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Math Help - Vector Spaces

  1. #1
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    Vector Spaces

    Let V be a complex Vector Space, and a:V->V be a linear map.

    a) Show that is a has finite order, then it is diagonalisable.
    b) Let v in V be non-zero. Show that if there exists a smallest integer s S.T. a^s(v) = 0, then v, a(v), a^2(v),..., a^s-1(v) are Lin. Independent.

    I'm really struggling with this one on a problem sheet here. Can anyone help?
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  2. #2
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    Quote Originally Posted by bumcheekcity View Post
    Let V be a complex Vector Space, and a:V->V be a linear map.

    a) Show that is a has finite order, then it is diagonalisable.
    In the Jordan normal form of A, the diagonal entries will all be roots of unity, and the off-diagonal elements must all be zero (because when you take powers of it, the off-diagonal elements can only get bigger). So A has a diagonal JNF and is therefore diagonalisable.

    Quote Originally Posted by bumcheekcity View Post
    b) Let v in V be non-zero. Show that if there exists a smallest integer s S.T. a^s(v) = 0, then v, a(v), a^2(v),..., a^s-1(v) are Lin. Independent.
    Suppose that \lambda_1Av + \lambda_2A^2v + \ldots + \lambda_{s-1}A^{s-1}v = 0. Multiply both sides by A^{s-2} and you see that \lambda_1A^{s-1}v = 0, hence \lambda_1 = 0.

    Therefore \lambda_2A^2v + \ldots + \lambda_{s-1}A^{s-1}v = 0. Multiply both sides by A^{s-3} and you see that \lambda_2 = 0. And so on. So all the coefficients are zero and thus the vectors are lin. ind.
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    Thanks!
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