# Jordan Basis for Differential Operator

by fishshoe
Tags: jordan basis, linear algebra
 P: 16 1. The problem statement, all variables and given/known data Let $V = P_n(\textbf{F})$. Prove the differential operator D is nilpotent and find a Jordan basis. 2. Relevant equations $D(Ʃ a_k x^k ) = Ʃ k* a_k * x^{k-1}$ 3. 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 $\left[ \begin{array}{ c c } 0 & 1 & 0 \\ 0 & 0 & 2 \\ 0 & 0 & 0 \end{array} \right]$ Is that the kind of basis they're looking for here?