I am confused about why x^2 and |x|x are linear independent in R. If the interval is R+ and R- , are they linear dependent?
$(a)$ In $\mathbb{R}.$ Suppose $\lambda_1 x^2+\lambda_2 x\left|x\right|=0.$ Then for $x=1$ and $x=-1$ we get
$\lambda_1+\lambda_2=0$ and $\lambda_1-\lambda_2=0$ which implies $\lambda_1=\lambda_2=0.$
$(b)$ In $\mathbb{R}^+,$ $x\left|x\right|=x^2$ so ...
$(c)$ In $\mathbb{R}^-,$ $x\left|x\right|=-x^2$ so ...