Can any vector be in orthonormal basis?

[catsarebad]

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.

 

[Simon Bridge]

Yes.

 

[catsarebad]

thanks :)

 

[Simon Bridge]

'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.

 

[catsarebad]

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.

 

[Simon Bridge]

Well done.

 

[HallsofIvy]

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