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- Jan 26, 2012

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**Problem**: Let $V=C^{\infty}(\mathbb{R})$ denote the $\mathbb{C}$-vector space of continuous functions $f:\mathbb{R}\rightarrow\mathbb{C}$ which have continuous $n^{\text{th}}$ derivatives for all $n\geq 1$. Let $D\in\text{End}_{\mathbb{C}}(V)$ denote the derivative operator. View $V$ as a $\mathbb{C}[X]$-module, in the usual way, so that $X\cdot v=D(v)$ for all $v\in V$.

- Describe $\text{Ann}\{e^x\}$ (the annihilator of $e^x$), and prove the accuracy of your description.
- It is a basic result in analysis that for any $f\in V$, and any $\lambda\in\mathbb{C}$, $Df=\lambda\cdot f$ if and only if $f(x)=C\cdot e^{\lambda x}$ for some $C\in\mathbb{C}$. Now assume that $W$ is a subspace of $V$. Define \[ [D-\lambda]^{-1}(W)=\{f\in V: Df-\lambda f\in W\}.\] Prove that $\dim([D-\lambda]^{-1}W)\leq\dim(W)+1$.
- Consider a homogeneous linear differential equation with constant coefficients, of the following form: \[D^nf+a_{n-1}D^{n-1}f+\cdots+a_nDf+a_0f=0.\] Prove that the set of solutions $f\in V$ to this equation is a subspace of $V$ with dimension at most $n$.

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