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Math Help - linear independent

  1. #1
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    linear 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?
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: linear independent

    Quote Originally Posted by jiahao View Post
    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 ...
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    Re: linear independent

    So, if the interval is R-{x=0}, it's still linear independent?
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  4. #4
    MHF Contributor FernandoRevilla's Avatar
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    Re: linear independent

    Quote Originally Posted by jiahao View Post
    So, if the interval is R-{x=0}, it's still linear independent?
    Yes.
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