dual homomorphism of the derivative
Let denote the vector space of real
polynomials of degree or less, and let denote the ordinary derivative. Linear forms
on
can be given in terms of evaluations, and so we introduce the
following notation. For every scalar , let denote the evaluation functional
Note: the degree superscript matters! For example:
whereas are
linearly independent. Let us consider the dual homomorphism
, i.e. the adjoint
of . We have the following
relations
:
In other words, taking as the basis of and as the basis of , the matrix that represents is just
Note the contravariant relationship between and . The
former turns second degree polynomials into first degree polynomials,
where as the latter turns first degree evaluations into second degree
evaluations. The matrix of has 2 columns and 3 rows
precisely because is a homomorphism
from a 2-dimensional
vector space to a 3-dimensional vector space.
By contrast, will be represented by a matrix. The
dual basis of is
and the dual basis of is
Relative to these bases, is represented by the transpose of the
matrix for , 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 |