The zero vector space
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The most elementary vector space of all is… simply the set {}, consisting solely of the zero vector. It really is a vector space! It satisfies all the required technical properties, since adding the zero vector to itself or multiplying it by any number always gives the same result: the zero vector. This trivial vector space is, of course, the simplest possible. But does that make it wholly uninteresting?
The zero vector space can be viewed as a subspace of every vector space. That is what makes it important. For example, linear maps have certain properties when their kernels, which are vector spaces, are trivial.
