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Linear Algebra: Orthonormal Basis

  1. Apr 14, 2011 #1
    1. The problem statement, all variables and given/known data

    Find an orthonormal basis for the subspace of R^4 that is spanned by the vectors: (1,0,1,0), (1,1,1,0), (1,-1,0,1), (3,4,4,-1)

    3. The attempt at a solution

    When I try to use the Gram-Schmidt process, I am getting (before normalization): (1,0,1,0), (0,1,0,0), (1,0,-1,2), (0,0,0,0). So obviously there is some mistake that I am making but I have checked this at least 3 times. Can someone help me and let me know if it is something on my end or the problem. Thank you.
     
  2. jcsd
  3. Apr 14, 2011 #2

    Dick

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    Without have actually worked it out fully, why do you think you made a mistake? The dimension of the subspace is less than 4. You are only going to get a number of orthonormal vectors equal to the dimension of the subspace.
     
  4. Apr 14, 2011 #3
    I thought that the basis had to span R^4? And since of of the elements was (0,0,0,0) the basis can't span R^4.
     
  5. Apr 14, 2011 #4

    Dick

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    It doesn't span R^4. It spans the subspace of R^4 spanned by the given vectors.
     
  6. Apr 14, 2011 #5
    I see. So will (0,0,0,0) be included in the orthonormal basis?
     
  7. Apr 14, 2011 #6

    Dick

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    (0,0,0,0) is in EVERY subspace. You throw that away. It's never part of a basis. A basis is a set of linearly independent vectors. And it certainly isn't normal.
     
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