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Math Help - vector space

  1. #1
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    vector space

    any ideas on how to go about conducting these please. i will attempt them once i have a clear idea on how to do this. thanx

    let V be the vector space of polynomials over C of degree <= 10 and let
    "D: V -----> V" be the linear map defined by

    D(f) = df/dx

    show
    (1) D^11=0
    (2) deduce 0 is the only eigenvalue of D
    (3) find a basis for the generalised eigenspaces v1(0), v2(0), and v3(0).
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  2. #2
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    Quote Originally Posted by smoothman View Post
    any ideas on how to go about conducting these please. i will attempt them once i have a clear idea on how to do this. thanx

    let V be the vector space of polynomials over C of degree <= 10 and let
    "D: V -----> V" be the linear map defined by

    D(f) = df/dx

    show
    (1) D^11=0
    (2) deduce 0 is the only eigenvalue of D
    (3) find a basis for the generalised eigenspaces v1(0), v2(0), and v3(0).
    Here are a few hints.
    (1) Each time you differentiate a polynomial, you reduce its degree by 1.
    (2) If λ is an eigenvalue, with Dx = λx, then D^{11}x = \lambda^{11}x.
    (3) If I understand this notation correctly, then v1(0) = constants, v2(0) = all linear functions, v3(0) = all quadratic functions. Their bases consist of monomial functions 1, x, x^2.
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