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Say we have a list of normalized vectors. Let q be a vector such that each kth component of q is the average of all the kth components of the normalized vectors. All vectors here have the same length. Is q normalized?

My intuition says yes, but I would like to see a proof.

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en.wikipedia.org/wiki/Root_mean_square –  wj32 Nov 7 '12 at 20:24
    
What prevents you from accepting an answer to this question? –  Did Nov 24 '12 at 11:35

2 Answers 2

up vote 3 down vote accepted

No, take the vectors $(0,1)$ and $(1,0)$. The average is $(\frac 12, \frac 12)$. For an even simpler example, try $(1,0)$ and $(-1,0)$

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Actually, depending on the norm, the first isn't necessarily a counterexample. The second one works, though. –  Cameron Buie Nov 7 '12 at 20:36

For a counterexample, use $(1,0)$ and $(0,1)$.${}{}{}{}{}{}{}{}$

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Actually, depending on the norm, this isn't necessarily a counterexample. –  Cameron Buie Nov 7 '12 at 20:37

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