The Derivative and the Integral as Infinite Matrices

A polynomial can be encoded as a vector using the coefficients of as the entries of . In this Demonstration column vectors are shown using round parentheses (like these) and row vectors using braces {like these}.
The vector space of 𝒫 of polynomials with coefficients over a field (like the real or complex numbers) has the infinite canonical basis .
The derivative and the integral on 𝒫 are linear transformations. Their infinite matrix representations have nonzero entries above or below the main diagonal. Imagine them as filling the fourth quadrant, with dimensions ∞×∞.
= and = .
The matrix product is the infinite identity matrix, but has a zero in the top-left spot. In a finite-dimensional vector space, if and are square and , then and is the unique inverse of . Linear transformations on infinite-dimensional vector spaces can be both familiar and strange!

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