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?

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- March 20th 2014, 09:30 PMjiahaolinear independent
**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?** - March 21st 2014, 05:23 AMFernandoRevillaRe: linear independent
$(a)$ 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 ... - March 23rd 2014, 06:33 PMjiahaoRe: linear independent
So, if the interval is R-{x=0}, it's still linear independent?

- March 24th 2014, 01:33 AMFernandoRevillaRe: linear independent