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Orthogonal complement question

  1. Apr 24, 2006 #1
    I have the set

    [tex]
    s = span ( [[0][1][-1][1]]^{T} )
    [/tex]

    And I need to find the orthogonal complement of the set.

    It seems like it should be straight foward, but I'm a bit confused. I know that S is a subspace of R^4, and that there should be three free vairables.

    What I did so far is to take the column vector given, and I need to find the null space of its transpose. The three free variables I picked are [tex]x_1= s, x_2=t, x_3=w, x_4=t-w[/tex].

    However, x_1=s is throwing me off because its always zero. I guess what I'm really asking is, what exactly is the solution space of the homogenous system,
    [tex]
    Ax=0
    [/tex]
    in this problem?

    Thanks
     
    Last edited: Apr 24, 2006
  2. jcsd
  3. Apr 24, 2006 #2

    0rthodontist

    User Avatar
    Science Advisor

    You row reduce it (actually it is already row reduced) and you get 3 free variables: x1, x3, and x4. You have a pivot for x2. Your equations will be
    x1 = x1
    x2 = (an expression involving some of x1, x2, x3)
    x3 = x3
    x4 = x4
     
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