Vector space/Generating system and spanning subspace/Fact/Proof/Exercise

Let $K$ be a field, and let $V$ be a $K$-vector space. Prove the following facts.

1. Let $U_j$, $j\in J$, be a family of linear subspaces of $V$. Prove that also the intersection

   $U=\bigcap _{j\in J}{U}_j$

   is a subspace.

2. Let $v_i$, $i\in I$, be a family of elements of $V$, and consider the subset $W$ of $V$ that is given by all linear combinations of these elements. Show that $W$ is a subspace of $V$.
3. The family $v_i$, $i\in I$, is a system of generators of $V$ if and only if

   $⟨v_i,i\in I⟩=V.$

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