Example of a subset of $\mathbb{R}^2$ such that U is closed under addition and is closed under scalar multiplication but is not a subspace of R2

Question

I think in $\mathbb{R}^2$ the only solution to this question would be the empty subset, but how do I define addition and multiplication for it?

If i have $\emptyset+\emptyset$ does it equal $\emptyset$ ? And if i do $a\cdot\emptyset$ does it equal $\emptyset$ ?

Would there be another example for the original question?