dual homomorphism of the derivative

Let 𝒫n denote the vector spaceMathworldPlanetmath of real polynomials of degree n or less, and let Dn:𝒫n𝒫n-1 denote the ordinary derivative. Linear formsPlanetmathPlanetmath on 𝒫n can be given in terms of evaluations, and so we introduce the following notation. For every scalar k, let Evk(n)(𝒫n)* denote the evaluation functionalPlanetmathPlanetmathPlanetmathPlanetmath


Note: the degree superscript matters! For example:


whereas Ev0(2),Ev1(2),Ev2(2) are linearly independentMathworldPlanetmath. Let us consider the dual homomorphism D2*, i.e. the adjointPlanetmathPlanetmath of D2. We have the following relationsMathworldPlanetmathPlanetmath:


In other words, taking Ev0(1),Ev1(1) as the basis of (𝒫1)* and Ev0(2),Ev1(2),Ev2(2) as the basis of (𝒫2)*, the matrix that represents D2* is just


Note the contravariant relationship between D2 and D2*. The former turns second degree polynomialsMathworldPlanetmathPlanetmath into first degree polynomials, where as the latter turns first degree evaluations into second degree evaluations. The matrix of D2* has 2 columns and 3 rows precisely because D2* is a homomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath from a 2-dimensional vector space to a 3-dimensional vector space.

By contrast, D2 will be represented by a 2×3 matrix. The dual basisMathworldPlanetmath of 𝒫1 is


and the dual basis of 𝒫2 is


Relative to these bases, D2 is represented by the transposeMathworldPlanetmath of the matrix for D2*, namely


This corresponds to the following three relations:

Title dual homomorphism of the derivative
Canonical name DualHomomorphismOfTheDerivative
Date of creation 2013-03-22 12:35:28
Last modified on 2013-03-22 12:35:28
Owner rmilson (146)
Last modified by rmilson (146)
Numerical id 4
Author rmilson (146)
Entry type Example
Classification msc 15A04
Classification msc 15A72