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I have a linear operator with its matrix in certain coordinates to be

$$ \begin{pmatrix} 1 & 0 & 0 & \cdots & 0 \\ 0 & \frac{1}{2} & 0 & \cdots & 0 \\ 0 & 0 & \frac{1}{3} & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \\ 0 & 0 & 0 & \cdots & \frac{1}{n} \end{pmatrix} $$

What is this linear operator? How could I construct it without referring to coordinates?

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Of course it could be any number of things, but one operator with this matrix is the one that assigns to every polynomial $p(x)$ of degree less than $n$ the polynomial $\frac1x\int_0^xp(t)\,\mathrm dt$.

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