# Vector space

> 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.

A vector space over a field $F$ is a non-empty <set> $V$ together with a [binary operation](https://wiki.g15e.com/pages/Binary%20operation.txt) call a *vector addition* and a [binary function](https://wiki.g15e.com/pages/Binary%20function.txt) 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 법칙>은 $(f + g)(x) = f(x) + g(f)$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: https://www.3blue1brown.com/lessons/abstract-vector-spaces