Re: linear algebra--can someone check my work

From: Jannick Asmus (jannick.news_at_web.de)
Date: 02/07/05 

Date: Mon, 07 Feb 2005 08:47:11 +0100
To: tsmith <tsmith76@yahoo.com>

On 07.02.2005 03:26, tsmith wrote:
> 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.

Ok.

>
> 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 }.

since ... (have a look at the equation D(p)= ... )

>
> 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
>

You could crosscheck this with the dimension formula of linear maps. And
perhaps you can give a reasoning for Dim(V) = k+1?

J.

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