Derivation as operator/Linear/Eigenvectors/Exercise

Let ${}V$ denote the real vector space that consists of all functions from ${}\mathbb {R}$ to ${}\mathbb {R}$ that are arbitrarily often differentiable.

a) Show that the derivation ${}f\mapsto f'$ is a linear mapping from ${}V$ to ${}V$.

b) Determine the eigenvalues of the derivation and determine, for each eigenvalue, at least one eigenvector.

c) Determine for every real number the eigenspace and its dimension.

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