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Does the vector space $V$ spanned by the set $(e^x,e^{-x})$ have any other invariant subspace under the differential operator apart from the trivial subspaces $V$ itself and the subspace $0$?

My answer is yes, and that the subspaces spanned by only $e^x$ and $e^{-x}$ are also invariant. I am not sure if I am right. Can someone help me?

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Please don't use [homework] as the only tag. – Arturo Magidin Oct 31 '11 at 2:11
What are you unsure about? Show that $D(e^x)\in\mathrm{span}(e^x)$, which shows that $\mathrm{span}(e^x)$ is invariant; similar for $\mathrm{span}(e^{-x})$. – Arturo Magidin Oct 31 '11 at 2:13
@ArturoMagidin. Thank you. – aponga Oct 31 '11 at 2:16
Are you assuming complex scalars, or did you mean $\{\sinh x, \ \cosh x\}$? Is this off a hyperbolic tangent? – Robert Israel Oct 31 '11 at 2:25
@Robert Yes, my previous comment was wrong. Thanks for spotting the mistake... I first thought of $\{ \sinh, \cosh \}$, but that is really the same example as here. $\{ \sin, \cos \}$ appeared more different, but it doesn't work (as I see now). I was looking for an example with real scalars, not complex. – Srivatsan Oct 31 '11 at 17:31

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