Differentiation as a linear map

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: $D(p + q) = D(p) + D(q)$ 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.