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Thread: norm definition..

  1. #1
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    norm definition..

    i cant understand this definition
    $\displaystyle
    \left \| x \right \|_p=(\sum_{i=1}^{n}\left | x_i \right |^p)^{\frac{1}{p}}
    $
    norm describes the length of a vector
    so if x is our vector

    what do we sum in the formula?
    why there is absolute value in a vector?
    what is x_i?
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  2. #2
    Moo
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    Hello,

    x is a vector of $\displaystyle \mathbb{R}^n$
    And in this case, we have $\displaystyle x=(x_1,x_2,\dots,x_n)$

    the xi's are the coordinates of the vector.
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  3. #3
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    so what are we doing whit those coordinates
    $\displaystyle
    x=(x_1,x_2,\dots,x_n)

    $
    ?
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  4. #4
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    Quote Originally Posted by transgalactic View Post
    so what are we doing whit those coordinates
    $\displaystyle
    x=(x_1,x_2,\dots,x_n)

    $
    ?
    You do exactly what the formula said:
    1) Take the absolute value of each of them.
    2) Take the $\displaystyle p^{th}$ power of each.
    3) Sum them all.
    4) Take the $\displaystyle p^{th}$ root of that sum.

    For example, if p= 1, that is just the sum of the absolute values.

    If p= 2, it is just the usual "Euclidean" norm on $\displaystyle R^n$.

    If p= 3 , n= 4, $\displaystyle x_1= 3$, $\displaystyle x_2= 2$, $\displaystyle x_3= 1$, and $\displaystyle x_4= 2$, the norm is $\displaystyle \left(3^3+ (-2)^3+ 1^3+ 2^3\right)^{1/3}= \left(27+ 8+ 1+ 8\right)^{1/3}= \sqrt[3]{44}$.

    If p= 2 or any even power, you don't need the absolute value. But with odd p, odd powers could cancel- and we don't want that. (1, -1) is not the 0 vector so it shouldn't have 0 norm. But if we used p= 3 without the absolute value, we would have $\displaystyle \sqrt[3]{1^3+ (-1)^3}= \sqrt[3]{0}= 0$. With the absolute value that becomes $\displaystyle \sqrt[3]{|1|^3+ |-1|^3}= \sqrt[3]{1+ 1}= \sqrt[3]{2}$
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  5. #5
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    Quote Originally Posted by transgalactic View Post
    i cant understand this definition
    $\displaystyle
    \left \| x \right \|_p=(\sum_{i=1}^{n}\left | x_i \right |^p)^{\frac{1}{p}}
    $
    norm describes the length of a vector
    so if x is our vector

    what do we sum in the formula?
    why there is absolute value in a vector?
    what is x_i?
    here is another definition
    http://i35.tinypic.com/qxqln9.jpg

    why smaller and equal?
    the inner priduct gives the same resolt

    for me its the same thing

    wher is my mistake?
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