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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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    http://en.wikipedia.org/wiki/Root_mean_square2012-11-07

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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.2012-11-07
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For a counterexample, use $(1,0)$ and $(0,1)$.${}{}{}{}{}{}{}{}$

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    Actually, depending on the norm, this isn't necessarily a counterexample.2012-11-07