Can any vector be in orthonormal basis?

In summary, the conversation discussed conceptual difficulty with regards to a vector space V with an orthonormal basis and a normalized vector x in V. The question was whether any normalized vector in V could belong to an orthonormal basis that spans V. The conclusion was that there can be more than one possible orthonormal basis for a vector space, or in other words, there can be multiple coordinate systems. It was also mentioned that the "Gram-Schmidt" procedure can be used to find an orthonormal basis containing any unit vector.
  • #1
catsarebad
72
0
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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  • #2
Yes.
 
  • #3
thanks :)
 
  • #4
'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.
 
  • #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.
 
  • #6
Well done.
 
  • #7
Given any unit vector you can use the "Gram-Schmidt" procedure to find an orthonormal basis containing it.
 

Related to Can any vector be in orthonormal basis?

1. What is an orthonormal basis?

An orthonormal basis is a set of vectors in a vector space that are all orthogonal (perpendicular) to each other and have a length of 1.

2. Can any vector be part of an orthonormal basis?

No, not every vector can be in an orthonormal basis. In order for a set of vectors to be orthonormal, they must be linearly independent, meaning that no vector in the set can be written as a linear combination of the other vectors.

3. How do you determine if a set of vectors is orthonormal?

To determine if a set of vectors is orthonormal, you can use the Gram-Schmidt process, which is a mathematical method for finding an orthonormal basis for a given set of vectors.

4. What is the significance of having an orthonormal basis?

Having an orthonormal basis is useful in many areas of mathematics and science. It simplifies calculations, allows for easier visualization of vectors, and is the basis for many important concepts such as the Gram-Schmidt process and the concept of orthogonal projections.

5. Can an orthonormal basis exist in any vector space?

Yes, an orthonormal basis can exist in any finite-dimensional vector space. However, in infinite-dimensional vector spaces, an orthonormal basis may not always exist.

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