linear algebra--can someone check my work

From: tsmith (tsmith76_at_yahoo.com)
Date: 02/07/05 

Date: Sun, 6 Feb 2005 18:26:48 -0800

Q. Consider the case with V being the kth order polynomials with real
coefficients. Let the derivative mapping D be the transformation which
assigns to each polynomial function its derivative. Show that D maps V into
V. What is the rank, nullity, nullspace, and range of D?
===============================================
This is what I did:

Let p = a_0 + a_1 x + a_2 x^2 +...+ a_k x^k in V.

D(p) = a_1 + 2 a_2 x + ... + k a_k x^(k-1).

So D(p) in V since it is a polynomial of at most k.

Now the thing with the rank and nullity, is there suppose to be a rigorous
way to show these? The only way I know how to find them is by "eyeballing"
the space.

I note that only constants and the zero polynomial have zero derivatives,
hence N(T) = { a_0 | a_0 in Reals }.

And the range R(T) = {p(x) | p(x) = a_1 + 2 a_2 x + k a_k x^(k-1) }

Rank(T) = k
Nullity(T) = 1
Dim(V) = k + 1

Thanks for checking my work.

Relevant Pages

  • Re: A Two-Level SOLVER ??
    ... Polynomials are the most easy to use and the most difficult to predict! ... examining/studying/observing the behaviour of derivatives of complex ... I'll seriously consider any non-Frontline VBA Solver algorithm. ... ..constraint: ...
    (microsoft.public.excel.programming)
  • Re: Secant method vs Newtons method
    ... I am presenting on my web site the NIH method, a natural generalization to higher order of the bracketed Secant Method for root finding, where the Secant Method occurs naturally as the zero order case. ... My main example is the computation of the roots of all the Legendre polynomials up to a certain given order which is a nice example which suits my purposes. ... I have chosen to call them Normalized Hermite Polynomails, but am open to suggstions and maybe Two-Point Normalized Hermite [Interpolation Polynomials] would be better. ... The two root finding methods which I am presenting use these same polynomials to interpolate the inverse of a given function (after transforming the derivatives according to the rules). ...
    (sci.math.num-analysis)
  • Re: linear algebra--can someone check my work
    ... > Q. Consider the case with V being the kth order polynomials with real ... What is the rank, nullity, nullspace, and range of D? ... > I note that only constants and the zero polynomial have zero derivatives, ...
    (sci.math)
  • Re: Polynomial derivative via pascal matrix
    ... it's coefficients given a black box that evaluates ffor given x. ... In matrix-notation you set up a vandermondematrix ... To get the numerical derivatives you create now a pascal- ... Example with elliptic division polynomials - pari/gp, ...
    (sci.math)
  • Re: polynomial diophantines and derivatives ?
    ... replace 'sequence' with the successive derivatives at 0 of a function f that is a taylor series with positive radius. ... ...and this function f is definable without directly referring towards the integer sequence ... ... no more derivates here nor single valued polynomials ...
    (sci.math)