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## Homework Statement

Let ##n## be a positive integer and let ##V = P_n## be the space of polynomials over ##R##. Let D be the differentiation operator on ##V## . Find a basis for the null space of the transpose operator ##D^t: V^*\to V^*##.

## Homework Equations

Let ##T:V\to W## be a linear transformation. Then ##(im T)^0=kerT^t##

Let let ##\{f_0,f_1,...f_n\}## be a dual basis for ##\{v_0,...,v_n\}## then ##f_i(v_j)=\delta_{ij}##

## The Attempt at a Solution

Let ##\{1,x,...x^n\}## be a basis for ##V## and let ##\{f_0,f_1,...f_n\}## be the dual basis. The image of ##D## is the set of polynomials with degree less than n. We know from ##f_i(v_j)=\delta_{ij}## that ##f_n## will annihilate any polynomial without an ##x^n## term. But that is the entire image of ##D## so ##\{f_n\}## is the basis we seek, where I used ##(im T)^0=kerT^t##.

I think this is correct, I'm just looking for tips to make it cleaner.

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