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T-invariant subspaces

  1. Feb 7, 2013 #1
    1. The problem statement, all variables and given/known data
    Prove that the intersection of any collection of T-invariant subspaces of V is a T-invariant subspace of V.


    2. Relevant equations



    3. The attempt at a solution
    Let W1 and W2 be T-invariant subspaces of V. Let W be their intersection.
    If v[itex]\in[/itex]W, then v[itex]\in[/itex]W1 and v[itex]\in[/itex]W2. Since v[itex]\in[/itex]W1, T(v)[itex]\in[/itex]W1 & v[itex]\in[/itex]W2, T(v)[itex]\in[/itex]W2. Therefore T(v)[itex]\in[/itex]W.
    For any x,y[itex]\in[/itex]W, x,y[itex]\in[/itex]W1 and x,y[itex]\in[/itex]W2, x,y[itex]\in[/itex]W and cx+y[itex]\in[/itex]W since W1 and W2 are subspaces.
    Thus the intersection of any collection of T-invariant subspaces is a T-invariant subspace.

    My answer was marked wrong. The grader's comment was to cross out "any" and replace it with "2". What I should have said? Was I supposed to explicitly point out that this applies from 2...n? :(
     
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  3. Feb 7, 2013 #2

    micromass

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    Your grader is right. Your proof shows that given two invariant spaces W1 and W2, that their intersection is invariant.
    Your proof does not shows that given n invariant subspaces W1, W2,..., Wn, that their intersection is invariant.
     
  4. Feb 7, 2013 #3

    HallsofIvy

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    In fact, the problem says "the intersection of any collection" so showing this for "n invariant subspaces" would not be sufficient. The collection does not have to be finite nor even countable.

    Your proof will work nicely if you modify it so that instead of "W1" and "W2", you use [itex]W\alpha[/itex] where [itex]\alpha[/itex] is simply some label distinguishing the different sets- that is, [itex]\alpha[/itex] is just from some index set, not necessarily {1, 2}, nor even a set of numbers.
     
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