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Complete to orthonormal basis question

  1. Feb 24, 2009 #1
    i got these vectors which are othronormal
    v1 (1/2,-1/2,1/2,-1/2)
    v2 (-1/2,1/2,1/2,-1/2)

    i need to compete them into orthonormal basis
    i did row reduction on them
    and added these independant vectors to the group
    v3(1,0,0,0)
    v4(0,1,0,0)

    now all four vectors are independant (orthogonal)
    how to make them orthonormal??
     
  2. jcsd
  3. Feb 24, 2009 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Use the Gram-Schmidt orthogonalization process- that's what it's for. Since you want the orthonormal basis to include v1 and v2, start in the middle of the process- You already have v1 and v2, so find a new vector, based on v3 that is orthogonal to both v1 and v2 and normalize it. Then find a fourth vector, using the Gram-Schmidt process on v4 that is orthogonal to those three and normalize it.
    Check http://en.wikipedia.org/wiki/Gram–Schmidt_process
     
  4. Feb 24, 2009 #3
    did i create correctly an orthogonal basis R^4 ??

    is is correct to say that in order to make them to orthonormal basis
    i need to divide each each coordinate of a give vector
    by the normal of this specific vector

    ??
     
  5. Feb 24, 2009 #4

    Mark44

    Staff: Mentor

    These vectors might be linearly independent (I haven't checked), but they are definitely not orthogonal. Do you know what it means for two vectors to be orthogonal?

    So no, you don't have an orthogonal set. Halls has told you what you need to do to get an orthogonal set. Once you have the orthogonal set, just divide each vector by its magnitude to get unit vectors.
     
  6. Feb 24, 2009 #5
    orthogonal means that the inner product of two vectors equal 0
    v1 and v2 are orthogonal to each other
    -1/4-1/4+1/4+1/4=0

    and so is v2 to v3
    so you are correct
    but if i will do the gram shmidet proccess then they are going to be orthogonal to each other.
    to make them to orthonormal basis
    i need to divide each each coordinate of a give vector
    by the normal of this specific vector

    ??
    (i heard that there are many versions to the gram shmidt formula)
    which one makes them only orthogonal

    which one makes them only orthonormal??
     
  7. Feb 24, 2009 #6

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Both Mark44 and I have answered that: One you have found two new vectors v2 and v3 so that all four vectors are orthogonal to one another, divide each by its length to "normalize" them.
     
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