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Is the set of all differentiable functions ƒ:ℝ→ℝ such that ƒ'(0)=0 is a vector space over ℝ? I was given the answer yes by someone who is better at math than me and he tried to explain it to me, but I don't understand. I am having difficulty trying to conceptualize this idea of vector spaces with functions because I can't really visualize it like a plane in 3d space. I am also wondering what is the importance of having vector spaces set over a field? It seems trivial or maybe its just me being brainwashed by years of elementary mathematics