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Thread: Help

  1. #1
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    Help

    I don't know if this the right thread to get help, but here it goes.

    Prove that:

    |x + y|^2 + |x-y|^2 = 2|x|^2 + 2|y|^2 if x is an element of R^k and y is an element of R^k.

    Interpret this geometrically, as a statement about parallelograms.
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  2. #2
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    Quote Originally Posted by Susie38
    I don't know if this the right thread to get help, but here it goes.

    Prove that:

    |x + y|^2 + |x-y|^2 = 2|x|^2 + 2|y|^2 if x is an element of R^k and y is an element of R^k.

    Interpret this geometrically, as a statement about parallelograms.
    Considering $\displaystyle \mathbb{R}^k$ as a vector space over $\displaystyle \mathbb{R}$.
    Then,
    $\displaystyle x=(a_1,a_2,...,a_k)$
    $\displaystyle y=(b_1,b_2,...,b_k)$
    Then,
    $\displaystyle x+y=(a_1+b_1,a_2+b_2,...,a_k+b_k)$
    Then,
    $\displaystyle |x+y|=\sqrt{(a_1+b_1)^2+...+(a_k+b_k)^2}$
    Thus,
    $\displaystyle |x+y|^2=(a_1+b_1)^2+...+(a_k+b_k)^2$
    Similarily,
    $\displaystyle |x-y|^2=(a_1-b_1)^2+...+(a_k-b_k)^2$
    When you add them together you have,
    $\displaystyle \sum_{i=1}^k(a_i^2+2a_ib_i+b_i^2)+\sum_{i=1}^k(a_i ^2-2a_ib_i+b_i^2)$=$\displaystyle \sum_{i=1}^k2a_i^2+2b_i^2=2|x|^2+2|y|^2$
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