Determine Which of the following Sets is a Vector Space Determine which of the following sets is a vector space.
$V$ is the line $y=x$, in the $xy$-plane: $V =\{[x,y]: \; y=x\}$
$W$ is the union of the first and second quadrants in the $xy$-plane: $W=\{[x,y]: \; y\geq0\}$
$U$ is the line $y = x+1$ in the $xy$-plane: $U=\{[x,y]: \; y=x+1\}$

I know that the answer is that "$V$" is the only vector space. However how do I come up with a quick proof to show that $V $is a vector space and that the other $2$ aren't?
 A: 
Hint: The first is a vector space, the rest are not. 
As msteve mentioned, all you have to do is to verify the properties of a vector space (to show that it is a vector space), or identify one property that does not hold. 
The first statement in the picture that you provided is: "For every $\mathbf{u},\mathbf{v}\in V$, we have $\mathbf{u}+\mathbf{v}\in V$." How can we prove this? Since $\mathbf{u}\in V$, $\mathbf{u}$ is of the form $\mathbf{u} = [u\,\,u]$, and similarly $\mathbf{v}$ is of the form $\mathbf{v} = [v\,\,v]$. Hence, $\mathbf{u}+\mathbf{v} = [u+v\,\,u+v]\in V$, which verifies the first statement. I will leave the rest to you. 
For the second one, we will try find a property that fails, so that it is not a vector space. Hint: Property 5 does not hold (but e.g. properties 1-4) hold.
For the third space, you may want to verify that Property 6 does not hold in general.
A: You can show that they don't obey the axioms of a vector space. 
For example, if U is to be a vector space it must be closed under scalar multiplication. That means that no matter what scalar we multiply a vector in U by, the resulting vector must still be in U. However, if we multiply (0,1) by 0 we get (0,0) which is not in U so it is not closed under scalar multiplication and thus not a vector space. Try the same thing with W.
As for showing that V is a vector space, you need to show for any 2 arbitrary vectors in V that if you add them the resulting vector will also be in V (closed under addition). Then you must show closure under scalar multiplication (like in U and W).
