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Math Help - Algebra Proof

  1. #1
    Newbie Sterwine's Avatar
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    Algebra Proof

    Suppose that u is a non-zero vector in R^n. Prove that ||u-v|| = ||u|| - ||v|| if and only if v=ku for k element of R with 0 <= k <= 1.

    I tried putting ku in the equation but seem to have hit a road block

    any ideas?
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    Well if v=ku then

    ||u-v||=||u-ku||=||u(1-k)||=(1-k)||u||=||u||-k||u||=||u||-||ku||=||u||-||v||.

    For the converse, it depends; is ||\bullet|| the standard norm in \mathbb{R}^n or a norm with respect to an arbitrary inner product? What are you familiar with? You have to be specific.
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