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Math Help - T-invariant basis problem

  1. #1
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    T-invariant basis problem

    Hello all, I am having trouble showing linear independence at the indicated step.

    The setup:
    Let v\in V be a fixed vector in a finite-dimensional vector space V. Let W=\{v,Tv,T^2v,...\}.

    Prove W is T-invariant. (Easy)

    Suppose that \dim(W)=k and show that B=\{v,Tv,T^2v,\ldots,T^{k-1}v\} is a basis for W.

    It seems obvious that checking linearly independent is easier than span, but I am getting stuck trying to get all of the coefficients to be zero. Thanks to any and all help ahead of time!
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  2. #2
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    Quote Originally Posted by nqramjets View Post
    Hello all, I am having trouble showing linear independence at the indicated step.

    The setup:
    Let v\in V be a fixed vector in a finite-dimensional vector space V. Let W=\{v,Tv,T^2v,...\}.

    Prove W is T-invariant. (Easy)

    Suppose that \dim(W)=k and show that B=\{v,Tv,T^2v,\ldots,T^{k-1}v\} is a basis for W.

    It seems obvious that checking linearly independent is easier than span, but I am getting stuck trying to get all of the coefficients to be zero. Thanks to any and all help ahead of time!


    Use the lemma that says that if \{v_1,v_2,\ldots,v_n\} is a lin. dep. set, then there's some vector lin. dep. on the PRECEEDING ones (i.e., the 2nd one on the 1st one, or the 3rd on the 1st and 3nd one., etc.)

    So: if \{v,Tv,\ldots\} is lin. dep. then there's some natural number r s.r. T^rv is lin. dep. on \{v,Tv,\ldots,T^{r-1}v\}\Longrightarrow T^rv=a_0v+a_1Tv+\ldots+a_{r-1}T^{r-1}v\,,\,\,a_i\in\mathbb{F}

    As \{v,Tv,\ldots\} is a generating set for W\,\,\,and\,\,\,\dim W=k, it's clear from the above that \{v,Tv,\ldots,T^nv\} cannot be lin. dep. for any n<k-1 (why?), so it must be that \{v,Tv,\ldots,T^{k-1}v\} is lin. independent...and now end the argument.

    Tonio
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