Problem 3

Let $V=\mathbb{R}^{4}$ be a vector space over $\mathbb{R}$.

1. Let the set

   $$R=\{(a,b,c,d)\in V\mid a+b+c+d=1\}.$$

   Show that $R$ is not a subspace of $V$.

2. Let the set

   $$S=\{(a,b,c,d)\in V\mid a+b+c+d=0\}.$$

   Show that $S$ is a subspace of $V$. Find a basis for $S$ and hence state the value of $\dim(S)$.