Linearly independent/Simple properties/Fact/Proof/Exercise

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 ${}V$. Prove the following facts.

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 vector ${}v$ is linearly independent if and only if ${}v\neq 0$.
6. Two vectors ${}v$ and ${}u$ are linearly independent if and only if ${}u$ is not a scalar multiple of ${}v$ and vice versa.

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