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Vector Spaces

A vector space is a set along with a field , endowed with two operations: vector addition and scalar multiplication. The elements of are called vectors, and the elements of are called scalars. A set of axioms must be satisfied for to be considered a vector space:

  1. Closure under addition: For any two vectors , their sum is also in .
  2. Commutativity of addition: For any two vectors , .
  3. Associativity of addition: For any three vectors , .
  4. Existence of additive identity: There exists a vector such that for any vector , .
  5. Existence of additive inverse: For every vector , there exists a vector such that .
  6. Closure under scalar multiplication: For any scalar and any vector , the product is also in .
  7. Distributivity of scalar multiplication with respect to vector addition: For any scalar and any two vectors , .
  8. Distributivity of scalar multiplication with respect to field addition: For any two scalars and any vector , .
  9. Associativity of scalar multiplication: For any two scalars and any vector , .
  10. Existence of multiplicative identity: For any vector , , where is the multiplicative identity in .

We can summarize the axioms into smaller statements. In particular, axioms 1-5 state that is an abelian group, and axioms 6-10 describe how scalar multiplication interacts with vector addition and field addition.