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Math Help - norm definition..

  1. #1
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    norm definition..

    i cant understand this definition
    <br />
\left \| x \right \|_p=(\sum_{i=1}^{n}\left | x_i \right |^p)^{\frac{1}{p}}<br />
    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
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    Hello,

    x is a vector of \mathbb{R}^n
    And in this case, we have 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
    <br />
x=(x_1,x_2,\dots,x_n)<br /> <br />
    ?
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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
    <br />
x=(x_1,x_2,\dots,x_n)<br /> <br />
    ?
    You do exactly what the formula said:
    1) Take the absolute value of each of them.
    2) Take the p^{th} power of each.
    3) Sum them all.
    4) Take the 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 R^n.

    If p= 3 , n= 4, x_1= 3, x_2= 2, x_3= 1, and x_4= 2, the norm is \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 \sqrt[3]{1^3+ (-1)^3}= \sqrt[3]{0}= 0. With the absolute value that becomes \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
    <br />
\left \| x \right \|_p=(\sum_{i=1}^{n}\left | x_i \right |^p)^{\frac{1}{p}}<br />
    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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