A vector space is an algebraic structure consisting of an additive Abelian group (elements of which are called vectors, and are denoted in bold), a field (elements of which are called scalars), and a scalar multiplication function following these properties:

  • Distributive property of scalar multiplication over vector addition: For all and , .
  • Distributive property of scalar multiplication over field addition: For all and , .
  • Associative law of combined scalar and field multiplication: For all and , .
  • Scalar multiplication identity: With 1 as the field multiplicative identity, for all , we have .

As both and each have their own respective additive identites, we will denote boldface to represent the additive identity in . We then say that is a vector space over the field .

Definitions

  • A subset is:
    • Linearly dependent if there exist (distinct) vectors and scalars , with at least one of these scalars non-zero, such that . Otherwise, we say is linearly independent.
    • A spanning set if, for any there exist vectors and scalars such that . That is to say, any vector in is a linear combination of vectors in .
    • A basis for if is linearly independent and a spanning set.
    • A subspace for if is also a subgroup and is closed under scalar multiplication: For any vector and scalar , .
  • Given two vector spaces and over a field , a function is a linear transformation if for any and , and . In abstract algebra, this is known as a homomorphism.

Theorems

  • If and are bases for , then they have the same cardinality. That is, there exists a bijective function (proof). As such, we may define the dimension of as the cardinality of any basis for .
  • Given a basis , and linear combinations and , if , then for each applicable . This is to say any vector in is a unique linear combination of vectors in .
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