# 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?

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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: $$a\mapsto b\\ b\mapsto -a \\ c\mapsto a+d\\ d\mapsto b-c$$
@user1131194 This is important for you to spend time thinking about so that you can understand what's going on. To get you started, the coordinates for $a$ are $[1,0,0,0]$, the coordinates of $b$ are $[0,1,0,0]$ etc for $c$ and $d$. Then you have to work out a matrix such that if you multiply these coordinate vectors, you get the results predicted above. –  rschwieb Nov 1 '12 at 1:05