider two arbitrary vectors made out of these numbers (which is then an arbitrary element in $\mathbb{R}^3$) and make sure that
$$
v + w = w + v
$$
Proof:
$$
\begin{bmatrix} a \\ b \\ c \end{bmatrix} + \begin{bmatrix} d \\ e \\ f \end{bmatrix} = \begin{bmatrix} a + d \\ b + e \\ c + f \end{bmatrix} = \begin{bmatrix} d + a \\ e + b \\ f + c \end{bmatrix} = \begin{bmatrix} d \\ e \\ f \end{bmatrix} + \begin{bmatrix} a \\ b \\ c \end{bmatrix}
$$
thus the claim is proven.
With all this said, if you don't understand after reading it once, I encourage you to reread it (this is a lot of information that is by no means trivial to learn completely in one night, in my opinion), but if you have questions feel free to ask.
Do you see how to go about trying to prove that $S$ as you defined is a vector space? (Note that if one of the conditions fails to be true for any arbitrary vector/scalar then $S$ is in fact not a vector space). Specifically I hope that you now see how to respresent an arbitrary vector in $S$ (hint: a vector in $S$ is just like a vector in $\mathbb{R}^3$, but this time the vectors are from $\mathbb{R}^2$ with an extra condition).
A: Because your set $S$ is a subset of $\mathbb{R}^2$, the properties you list will all be satisfied when they are defined on $S$. So the issue here is not, for instance (a), whether addition is commutative for the set $S$, but rather whether it is defined.
Specifically, if $\mathbf{x}$ and $\mathbf{y}$ are vectors in $S$, is it always the case that $\mathbf{x}+\mathbf{y} \in S$? If not, then it isn't a vector space, not because any of the conditions you list fail, but because the addition they refer to isn't always defined in a way that stays in $S$. Similarly, if $\mathbf{x} \in S$, and $c$ is a scalar, is $c\mathbf{x}\in S$?
So a hint: check to see if these two properties hold. If they do, show this. If they don't, give an example where they don't.
A: In order for these axioms to be valid statements, the elements they're describing have to exist in $S$. So, for instance, c) uses the zero vector. Does the zero vector exist in $S$? d) uses additive inverses. Does $S$ even have additive inverses? Does $S$ have the multiplicative identity used in f)?
