Subspaces of $\mathbb R^3$? Which of the following sets are subspaces of R^3? 
A.  $\{ (x,y,z)\mid x + y + z = 0 \}$
B.  $\{ (x,y,z) \mid x < y < z \} $
C.  $\{ (7 x  - 5 y, 2 x  - 6 y, 9 x + 6 y ) \mid  x,y \text{ arbitrary numbers}\}$
D.  $\{ (-4, y, z) \mid y, z \text{ arbitrary numbers} \}$
E.  $\{ (x,y,z)  \mid x + y + z = -7 \}$ 
F.  $\{ (x, y, z) \mid  4 x  - 3 y = 0, 8 x + 7 z = 0\}$
I have tried to prove A and B are subspaces but I am slightly confused how to prove the others are subspaces or not. 

Notice that if $V\subset\mathbb R^3$ is a vector subspace, then necessarily $(0,0,0)$ belongs to $V$, hence B., D. and E. are not subspaces. 
$C$ can be written as the range of a linear map while $A$ and $F$ can be written as the kernel of a linear map.
