Linear and multilinear algebra; matrix theory Example

Differentiation as a linear map

Builds on Linear map Space of polynomials of bounded degree

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.

Reading path

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Space of polynomials of bounded degree example
Vector space definition
Linear map definition
Differentiation as a linear map example

Tag	`0SS770NS8B`
URI	`m2://0SS770NS8B`
Kind	example
Writing	W0 stub
MSC	15A04, 12E10
Instance of	Linear map
Uses	Linear map, Space of polynomials of bounded degree

Differentiation as a linear map

Example

Remarks

Reading path

Further reading

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