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Orthogonal Complements of complex and continuous function subspaces

  1. Dec 22, 2011 #1
    1. The problem statement, all variables and given/known data

    I'm having a tough time figuring out just how to get the orthogonal complement of a space. The provlem gives two separate spaces:

    1) span{(1,0,i,1),(0,1,1,-i)},
    2) All constant functions in V over the interval [a,b]

    2. Relevant equations

    I know that for a subspace W of an inner product space, the orthogonal complement is defined as:
    W_perp = {vectors in v[itex]\in[/itex]V: <v,w> = 0 for all w [itex]\in[/itex] W}
    <v,w> is the standard dot product between two vectors;
    In the case of constant functions, the dot product is [itex]\int[/itex]f(x)g(x)dx;

    3. The attempt at a solution

    1) I tried putting it in matrix form:
    [1 0 i 1]
    [0 1 1 -i]
    but don't know how to row reduce with complex variables. I actually don' think the matrix needs to be simplified more than it is, but still don't know how to plug in to get two orthogonal vectors (I would like the result to be an orthogonal set)

    2) I don't even know where to start ... the book doesn't cover inner products of functions much , let alone how to find the orthogonal complement of them.
    Last edited: Dec 22, 2011
  2. jcsd
  3. Dec 22, 2011 #2


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    You doing a pretty good job of ignoring any information in your "Relevant equations" section. If you don't do that the second question should be pretty easy. You can factor a constant function outside of the integral. For the first one you should remember, if you weren't told, that <u,v> for complex vectors involves taking a complex conjugate of one of the vectors. Suppose v=(A,B,C,D). Then what two equations do you have to solve for the four unknowns A, B, C and D?
  4. Dec 22, 2011 #3
    So, for the first part, the orthogonal complement would just be all f(x) where by ∫f(x) = 0 over the given interval [a,b]?

    As for the second, I'm not quite sure what you mean. If I have those equations, and let x =[x_1, x_2, x_3, x_4] be my unknowns, I have:

    [x_1,x_2,x_3,x_4] = x_3[-i, -1, 1, 0] + x_4[-1,i,0,1]

    Is that correct? I would then plug in x_3=1 and then x_4=1 to each of those vectors to get my orthogonal complement.
  5. Dec 23, 2011 #4


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    That's the answer to the first one alright. For the second one, take w=(w1,w2,w3,w4) to be your unknown vector in the orthogonal complement. Now since w need to be orthogonal to the span{(1,0,i,1),(0,1,1,-i)}, it has to be orthogonal to v1=(1,0,i,1) and v2=(0,1,1,-i). So you must have <v1,w>=0 and <v2,w>=0. What do those equations look like when you write them out in terms of w1, w2, w3 and w4?
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