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

Evk(n):pp(k),p𝒫n.

Note: the degree superscript matters! For example:

Ev2(1)=2Ev1(1)-Ev0(1),

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:

D2*(Ev0(1))=-32Ev0(2)+2Ev1(2)-12Ev2(2),D2*(Ev1(1))=-12Ev0(2)+12Ev2(2).

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

(-32-1220-1212)

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

-x+1,x

and the dual basis of 𝒫2 is

12(x-1)(x-2),x(2-x),12x(x-1).

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

(-322-12-12012)

This corresponds to the following three relations:

D2[12(x-1)(x-2)]=-32(-x+1)-12xD2[x(2-x)]=2(-x+1)+0xD2[12x(x-1)]=-12(-x+1)+12x
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