Vector Space

definitionvector-spaces

Definition

A vector space (over a field $\mathbb{F}$) is a set $V$ along with an addition and scalar multiplication satisfying the following:

1. Commutativity. $u+v=v+u$ for all $u,v\in V$.
2. Associativity. $u+(v+w)=(u+v)+w$, and $a(bv)=(ab)v$ for all $u,v,w\in V$ and all $a,b\in \mathbb{F}$.
3. Additive Identity. $\exists 0\in V$ such that $0+v=v$ for all $v\in V$.
4. Additive Inverse. $\forall v\in V$, $\exists w\in V$ such that $v+w=0$.
5. Multiplicative Identity. $1_{\mathbb{F}}v=1v=v$ for all $v\in V$.
6. Distributivity. $a(u+v)=au+av$ and $(a+b)v=av+bv$ for all $a,b\in \mathbb{F}$ and $u,v\in V$.

Addition

An addition on a set $V$ is a function ($+$) that assigns an element $u+v\in V$ to each pair of elements $u,v\in V.$

Scalar Multiplication

A scalar multiplication on $V$ is a function that assigns an element $\lambda v\in V$ to each $\lambda \in \mathbb{F}$, and $v\in V$.

Properties

• A vector space has a unique additive identity. (Proof)
• Every element of $V$ has a unique additive inverse. (Proof)

Examples

1. Let ${\mathbb{F}}^{\infty }=\{(x_1,x_2,\dots )\mid x_k\in \mathbb{F}\}$ where vector addition and scalar multiplication are defined coordinate-wise is a vector space (over $\mathbb{F}$).
2. Let $S$ be any set. Define ${\mathbb{F}}^S=\{f\mid f:S\to \mathbb{F}\}$. For $\lambda \in \mathbb{F}$, $f,g\in {\mathbb{F}}^S$, define
   • $(f+g)(x)=f(x)=g(x)$.
   • $(\lambda f)(x)=\lambda f(x)$.

Related

• Subspace