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Math Help - transformation question

  1. #1
    Junior Member ibnashraf's Avatar
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    transformation question

    Question:

    If T is a linear operator on a vector space V(F) such that T^2-T+I=Z (zero map), then show that T is invertible.


    i know that to show invertible i have to show 1-1 and onto, but i have no idea how to begin showing this ... any help?
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  2. #2
    Member ModusPonens's Avatar
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    Re: transformation question

    T^2-T=-I. Now see if the kernel has any vectors different from zero.
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  3. #3
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    Re: transformation question

    Slight variation: T- T^2= T(I- T)= (I- T)T= I
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  4. #4
    Member ModusPonens's Avatar
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    Re: transformation question

    Can you commute T and I-T?
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  5. #5
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    Re: transformation question

    Yes, of course: T- T^2= I*T- T*T= T*I- T*T. Any linear operator from a vector space to itself commutes with itself and the identity.
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  6. #6
    MHF Contributor Drexel28's Avatar
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    Re: transformation question

    Quote Originally Posted by HallsofIvy View Post
    Slight variation: T- T^2= T(I- T)= (I- T)T= I
    Quote Originally Posted by ModusPonens View Post
    Can you commute T and I-T?
    Moreover, judging from the level of the question one would guess that one can assume that the vector space is finite dimensional from where T(\mathbf{1}-T)=\mathbf{1} implies T is surjective from where (by finite dimensionality) bijectivity follows.
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