Family of vectors/Scaling/Generating system and linearly independent/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 ${}V$. Let ${}\lambda _{i}$, ${}i\in I$, be a family of elements ${}\neq 0$ in ${}K$. Prove that the family ${}v_{i}$, ${}i\in I$, is linearly independent (a system of generators of ${}V$, a basis of ${}V$), if and only if the same holds for the family ${}\lambda _{i}v_{i}$, ${}i\in I$.

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