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Example of Projection

  1. Nov 2, 2009 #1
    1. The problem statement, all variables and given/known data
    Give an example of a subspace W of a vector space V such that there are two projections on W along two distinct subspaces.


    2. Relevant equations



    3. The attempt at a solution
    I tried looking into Euclidean geometry spaces (R3 and R2) but no matter what subspace W I choose, there is only one subspace along which W projects. For example, if my vector space is (x,y,z) and my subspace W is (x,y,0), then by the properties of subspaces in projection, the other subspace must be (0,0,z). How is it possible to get two distinct subspaces along which W projects, and still have a direct sum of the same vector space?
     
  2. jcsd
  3. Nov 3, 2009 #2

    HallsofIvy

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    What? One of us is completely misunderstanding the problem. You said "Give an example of a subspace W of a vector space V such that there are two projections on W along two distinct subspaces." Let V= R3, W= {(x,y,0}). Then the projections (x,y,z)->(x, 0, 0) and (x,y,z)-> (0, y, 0) are projections on W along different subspaces. The orthogonal complement of W, (0, 0, z) has nothing to do with the problem.
     
  4. Nov 3, 2009 #3
    I thought that by definition, projection only works if V (vector space) = W (one subspace) (+) W' (another subspace) [i.e. V is the direct sum of two subspaces] - so when the question asks for two projections on W along two distinct subspaces, wouldn't the "distinct subspaces" each have to add to W to yield V?
     
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