Linear independence of a set with absolute values Consider the vector space $V$ of all continues functions on $R$ over the field $R$.    
Let $S= \{ \vert x \vert, \vert x-1 \vert, \vert x-2 \vert \}$.  
Is the set linearly dependent or independent and span the vector space?   
$|x|$ is either $x$ or $-x$  i.e. dependent implying that the whole set is dependent.
But I'm not sure about it.

Can we achieve $|x|=a|x-1|+b|x-2|$ ?
If yes, the equality must hold for $x=1,2$, giving
$$b=1,a=2.$$
But then, with $x=0$ we get
$$a+2b=0,$$ a contradiction.
