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Can any vector be in orthonormal basis?

  1. Apr 8, 2014 #1
    okay so i'm having some conceptual difficulty

    given some vector space V (assume finite dimension if needed)

    which has some orthonormal basis

    i'm given a vector x in V (assume magnitude 1 so it is normalized)

    now my question is:

    can x belong to some orthonormal basis of v? basically can any normalized vector in V belong to some orthonormal basis that spans V.
     
  2. jcsd
  3. Apr 8, 2014 #2

    Simon Bridge

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  4. Apr 8, 2014 #3
    thanks :)
     
  5. Apr 8, 2014 #4

    Simon Bridge

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    'scuse I was in a bit of a hurry all of a sudden.
    Basically what you've noticed is that there is more than one possible orthonormal basis for a vector space.
    Another way of putting it is that there is more than one possible coordinate system.
     
  6. Apr 9, 2014 #5
    yes, excellent, that is what i was thinking. it was pretty helpful for you to just confirm it then and there and i did use that information right away.
     
  7. Apr 9, 2014 #6

    Simon Bridge

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    Well done.
     
  8. Apr 9, 2014 #7

    HallsofIvy

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    Given any unit vector you can use the "Gram-Schmidt" procedure to find an orthonormal basis containing it.
     
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