Representation of a Linear Functional

Reference: The following project was suggested by two exercises in chapter 6 of

Sheldon Axler's "Linear Algebra Done Right, Second Edition," Springer, 2000.

Let's define a linear functional on the space of all real polynomials of degree two or less:

φ is the functional that evaluates such a polynomial at the point a == 2/3.

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Apply that functional to two polynomials.

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Now, find an orthonormal basis for the space spanned by the polynomials {1, x, }.

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Representation of the Linear Functional φ

Now, calculate the coefficients of the polynomial which will represent the linear functional φ.

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Use those coefficients to construct the representing polynomial.

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Use the representing polynomial to define a new functional ψ.

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Now ψ == φ, since they both agree on the basis β.

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Check that φ and ψ agree on the two polynomials p and q.

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Let's define another linear functional on the space of all polynomials of degree two or less.

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Apply that functional to two polynomials.

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Representation of the Second Linear Functional ξ

Now, calculate the coefficients of the polynomial which will represent the second linear functional ξ.

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Use those coefficients to construct the representing polynomial.

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Use the representing polynomial to define a new functional ρ.

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Now ρ == ξ, since they both agree on the basis β.

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Check that ρ and ξ agree on the two polynomials p and q.

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Created by Mathematica (April 14, 2005) |