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Orthonormal vector question

  1. Feb 24, 2009 #1
    there is
    W=span{v1=(1,0,i),v2=(2,1,1+i)}
    find the orthonormal basis of
    [tex]
    W^\perp
    [/tex]

    i can do a row reduction and add another vector
    which is independant to the other two.
    so thy are othogonal.and then divide each coordinate of a given vector by the normal
    of that vector

    but what is
    [tex]
    W^\perp
    [/tex]
    ??
     
  2. jcsd
  3. Feb 24, 2009 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    [tex]W^\perp[/tex] is the set of all vector in the space C3 orthogonal to every vector in W. It's easy to show it is also a subspace.

    In order that a vector, u= (a, b, c), be orthogonal to every vector in W, it is sufficient that it be orthogonal to v1 and v2 and that is true as long as their dot products are 0:
    (a, b, c).(1, 0, i)= a+ ci= 0 and (a,b,c).(2,1,1+i)= 2a+ b+ (1+i)c= 0. That gives you two equations in three unknowns. You can solve for two of the unknowns, say a and b, in terms of the other, c. "Normalize" the vector by choosing c so that its length is 1 and that will be your "orthonormal" basis. C3, the set of all ordered triples of complex numbers, has dimension 3. Since W has dimension 2, its "orthogonal complement" has dimension 1 so a basis consists of a single vector and you don't have to worry about the "ortho" part of orthonormal.
     
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