Vector space

Definition

A vector space $V$ over a field $F$ consists of a set on which two operations $+$ (addition) and $\cdot$ (scalar multiplication) are defined so that the followings hold.

For any $\mathbf{x},\mathbf{y},\mathbf{z}\in V$ and $c,d\in F$:

(A1) $\mathbf{x},\mathbf{y}\in V\Rightarrow \mathbf{x}+\mathbf{y}\in V$

(A2) $\mathbf{x}+\mathbf{y}=\mathbf{y}+\mathbf{x}$ (Commutativity of addition)

(A3) $(\mathbf{x}+\mathbf{y})+\mathbf{z}=\mathbf{x}+(\mathbf{y}+\mathbf{z})$ (Associativity of addition)

(A4) $\exists !0\in V$ such that $\forall \mathbf{x}\in V:\mathbf{x}+0=\mathbf{x}$ (Existence of additive identity)

(A5) $\forall \mathbf{x}\in V:\exists !(-\mathbf{x})\in V$ such that $\mathbf{x}+(-\mathbf{x})=0$ (Existence of additive inverse)

(M1) $\mathbf{x}\in V\Rightarrow c\mathbf{x}\in V$

(M2) $c(d\mathbf{x})=(cd)\mathbf{x}$

(M3) $c(\mathbf{x}+\mathbf{y})=c\mathbf{x}+c\mathbf{y}$

(M4) $(c+d)\mathbf{x}=c\mathbf{x}+d\mathbf{x}$

(M5) $\forall \mathbf{x}\in V:1\mathbf{x}=\mathbf{x}$

Examples

The space of $m\times n$ matrices

Let $F$ be any field and $m,n\in \mathbb{N}$. Define $F^{m\times n}$ as the set of all $m\times n$ matrices over the field $F$. Then, for any matrices $A,B\in F^{m\times n}$ and scalar $c\in F$,

• $(A+B)\_ij=A\_ij+B\_ij$
• $(cA)\_ij=cA\_ij$

The space of functions from a set to a field

Let $F$ be any field, and $S$ be any nonempty set. Define $V$ as the set of all functions $S\to F$. Then, for any functions $f,g\in V$ and scalar $c\in F$, $\forall s\in S$:

• $(f+g)(s)=f(s)+g(s)$
• $(cf)(s)=cf(s)$
• The zero vector is the function $0:S\to \{0\}$.
• The inverse vector $(-f)$ of an arbitrary function $f\in V$ is given as $(-f)(s)=-f(s)$.

The space of polynomial functions over a field

Let $F$ be any field. Define $V$ as tje set of all polynomials $F\to F$, that is, all functions $f:F\to F$ in the form of $f(x)=\sum _{i=0}^n{c}_i{x}^i$ where $c_i\in F$ $(i=0,\cdots ,n)$ are independent of $x$.

Related Concepts

• Norm: Defines a distance-like property in vector spaces
• Inner product: Allows the concept of ‘direction’ to be considered in vector spaces