Can any vector be in orthonormal basis?

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Discussion Overview

The discussion centers on the question of whether any normalized vector in a finite-dimensional vector space can belong to some orthonormal basis that spans that space. Participants explore the implications of having multiple orthonormal bases and the methods to construct them.

Discussion Character

  • Conceptual clarification, Technical explanation

Main Points Raised

  • One participant questions if a normalized vector can belong to an orthonormal basis of a vector space.
  • Another participant confirms that any normalized vector can indeed belong to some orthonormal basis of the vector space.
  • A later reply emphasizes that there are multiple possible orthonormal bases for a vector space, suggesting a relationship to different coordinate systems.
  • Another participant mentions the "Gram-Schmidt" procedure as a method to construct an orthonormal basis that includes a given unit vector.

Areas of Agreement / Disagreement

Participants generally agree that any normalized vector can be part of an orthonormal basis, but the discussion includes various perspectives on the construction of such bases and the implications of having multiple bases.

Contextual Notes

The discussion assumes a finite-dimensional vector space and does not delve into the specifics of the Gram-Schmidt procedure or the conditions under which it applies.

catsarebad
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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.
 
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Yes.
 
thanks :)
 
'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.
 
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.
 
Well done.
 
Given any unit vector you can use the "Gram-Schmidt" procedure to find an orthonormal basis containing it.
 

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