Give an example of a nonempty subset $U$ of $\mathbf R^2$ such that $U$ is closed under scalar multiplication, but $U$ is not a subspace of $\mathbf R^2$.

We must construct a $U$ that isn't closed under addition (see 1c/7). Let $U$ be the union of two lines through the origin

Clearly $U \ne \emptyset$ and $\lambda u \in U$ but any nonzero $(x,0) + (0,y) = (x,y) \notin U$ hence $U$ is not a subspace.