Monic Generator (Minimal Polynomial)

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SUMMARY

The discussion focuses on finding the monic generator, denoted as d, for the ideal M generated by the transformation H, which maps polynomials from the space V (degree ≤ 2) to R[x] via integration. The minimal polynomial is identified as the polynomial that annihilates H. The process involves determining a basis for V, constructing the matrix representation of H, and deriving the characteristic equation. A key insight is that the range of H should be considered as a three-dimensional subspace of R[x], leading to a 3x3 matrix for accurate calculations.

PREREQUISITES
  • Understanding of polynomial spaces, specifically V (degree ≤ 2).
  • Knowledge of linear transformations and their matrix representations.
  • Familiarity with characteristic polynomials and minimal polynomials.
  • Basic integration techniques in the context of polynomial functions.
NEXT STEPS
  • Study the construction of matrix representations for linear transformations in polynomial spaces.
  • Learn about the relationship between characteristic polynomials and minimal polynomials.
  • Explore integration of polynomials and its implications in linear transformations.
  • Investigate the properties of monic polynomials and their role in generating ideals.
USEFUL FOR

Students and educators in advanced algebra, particularly those studying linear algebra and polynomial theory, as well as mathematicians interested in the properties of transformations in polynomial spaces.

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1. Homework Statement [/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 (Hp)(x)=\int^x_{-1}p(t)dt\\. a) Find the monic generator, d, which generates the ideal, M, containing the range of H.
b) If f\in{M\cap Range(H) then f=dg where d is given above. Show that g\in{V}.

Homework Equations





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.
 
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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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