Linear independence of $x, x^3, |x^3|$ How do I make sure whether $x, x^3, |x^3|$ are linearly independent? I know I have to show that $$kx+lx^3+m|x^3|=0\quad\forall x\in\mathbb{R}\quad \Rightarrow k=l=m=0,$$ but I'm not sure how to do this. Should I plug in different values for $x$? 

Yes, the idea is good! For $x=1$ you get
$$
k+l+m=0
$$
For $x=-1$ you get
$$
-k-l+m=0
$$
Can you finish?
