6.3.* Prove that in a vector space $((V, +), K,)$, for every $v, u \in V$ and $\lambda \in K$, $\lambda \cdot (v - u) = \lambda \cdot v - \lambda \cdot u$.
6.4. The intersection of any collection of subspaces of some linear space $((V, +), K,)$ is a subspace of
$((V, +), K,)$. Show that this is not necessarily true for the union of subspaces.
6.5. The real plane $\mathbb{R}^2$ is a vector space over $\mathbb{R}$. Describe, in geometrical terms, all subspaces of $\mathbb{R}^2$.
6.6.* Let $((V, +), K,)$ be a vector space and $S, T \subseteq V$. Prove that
(a) $S \subseteq \text{span}(S)$
(b) $S \subseteq T \implies \text{span}(S) \subseteq \text{span}(T)$
(c) $\text{span}(\text{span}(S)) = \text{span}(S)$
(d) $v \in \text{span}(S) \iff \text{span}(S) = \text{span}(S \cup \{v\})$.
