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Invariant subspaces under linear operators

  1. Sep 30, 2008 #1
    1. The problem statement, all variables and given/known data
    Prove or give a counterexample: If U is a subspace of V that is invariant under every operator on V, then U = {0} or U = V.

    2. Relevant equations
    U is invariant under a linear operator T if u in U implies T(u) is in U.

    3. The attempt at a solution
    Assume {0} does not equal U does not equal V. Let {u1,...,un} be a basis for U. Extend to a basis for V: {u1,...,un,v1,...,vm}. Since V does not equal {0}, m is greater than or equal to 1. Define a linear operator by T=v1, i=1,...n and T(vi)=v1, i=1,...,m. Then U is not invariant under T.

    I think this is a counterexample to the contrapositive of the statement. does it work? (the contrapositive is If U does not equal {0} does not equal V, then U is not invariant under every operator on V, right?)
  2. jcsd
  3. Oct 1, 2008 #2


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    I think you have the right idea, but your write up has some typos in it so I'm not sure.
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