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!
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