Definition: abstract vector space
A (real) vector space is a non-empty set
of objects called vectors with the following operations:
- Vector addition operation
. - Multiplication by scalar operation
. The vector space must satisfy the following properties, known as vector space axioms:
- Associativity of addition:
. - Commutativity of addition:
. - Existence of additive identity:
. - Existence of additive inverse:
. - Compatibility of multiplication by scalar with real number multiplication:
. - Identity element of multiplication by scalar:
. - Distributivity of multiplication by scalar over vector addition:
. - Distributivity of multiplication by scalar over real number sum:
.