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Math Help - Direct sum question

  1. #1
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    Direct sum question

    Prove or give a counter example:
    Let V be a finite dimensional vector space over F.
    Let T be a linear operator from V to V.
    Let W be a T invariant subspace of V.
    Then there exists a subspace U that is also T invariant and V = W (+) U (direct sum).

    I know there always exists a subspace U such that the direct sum of W + U = V but I don't know if there exists a T invariant U.

    Thanks
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  2. #2
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    Re: Direct sum question

    Quote Originally Posted by durrrrrrrr View Post
    Prove or give a counter example:
    Let V be a finite dimensional vector space over F.
    Let T be a linear operator from V to V.
    Let W be a T invariant subspace of V.
    Then there exists a subspace U that is also T invariant and V = W (+) U (direct sum).

    I know there always exists a subspace U such that the direct sum of W + U = V but I don't know if there exists a T invariant U.
    Hint: What would happen if T maps U into W? and can you construct an example of a T which does that?
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