Finding an orthonormal basis

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The black vegetable
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Trying to find your Orthonormal basis
243412

Is this correct? If not any hints on how to find
243413

Many thanks
 
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It seems to me you are missing some sign in your definition of M. Otherwise notationally dubious but in essence fine.

Also, wouldn’t something like ”the black adder” sound better?
 
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thanks Orodruin very helpful, But I think I have a better idea, I think I'm missing a sign in my definition of M :)
 
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