## Jordan Basis for Differential Operator

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?

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 Recognitions: Homework Help Science Advisor 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?

 Tags jordan basis, linear algebra