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Math Help - linear operator

  1. #1
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    Arrow linear operator

    Can you please, help me with this:
    Let T:V->V be a linear operator on the vector space over the field F. Let v is in V and let m be a positive integer for which v is not equal to 0, T(v) is not equal to 0, ...,T^(m-1)(v) is not equal to 0. Show that {v, T(v), ... , T^(m-1)(v)} is linearly independent set.
    Thank you in advance!
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  2. #2
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    Quote Originally Posted by maria_stoeva View Post
    Can you please, help me with this:
    Let T:V->V be a linear operator on the vector space over the field F. Let v is in V and let m be a positive integer for which v is not equal to 0, T(v) is not equal to 0, ...,T^(m-1)(v) is not equal to 0. Show that {v, T(v), ... , T^(m-1)(v)} is linearly independent set.
    Thank you in advance!
    I think you need the condition T(\bold{v}) \not = \bold{v} because otherwise if T is the identity operator than it satisfies the hypothesis but the conclusion fails.
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  3. #3
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    May be is a typo in the book, but I still don't get it
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  4. #4
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    Quote Originally Posted by maria_stoeva View Post
    May be is a typo in the book, but I still don't get it
    And even with this assumption the problem as stated is still not true. Let F = \mathbb{R} and V = \mathbb{R}^2. Define T:V\to V to be a rotation operator by \tfrac{\pi}{2}. If your problem is true it would mean \{ \bold{i}, T(\bold{i}), T^2 (\bold{i})\} is linearly independent. But it is not.
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  5. #5
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    Wink Corrected Linear Operator problem

    I asked the teacher and the correct settings are v is not equal to 0, T(v) is not equal to 0, ...,T^(m-1)(v) is not equal to 0 BUT T^m(v) IS equal to 0.
    Would you help me?
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