$\vec{u}+\vec{v}-\vec{w},\;\vec{u}-\vec{v}+\vec{w},\;-\vec{u}+\vec{v}+\vec{w} $ are linearly independent if and only if $\vec{u},\vec{v},\vec{w}$ are I'm consufed: how can I prove that $$\vec{u} + \vec{v} - \vec{w} , \qquad  \vec{u} - \vec{v} + \vec{w},\qquad - \vec{u} + \vec{v} + \vec{w} $$ are linearly independent vectors if, and only if $\vec{u}$, $\vec{v}$ and $\vec{w}$ are linearly independent?
Ps: sorry for my poor English!

Call the three vectors $a, b, c$ then you find $2u=a+b, 2v=a+c, 2w=b+c$
If there is a linear dependence in $a, b, c$ the equations translate into a (nontrivial) dependence in $u, v, w$ and vice versa - the two sets of vectors are related by an invertible linear transformation.
