Vector space

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A vector space over a field (frequently the real numbers) is an object which arises in linear algebra and abstract algebra. A vector space over a field consists of a set (of vectors) and two operations, vector addition and scalar multiplication, which obey the following rules:

Axioms of a vector space

• Under vector addition, the set of vectors forms an abelian group. Thus, addition is associative and commutative and there is an additive identity and additive inverses.

• Scalar multiplication is associative, so if and then .

• Scalar multiplication is distributive, so if and then .