---
tag: 0SS770NS8B
kind: example
name: Differentiation as a linear map
msc: [15A04, "12E10"]
instance_of: [0NRYHJPQG2]
uses: [0RJF668P1T, 0JGZCDT5DN]
sources:
  - url: https://en.wikipedia.org/wiki/Linear_map
    locator: Examples
---

## Example

Let $F$ be a field and let $F[x]$ be the vector space of polynomials over $F$.
The *formal derivative* is the map $D \colon F[x] \to F[x]$ with
$$D\left(\sum_{k=0}^{n} a_k x^k\right) = \sum_{k=1}^{n} k\, a_k x^{k-1}.$$
It is a [[linear map|0NRYHJPQG2]]: $D(p + q) = D(p) + D(q)$ and $D(cp) = c\, D(p)$ for all polynomials $p, q$ and all $c \in F$.

## Remarks

Over a field of characteristic $0$, the kernel of $D$ is the space of constant polynomials.
On the space $F[x]_{<n}$ of polynomials of degree less than $n$, $D$ is nilpotent: $D^n = 0$.
In the basis $1, x, x^2$ of $F[x]_{<3}$, the matrix of $D$ is $\begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 2 \\ 0 & 0 & 0 \end{pmatrix}$.
Over $\R$, differentiation is also a linear map on the space of differentiable functions.

## Explanations

### L1 human

Differentiation is linear: the derivative of a sum is the sum of the derivatives, and constants come out.
So calculus gives important examples of linear maps.
On polynomials of bounded degree, we can even write differentiation as a matrix.