Determine whether each of the following subsets of $\mathbb R^3$ is a subspace of $\mathbb R^3$ The following problem:


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*Determine whether each of the following subsets of $\mathbb R^3$ is a subspace of $\mathbb R^3$


a) {(x,y,z) $\in$ $\mathbb R^3$ : x = 0}
b) {(x,y,z) $\in$ $\mathbb R^3$ : x + y = 0} 
c) {(x,y,z) $\in$ $\mathbb R^3$ : xz = 0}
d) {(x,y,z) $\in$ $\mathbb R^3$ : y $\geq$ 0}
e) {(x,y,z) $\in$ $\mathbb R^3$ : x = y = z}
I've attempted problem (a):
Let W = {(x,y,z) $\in$ $\mathbb R^3$ : x = 0}
Let $w_1$ = (0, $y_1$, $z_1$)
Let $w_2$ = (0, $y_2$, $z_2$)
1) $w_1$ + $w_2$ = (0, $y_1$, $z_1$) + (0, $y_2$, $z_2$) = (0, $y_1 + y_2$, $z_1+ z_2$)
2) c$w_1$ = c(0, $y_1$, $z_1$) = (0, $cy_1$, $cz_1$) = (0, $cy_2$, $cz_2$) $\in$ W
if c = 0 then 0 $*$(0, $y_1$, $z_1$) = (0,0,0) = 0
We therefore have infinite subspaces in $\mathbb R^3$. Note: Any plane or line that intersects the origin is a subspace of $\mathbb R^3$.
Is this correct? Am I on the right track? How do I complete the other problems, b-e?
 A: Your answer for a) is correct, although I don't understand why you bothered with the "if $c=0$ then..." line. Your points 1) and 2) are enough.
For b), an element in that set can be written as $(x,-x,z)$. Use your points 1) and 2) to show these elements are a subspace.
For c), consider the points $(1,0,0)$ and $(0,0,1)$. They are both in the subset, but their sum is not. Therefore this is not a subspace.
For d), consider the point $(0,1,0)$. This is in the subset, but the multiple $-1\cdot(0,1,0)$ is not. Therefore this is not a subspace.
For e), every element can be written as $(0,0,0)$. Use your points 1) and 2) to show this is a subspace. This is in fact the smallest possible subspace, the trivial subspace, containing only the zero vector. (NOTE: This answer was for the original version of e), $x=y=z=0$. For the current version $x=y=z$, each vector can be written $(x,x,x)$. Show this is a subspace.)
A: This is certainly the way to check whether subsets are subspaces, you have correctly identified the form of any vector in your set, and checked whether elements of this form satisfy the subspace axioms. 
I will demonstrate how you may want to solve b), and hopefully that will help you as to how the other problems can be attempted. 
Note that if we have $(x,y,z)$ such that $x+y=0$, we can write this vector as $(a_1,-a_1,a_2)$. We then write a second vector as $(b_1,-b_1,b_2)$. 
Now you need to check the subspace axioms once again.
$1)\: (a_1,-a_1,a_2)+(b_1,-b_1,b_2)=(a_1+b_1,-(a_1+b_1),a_2+b_2)\in W$ since $x+y=0$
I will leave the others as exercises for you. 
