 The Derivative and the Integral as Infinite Matrices

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram CDF Player or other Wolfram Language products.

Requires a Wolfram Notebook System

Edit on desktop, mobile and cloud with any Wolfram Language product.

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}.

[more]

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!

[less]

Contributed by: George Beck (March 2017)
Open content licensed under CC BY-NC-SA

Snapshots   Permanent Citation

George Beck

 Feedback (field required) Email (field required) Name Occupation Organization Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback. Send