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

   ${}\langle v_{i},\,i\in I\rangle =V\,.$

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