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Monic Generator (Minimal Polynomial)

  1. Jun 11, 2010 #1
    1. The problem statement, all variables and given/known data[/b]
    Let V be the space of all polynomials of degree less than or equal to 2 over the reals. Define the transformation, H, as a mapping from V to R[x] by [tex](Hp)(x)=\int^x_{-1}p(t)dt\\[/tex]. a) Find the monic generator, d, which generates the ideal, M, containing the range of H.
    b) If [tex]f\in{M\cap Range(H)[/tex] then [tex]f=dg[/tex] where d is given above. Show that [tex]g\in{V} [/tex].

    2. Relevant equations



    3. The attempt at a solution

    I know the monic generator is the minimal polynomial which annihilates H. The way I would find the minimal polynomial is
    1. find a basis for V
    2. find the matrix representation for H with respect the basis
    3. find the characteristic eqn of the matrix
    4. the minimal polynomial divides the characteristic polynomial.

    But when I try step 2, I keep getting a 4 x 3 matrix which doesn't have a determinant. So i don't know how I would find the characteristic eqn.
     
  2. jcsd
  3. Jun 11, 2010 #2

    HallsofIvy

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    H maps V into a 3 dimensional subspace of R[x]. Use that three dimensional subspace as the "range" space rather than R[x] and you will get a 3 by 3 matrix.
     
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