Chapter 4 - Vector Spaces - 4.3 Subspaces of Vector Spaces - 4.3 Exercises - Page 167: 9

$W$ is not a vector subspace of $R^2$.

Let $u=(1,2)\in W$, so the components of $u$ are rational but the vector $\sqrt{2}u=\sqrt{2}(1,2)=(\sqrt{2,2\sqrt{2}})$ has irrational components. Hence $W$ is not vector subspace of $R^2$.