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 ∞×∞.
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!