- #1

fishshoe

- 16

- 0

## Homework Statement

Let [itex] V = P_n(\textbf{F}) [/itex]. Prove the differential operator D is nilpotent and find a Jordan basis.

## Homework Equations

[itex] D(Ʃ a_k x^k ) = Ʃ k* a_k * x^{k-1} [/itex]

## The Attempt at a Solution

I already did the proof of D being nilpotent, which was easy. But we haven't covered what a "Jordan basis" is in class and it's not in either of my textbooks. I know what Jordan Canonical Form is, and Jordan blocks, but I don't know what a Jordan basis is.

Earlier I did a problem that showed that the matrix form of the differential operator on polynomials of order 2 or less. It was

[itex]

\left[

\begin{array}{ c c }

0 & 1 & 0 \\

0 & 0 & 2 \\

0 & 0 & 0

\end{array} \right]

[/itex]

Is that the kind of basis they're looking for here?