Linear Algebra – Sufficient Conditions for Proving $V$ is a Vector Space

linear algebrasoft-question

Often times in introductory linear Algebra the statement Prove that $V$ is vector space comes up.

My question is whether proving closure under addition and scalar multiplication along with the existence of an additive identity is sufficient to show that some set is a vector space or do we need to prove the other plethora of properties that go along with the definition of a vector space.

I ask this because generally those other properties can be inferred from the basic properties of closure under addition and scalar multiplication.

Best Answer

A minimal set of axioms for vector spaces is:

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This is discussed and proved in this paper:

Independent Axioms for Vector Spaces, J. F. Rigby and James Wiegold. The Mathematical Gazette Vol. 57, No. 399 (Feb., 1973), pp. 56-62.

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