Vector space/Generating system and spanned subspace/Fact

Let ${}K$ be a field, and let ${}V$ be a ${}K$-vector space. Then the following statements hold.

1. For a family ${}v_{i}$, ${}i\in I$, of elements in ${}V$, the linear span is a linear subspace of ${}V$.
2. The family ${}v_{i}$, ${}i\in I$, is a spanning system of ${}V$, if and only if

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

Write a proof