Is the empty set a vector space?

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I think the empty set satisfies all of the axioms of a vector space except the one about the existence of an additive identity. Is this right?

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2 Answers

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The empty set is empty (no elements), hence it fails to have the zero vector as an element.

Since it fails to contain zero vector, it cannot be a vector space.

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No! If $(E,+,\cdot)$ is a vector space then $(E,+)$ is an abelian group so it contains a neutral element which is the zero vector hence $E\ne\varnothing$.

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