How to check whether the sets U1, U2, U3 are subspaces of the vector space V. In the $\mathbb{R}$-vector space $V = \mathbb{R}^3$
, we consider the following subsets:
$U_1 := \{x\in\mathbb{R}^3\ |\ x_1 + x_2 − x_3 = 0\}$,
$U_2:= \{x\in\mathbb{R}^3\ |\ \exists n\in\mathbb{Z}\ :\ x_1 = n, x_2 = 2n, x_3 = 3n\}$,
$U_3:= \{x\in\mathbb{R}^3\ |\ (x_1 = 0)\vee(x_2 = 0)\vee (x_3 = 0)\}$
I need to check whether the sets $U_1,\ U_2$, and $U_3$ are subspaces of the vector space $V$, How can I properly do that?

As commented above you can check whether they are subspaces. You can also check by just looking at the problem of such types. How this happens?
If $W$ be any plane in $\Bbb{R}^3$ passing through the origin, then they satisfy the subspace conditions (You can see here).
Good luck!
