# 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