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A strange kind of subspace

  1. Sep 20, 2009 #1
    Suppose U and V are finite-dimensional subspaces of some finite-dimensional space W (over the field Q in my actual case, but probably irrelevant)

    I've got a subspace in mind that I can't quite define well. I'll start with a concrete example:

    Consider R^3 with orthonormal basis {x, y, z}. Let U = Span{x, y} and V = Span{y, z}. The symmetric difference of {x, y} and {y, z} is {x, z}, and the strange thing I want is Span{x, z}. I resorted to using an "obvious" basis set here, but I want to know if the thing I'm looking for is well-defined. What if I haven't picked a basis; what if it's not a normed space; what if it's not even an inner product space; etc.

    A bit more formally: Let [tex]U \cap V = I[/tex] have dimension k > 0. Then I can pick an orthonormal basis for I, Bi = {b1, ..., bk}, and then choose an orthonormal basis for [tex]U + V = Span(U \cup V)[/tex] (call it simply B) that has Bi as a subset. Now Span(B \ Bi) is uniquely defined (I think).

    Is that well-defined, or is my intuition wrong? If it is, is there a cleaner way of specifying it?

    Even if this is true, my problems have only just begun: the W I have in mind is not an inner product space; it's R over Q, with U and V being subspaces of some n-dimensional subspace S.

    S is isomorphic to Q^n, so I can define an inner product and norm, but this seems like a painful way to go about things.
     
    Last edited: Sep 20, 2009
  2. jcsd
  3. Sep 20, 2009 #2
    The orthocomplement of Bi in U+V ...

    In general, the orthocomplement of K in H (K a subspace of H) is the set of all elements of H orthogonal to K. But you probably need real scalars, not rational.
     
  4. Sep 27, 2009 #3
    Thanks g_edgar! The following then seems like it should hold, although my proof may be wrong.

    Let [tex]U \cap V = I[/tex] have dimension k > 0. Let O be the orthocomplement of [tex]I[/tex] in [tex]W = U + V[/tex], [tex]O_{U}[/tex] be the orthocomplement of [tex]I[/tex] in [tex]U[/tex], and [tex]O_{V}[/tex] be the orthocomplement of [tex]I[/tex] in [tex]V[/tex].

    Then [tex]O = O_{U} + O_{V}[/tex].

    Seems straightforward: If U has dimension u, V has dimension v, and I has dimension i, then O is a subspace of W with dimension u+v-2i. Meanwhile, O_u has dimension u-i, and O_v has dimension v-i. Since O_u and O_v are linearly independent subspaces of O and have dimensions summing to u+v-2i, O_u + O_v = O.

    Maybe?
     
    Last edited: Sep 27, 2009
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