Matrix Representation of a Differential Operator on an Exponential Basis

The set \( S = \{e^{3t}, te^{3t}, t^2e^{3t}\} \) is a basis of a vector space \( V \) of functions \( f: \mathbb{R} \to \mathbb{R} \). Let \( \mathbf{D} \) be the differential operator on \( V \); that is, \( \mathbf{D}(f) = df/dt \). Find the matrix representation of \( \mathbf{D} \) relative to the basis \( S \).