Let D : R_{≤3}[x] -> R_{≤3}[x] be the dierentiation 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.