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  "body": "\n## Example\n\nLet $F$ be a field and let $F[x]$ be the vector space of polynomials over $F$.\nThe *formal derivative* is the map $D \\colon F[x] \\to F[x]$ with\n$$D\\left(\\sum_{k=0}^{n} a_k x^k\\right) = \\sum_{k=1}^{n} k\\, a_k x^{k-1}.$$\nIt 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 $c \\in F$.\n\n## Remarks\n\nOver a field of characteristic $0$, the kernel of $D$ is the space of constant polynomials.\nOn the space $F[x]_{\u003cn}$ of polynomials of degree less than $n$, $D$ is nilpotent: $D^n = 0$.\nIn the basis $1, x, x^2$ of $F[x]_{\u003c3}$, the matrix of $D$ is $\\begin{pmatrix} 0 \u0026 1 \u0026 0 \\\\ 0 \u0026 0 \u0026 2 \\\\ 0 \u0026 0 \u0026 0 \\end{pmatrix}$.\nOver $\\R$, differentiation is also a linear map on the space of differentiable functions.\n\n## Explanations\n\n### L1 human\n\nDifferentiation is linear: the derivative of a sum is the sum of the derivatives, and constants come out.\nSo calculus gives important examples of linear maps.\nOn polynomials of bounded degree, we can even write differentiation as a matrix.\n",
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