We have three vectors $T=\{{v}_{1},{v}_{2},{v}_{3}\}$, and the set of their sums $S=\{{v}_{1}+{v}_{2},{v}_{1}+{v}_{3},{v}_{2}+{v}_{3}\}$ for ${v}_{i}\in V$, for V any vector space. I have to prove that for rational coefficients $a,b,c\in \mathbb{Q}$, the following statement is true:

T linearly independent $\iff $ S linearly independent

$\Rightarrow $ was quite easy and I had no problem calculating it.

However, $\Leftarrow $ is where I faced the problem and I would appreciate any help and besides that, does $\Leftarrow $ is true for real coefficients.