벡터 공간

  • 2014-01-01 (modified: 2025-10-02)

A vector space over a field FF is a non-empty set VV together with a binary operation call a vector addition and a binary function call a scalar multiplication that satisfy the axioms listed below. In this context, the elements of VV are commonly called vectors, and the elements of FF are called scalars.

Axioms

For every uu, vv and ww in VV, and aa and bb in FF, the following axioms must be satisfied:

  • Associativity of vector addition: u+(v+w)=(u+v)+wu + (v + w) = (u + v) + w.
  • Commutativity of vector addition: u+v=v+uu + v = v + u.
  • Identity element of vector addition: There exists an element 0V0 \in V, called the zero vector u+0=uu + 0 = u for all vVv \in V.
  • Inverse element of vector addition: There exists an element vV-v \in V, called the additive inverse of vv, such that v+(v)=0v + (-v) = 0.
  • Compatibility of scalar multiplication with field multiplication: a(bv)=(ab)va(bv) = (ab)v (This axiom is not an associative property, since it refers to two different operations, scalar multiplication and field multiplication.)
  • Identity element of scalar multiplication: 1v=v1v = v, where 11 denotes multiplicative identity in FF.
  • Distributivity of scalar multiplication with respect to vector addition: a(u+v)=au+uva(u + v) = au + uv
  • Distributivity of scalar multiplication with respect to field addition: (a+b)v=av+bv(a + b)v = av + bv

Abstract Vector Space

보통은 필드 FF에 대한 벡터 VVFnF^n 으로 정의하기 때문에(예: V=R2,F=RV = R^2, F=R) VV에 속하는 모든 vv에 대하여, 각 vv를 구성하는 컴포넌트들은 항상 FF에 속한다. 하지만 위의 8개 공리에 의한 정의에 따르면 꼭 그럴 필요는 없다.

예:

  • F=RF = R
  • VV is the set of all countinous real-valued function defined on the interval [0, 1].

위 정의에서 VV에 속한 각 벡터는 함수이고 이 경우, 예를 들어 결합 법칙(f+g)(x)=f(x)+g(f)(f + g)(x) = f(x) + g(f)와 같이 성립한다. 다만 “벡터 vv를 구성하는 컴포넌트”라는 개념은 함수에 대해서는 존재하지 않기 때문에 vv를 구성하는 원소가 항상 FF에 속한다는 식으로 말할 수 없다.

참고: 3blue1brown.com/lessons/abstract-vector-spaces