Problem 1

Let $U$ and $W$ be subspaces of a vector space $V$ over the field $F$.

1. Prove that $U\cap W$ is a subspace of $V$.

2. Define the set

   $$U+W=\{\mathbf{u}+\mathbf{w}\mid\mathbf{u}\in U\text{ and }\mathbf{w}\in W\}.$$

   Prove that $U+W$ is a subspace of $V$.