Determine whether the set is a vector space. So I have a final tomorrow and I have no clue how to determine whether a set is vector space or not. I've looked online on how to do these proofs but I still don't understand how to do them. Can any one help me with a question like this?
Let V be the set of all positive real numbers. Determine whether V is a vector space with the operations below.
$x + y = xy$
$cx=x^c$
If it is, verify each vector space axiom; if not, state all vector space axioms that fail.
Edit: Turns out I'm going to fail the exam based on what you guys are saying.
 A: Choose another notation $x \oplus y := xy$ and $c \otimes x := x^c$. Then the exponential map gives an isomorphism of structures $(\mathbb{R},+,*) \cong (\mathbb{R}^+,\oplus,\otimes)$. Since the first is a vector space, the same is true for the latter. And this way the creator of this "exercise" came up with this artificial vector space (he wanted that you waste your time with computations ...).
A: Here is a start. 


*

*Additive axioms:  For every $x,y,z \in \mathbb{R^+}$, 


i) $x+y=xy = yx = y+x\, $ ( since real numbers commute) 
ii) $(x+y)+z=xy+z = xyz = x+yz= x + (y+z).$
Can you continue?
A: You need to translate the usual vector space axioms, expressed using '+' and '$\cdot$', into your problem's notation. For example, the vector space axiom $$c\cdot(x+y)=c\cdot x+c\cdot y$$becomes$$(xy)^c=(x^c)(y^c)$$
As other comments have indicated, the closure axioms need to be dealt with. If $x,y\in V$, then you need to show that $xy$ is in $V$, since $xy$ is the 'vector sum' of $x$ and $y$.
One last point: do you know about vector space isomorphisms? There is an isomorphism between your vector space and a much more familiar vector space (call it $W$). If $\psi:V\rightarrow W$ stands for this isomorphism, then we have $$\psi(xy)=\psi(x)+\psi(y)$$for example.
By the way, you didn't not explicitly state what possible values $c$ could have.
A: Lucky I got the same question in one of my Assignments. And here's what I got:
http://www.math.tamu.edu/~dallen/linear_algebra/chpt3.pdf
The same question in here!
Hope it helped :)
A: What about closure under scalar multiplication? For example, if c = 1/2 and x = -1, we have:  1/2*(-1) = (-1)^(1/2) which is not real. So, it's not closed under scalar multiplication, correct? Therefore, not a vector space.
