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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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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.

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I thought that it had to be a square matrix to get a characteristic equation? I don't know how to make this into a square matrix? – user1131194 Nov 1 '12 at 0:11
@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

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