# Math Help - Differentiation Operator

1. ## Differentiation Operator

Let D : R3[x] -> R3[x] be the di erentiation operator. Does there exist a basis of R3[x] with respect to
which the matrix representation of D is:
 Orthogonal?
 Diagonalizable?
 Symmetric?

This is part of our review for my Linear Algebra class. How do I go about finding a basis (if there is one) for D to be diagonalizable and symmetric?

Before you ask, no this is not homework. It is questions my teacher made up for the exam and decided to not use so he gave it to us as a study guide.

2. ## Re: Differentiation Operator

Have you tried doing any calculations at all? What is " R≤3[x] "? What do derivatives of those functions look like?

If you tried the simpler example, derivative operator on linear functions, any such function would be of the form y= ax+ b and its derivative would be dy/dx= a. A standard basis would be $v_1= x$ and $v_2= 1$ so that $ax+ b= av_1+ bv_2$ and the derivative would be $a[v_2]$. Using that basis, the matrix representation would be $\begin{bmatrix}0 & 0 \\ a & 0\end{bmatrix}$. Can that be diagonalized or orthogonalized.