Linearly independent/Simple properties/Fact

Let $K$ be a field, let $V$ be a $K$-vector space, and let $v_i$, $i\in I$, be a family of vectors in $V$. Then the following statements hold.

1. If the family is linearly independent, then for each subset $J\subseteq I$, also the family $v_i$ , $i\in J$, is linearly independent.
2. The empty family is linearly independent.
3. If the family contains the null vector, then it is not linearly independent.
4. If a vector appears several times in the family, then the family is not linearly independent.
5. A single vector $v$ is linearly independent if and only if $v\ne 0$.
6. Two vectors $v$ and $u$ are linearly independent if and only if $u$ is not a scalar multiple of $v$ and vice versa.

Write a proof