Give an example of a nonempty subset $U$ of $\mathbf R^2$ such that $U$ is closed under addition and taking additive inverses but $U$ is not a subspace of $\mathbf R^2$.

Recall we could have replaced the first condition in 1.34 with the condition that $U$ is nonempty, thus the first two conditions are satisfied meaning the only way $U$ can fail to be a subspace is for $\lambda u$ to be outside $U$ for some $u\in U,\lambda \in \mathbf R$.

Let $U = \mathbf Q^2 \cap \mathbf R^2 = \{(p,q) : p,q \in \mathbf Q\}$. then $U$ is closed under addition and taking inverses, but not under scalar multiplication since $\sqrt 2 u \notin U$ for any nonzero $u \in U$.