Linear Algebra Linear Independence Proof

How would I prove the following?

$V$ is a vector space and $S$ a subset of $V$ containing at least $2$ elements. Then $S$ is linearly independent iff no elements of $S$ can be written as a linear combination of the remaining elements of $S$.

I know linearly independent means that all coefficients $a_i$ in field $\mathbb{F}$ equal $0$, but I'm not sure how or if this is used for this proof.