[Math] Is a set consisting solely of the zero vector a vector space

Question

For a set to be a vector space, it needs to be:

1. Closed under addition.
2. Closed under scalar multiplication.

If I'm correct, the zero vector satisfies these two conditions:

1. $0+0=0$
2. $c\cdot 0=0$

Hence, my question narrows down to:

Is the zero vector itself considered a vector space? Or is a non-empty vector space considered to be the zero vector plus some other vectors?

Thanks in advance.

Answer 1

Yes. It is the zero space, which is zero-dimensional and which consist of one (very lonely) element.