Find the dimension of $V = {f in C^k [0, 1] : a_n f^{(n)}(t)+cdots+a_0 f (t) = 0 textrm{ for all } t in [0, 1]}$.

Using the additive properties of differentiation I proved that $V$ is a subspace. The fact that $overline{O} in V$ is also trivial. However, I am stuck as to how to use the property of the continuity and differentiability of $f in V$ to find out the dimension of $V$.

Any help would be appreciated.

EDIT: The $a_i$ from $i = 0$ to $i = n $ are fixed and are $in Bbb{R}$.

Mathematics Asked by Rajesh Sri on November 21, 2021

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