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) | ![]() |