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?(adsbygoogle = window.adsbygoogle || []).push({});

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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.

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# Can someone check if I did this right?

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