Q: Linear Map Given by $D(f) = f'$ Let $V$ be the complex vector space given by $$V = \{a\sin(x) + b\cos(x) +cx\sin(x) + dx\cos(x)\ \; | \; a, b, c, d \in \mathbb{C}\}$$
Let $D: V \rightarrow V$ be the linear map given by $D(f) = f'$.
I need to work out the characteristic equation and complex eigenvalues of D and $m_D x$.
I thought the first step would be to calculate the derivative but I don't know where to go from here? 

A: Well, you'll have to find a matrix representing the tranformation, won't you.
You have that $\sin(x),\cos(x), x\sin(x), x\cos(x)$ form a basis for your vector space. I'll call these $a,b,c,d$ respectively. Observe the effects of $D$ on the basis:
$$a\mapsto b\\
b\mapsto -a \\
c\mapsto a+d\\
d\mapsto b-c$$
Find the matrix in terms of this basis, and you can begin to find eigenvalues.
