Let $V={10}$ be the set of one element only. Is it possible to make $V$ a vector space...

Question

Let $V={10}$ be the set of one element only. Is it possible to make $V$ a vector space by introducing proper operations $+$ and scalar multiplication? If yes, enter 10+10. Here + is the vector space addition, not the usual addition of real numbers. If this is not possible, please explain why.

Answer 1

Yes, it is possible: $10+10=10$ and, for each real $\lambda$, $\lambda10=10$. It becomes a $0$-dimensional real vector space.

Answer 2

Any nonempty set $X$ can be made a (trivial) vector space over any field $F$ as follows.

Select an element $z\in X$ (this will be the "origin").

For every $x,y\in X$, define $x+y=z$.

For every $x\in X$ and every $\lambda\in F$, define $\lambda x = z$.

EDIT: This is incorrect, striking out. See comments.