1. The problem statement, all variables and given/known data Let [itex] V = P_n(\textbf{F}) [/itex]. Prove the differential operator D is nilpotent and find a Jordan basis. 2. Relevant equations [itex] D(Ʃ a_k x^k ) = Ʃ k* a_k * x^{k-1} [/itex] 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 [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?
No - in a Jordan basis, all entries in the superdiagonal (i.e. the line above the diagonal) have to be either 1 or zero.
So do I need something like \begin{array}{ccc} 0 & 1 & 0 & \dots & 0 \\ 0 & 0 & 1 & \dots & 0 \\ \dots \\ 0 & 0 & 0 & \dots & 1 \\ 0 & 0 & 0 & \dots & 0 \end{array} as an n-vector Jordan basis for the polynomials of order up to n?