TITLE: Is the zero vector in $\mathbb{R}^n$ by itself a subspace of $\mathbb{R}^n$?
QUESTION [9 upvotes]: W is a subspace of $\mathbb{R}^n$ iff

The zero vector ∈ W.
X + Y ∈ W for any X, Y ∈ W.
aX ∈ W for any X ∈ W and a ∈ R.

So, given W = { X : X = [x1...], x1 = 0, x2 = 0, ... xn = 0 } ∈ Rn

The zero vector ∈ W because each X in W is the zero vector by definition,
X + Y ∈ W because [0...] + [0...] = [0...]
aX ∈ W because a[0...] = [0...]

So if I understand this correctly, the zero vector is itself a subspace of $\mathbb{R}^n$. Is this correct? 
In addition, can this be extended to say that for any W = { X : X = [$x_1$...] } ∈ $\mathbb{R}^n$, assuming W is a subspace of $\mathbb{R}^n$, $x_i$ (an arbitrary component of W) can only be a constant if it is 0? (I.e., $x_i$ can't be 1 or 2, but can be 0)

REPLY [4 votes]: Yes. Took me sometime to get this. 

Zero is a member of ${0}$.
${0}$ is closed under addition. Let u and v be members of ${0}$. $u=v=0$ hence $u+v=0$.
${0}$ is closed under scalar multiplication.