I don't understand why the null set ∅ is not a vector space like it obeys all the vector space axioms namely commutativity of addition, associativity of addition and multiplication (with scalar from F),existence of additive inverse, multillicative identity (1v = v for all v ∈ V (vector space)), distributive proerties and existence of additive identity.
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3What is the zero vector/additive identity? – Randall Jul 04 '23 at 20:27
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1By definition, a vector space is nonempty: it has an element called $0$. – BrianO Jul 04 '23 at 20:28
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More generally, ${0}$ plays the same role for vector spaces as $\varnothing$ for the sets. It also plays the same (dual) role as the one element set ${\ast}$ – julio_es_sui_glace Jul 04 '23 at 20:37
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There exists an element 0 ∈ V, called the zero vector, such that v+0 = v for all v ∈ V. I'm unable to see why this implies that 0 must belong to the vector space V – Abelian Group Jul 04 '23 at 20:42
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3@AbelianGroup It says so right in your statement: $0\in V$. – Vercassivelaunos Jul 04 '23 at 20:44
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Moreover $V$ is an abelian group, hence nonempty by definition – julio_es_sui_glace Jul 04 '23 at 20:47
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Okay I get it know. I think I was confused by the use of quantifiers here. – Abelian Group Jul 04 '23 at 21:04
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${0}$ is the smallest Vector Space by definition – kotsos24919 Jul 04 '23 at 21:47
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$\{0\}$ is a vector space over any field, but $\emptyset $ is none since it does not contain the additive neutral element.
Marius S.L.
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