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Math Help - invertible operator

  1. #1
    jax
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    invertible operator

    I'm stuck on this h/w problem...any help would be appreciated!!! thank u

    Let V be a vector space of dimension n, and T: V --> V an invertible operator. Prove that if U is a T-invarient subspace of V, then U is also invariant under T^-1. Justify each step.
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  2. #2
    A Plied Mathematician
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    What ideas have you had so far?
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  3. #3
    jax
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    Well I don't know too much about T-invarient subspaces, but I figured it means eigenvalues. So here is my prrof so far-

    First suppose that U is an eigenvalue of T. Thus there exists a nonzero vector v exists in V such that: Tv=Uv
    Appyling T^-1 to both sides of the equation above and we get v=UT^-1v, which is equivantly to the equation T^-1v=(1/U)v. Thus (1/U) is an eigenvalue of T^-1. Therefore proving that if U is a T-invarient subspace of V, then U is also invariant under T^-1.

    What do u think??
    Thank you.
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  4. #4
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    No, I don't think you understand what's going on. A T-invariant subspace U is a subspace of V that remains the same under action from T. That is, T(U) \subseteq U. This relation says that \forall\,u\in U, it is the case that T(u)\in U.

    Can you write down, now, a mathematical statement that represents what you need to prove?
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  5. #5
    jax
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    ok that helps thank you!! I will play around with the proof now.
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  6. #6
    A Plied Mathematician
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    Ok, let me know how it goes.
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