벡터 공간

A vector space over a field $F$ is a non-empty set $V$ 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 $V$ are commonly called vectors, and the elements of $F$ are called scalars.

Axioms

For every $u$, $v$ and $w$ in $V$, and $a$ and $b$ in $F$, the following axioms must be satisfied:

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

Abstract Vector Space

보통은 필드 $F$에 대한 벡터 $V$를 $F^n$ 으로 정의하기 때문에(예: $V=R^2,F=R$) $V$에 속하는 모든 $v$에 대하여, 각 $v$를 구성하는 컴포넌트들은 항상 $F$에 속한다. 하지만 위의 8개 공리에 의한 정의에 따르면 꼭 그럴 필요는 없다.

예:

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

위 정의에서 $V$에 속한 각 벡터는 함수이고 이 경우, 예를 들어 결합 법칙은 $(f+g)(x)=f(x)+g(f)$와 같이 성립한다. 다만 “벡터 $v$를 구성하는 컴포넌트”라는 개념은 함수에 대해서는 존재하지 않기 때문에 $v$를 구성하는 원소가 항상 $F$에 속한다는 식으로 말할 수 없다.

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