Vector space/Generating subspace/Remove representable vector/Exercise

Let ${}K$ be a field, and let ${}V$ be a ${}K$-vector space. Let ${}v_{i}$, ${}i\in I$, be a family of vectors in ${}V$, and let ${}w\in V$ be another vector. Assume that the family

$w,v_{i},i\in I,$

is a system of generators of ${}V$, and that ${}w$ is a linear combination of the ${}v_{i}$, ${}i\in I$. Prove that also ${}v_{i}$, ${}i\in I$, is a system of generators of ${}V$.

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