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Math Help - [SOLVED] Linear Algebra, basis of eigenvectors

  1. #1
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    [SOLVED] Linear Algebra, basis of eigenvectors

    Let V be a finite dimensional vector space over a field \mathbb{F}. Let T be a linear
    transformation on V satisfying T^{2} = T. Show that V admits a basis of eigenvectors
    for T.
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  2. #2
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    Quote Originally Posted by JJMC89 View Post
    Let V be a finite dimensional vector space over a field \mathbb{F}. Let T be a linear
    transformation on V satisfying T^{2} = T. Show that V admits a basis of eigenvectors
    for T.

    T^2=T\iff T(T-I)=0 \Longrightarrow the minimal polynomial of T divides x(x-1)\Longrightarrow the min. pol. of T is the product of different linear factors \iff T is

    diagonalizable \iff there's a basis of V of eigenvectors of T.

    Tonio
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