Q:

$\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!

A:

Hint:$$a(\vec{u} + \vec{v} - \vec{w})+ b (\vec{u} - \vec{v} + \vec{w})+c( - \vec{u} + \vec{v} + \vec{w})=0\\
\implies(a+b-c)\vec{u}+(a-b+c)\vec{v}+(-a+b+c)\vec{w}=0$$ then we have $$a+b-c=0 ,a-b+c=0 ,-a+b+c=0$$ From which we can easily conclude  $a=b=c=0$.

