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Linear independence of orthogonal and orthonormal sets?

  1. Apr 20, 2010 #1
    (Note: this isn't a homework question, I'm reviewing and I think the textbook is wrong.)

    I'm working through the Gram-Schmidt process in my textbook, and at the end of every chapter it starts the problem set with a series of true or false questions. One statement is:

    -Every orthogonal set is linearly independent. ->My answer:True; Text: False

    What's the deal? I thought orthogonality => linear independence. I know if the statement was the other way around then it would be false, since Linear independence =/> orthogonality.
    I'd usually write it off as a typo, but the next statement is:

    -Every orthonormal set is linearly independent,

    which is true in my opinion and the text's, and that makes me think that there's a distinction being pointed out between orthogonal sets and orthonormal sets that I've missed.
  2. jcsd
  3. Apr 20, 2010 #2


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    Take an orthogonal set of vectors. Add the zero vector to it. What happens?
  4. Apr 20, 2010 #3
    Oh snap! Good one.
  5. Apr 20, 2010 #4


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    Does the book's definition of orthogonal sets allow the 0 vector to be a member?
  6. Apr 20, 2010 #5
    Yes, the book takes the stance that the zero vector is orthogonal to every vector.
  7. Apr 21, 2010 #6


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  8. Apr 25, 2010 #7
    Need some clarification myself as well:

    An orthogonal set is not always linearly independent because you could have a 0 vector in it, which would make the set dependent.

    But an orthonormal set must contain vectors that are all orthogonal to each other AND have length of 1, which the 0 vector would not satisfy.

    Is that the right logic?
  9. Apr 26, 2010 #8


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  10. Apr 26, 2010 #9
    Or perhaps you could argue that every orthonormal set contains vectors which are orthogonal with each other and this set is also a basis. Every basis is linearly independent. ==> every orthonormal set is L.I.
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