TITLE: Linear independence of $x, x^3, |x^3|$
QUESTION [2 upvotes]: 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 $x$?

REPLY [1 votes]: Idea: if they were dependent, $|x|^3$ was a polynomial as a linear combination of $x$ and $x^3$. This is of course impossible.
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