Linear independence of vectors and set of sums I came across a problem while proving a very simple statement. 
We have three vectors $T = \left \{ v_1, v_2, v_3 \right \}$,  and the set of their sums $S = \left \{ v_1 + v_2, v_1 + v_3, v_2 + v_3 \right \}$ 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 $\Leftrightarrow$ $S$ 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.

Note that the proof of $\Leftarrow$ is the same as that of $\Rightarrow$ only that $$T = \Big\lbrace \frac{1}{2} (w_1 + w_2 - w_3),\frac{1}{2} (w_1 + w_3 - w_2),\frac{1}{2} (w_2 + w_3 - w_1) \Big\rbrace$$
where $S = \lbrace w_1,w_2,w_3\rbrace$
