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Say I have a basis for $\mathbb{c}^{2}$ composed of the vectors $(1,1), (4i,2i )$ with complex inner product. When I construct my orthogonal basis using the Gram-Schmidt process how do I make a choice of which of these vectors are $u_{1}$ and $u_{2}$ because this will obviously have an impact when we carry out the inner product and swapping would give different answers.

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    $\begingroup$ Why does getting different answers matter? $\endgroup$ – amd Jan 9 '19 at 20:01
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    $\begingroup$ You pick whichever you think will make the calculations easier. There is no "correct" choice, because there is no "correct" or unique orthonormal basis. $\endgroup$ – Arturo Magidin Jan 9 '19 at 20:02
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    $\begingroup$ The Gram-Schmidt process operates on an ordered basis, so whatever order your ordered basis is in. $\endgroup$ – user3482749 Jan 9 '19 at 20:06
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You can take the vectors in any order you like. You will still get an orthonormal basis from the Gram-Schmidt process (though, in general, a different one).

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