Vector space

An introduction to the important concepts of vector spaces and linear algebra may be found on the Basis and coordinates page

Definition

A vector space $V$ over a field $F$ is a non empty set $V$ and the binary operations:

$+:V\times V\rightarrow V$ given by $+(x,y)=x+y$ - vector addition
$\times:F\times V\rightarrow V$ given by $\times(\lambda,x)=\lambda x$ - scalar multiplication

Such that the following 8 "axioms of a vector space" hold

$(x+y)+z=x+(y+z)\ \forall x,y,z\in V$
$x+y=y+x\ \forall x,y\in V$
$\exists e_a\in V\forall x\in V:x+e_a=x$ - this $e_a$ is denoted $0$ once proved unique.
$\forall x\in V\ \exists y\in V:x+y=e_a$ - this $y$ is denoted $-x$ once proved unique.
$\lambda(x+y)=\lambda x+\lambda y\ \forall\lambda\in F,\ x,y\in V$
$(\lambda+\mu)x = \lambda x+\mu x\ \forall\lambda,\mu\in F,\ x\in V$
$\lambda(\mu x)=(\lambda\mu)x\ \forall\lambda,\mu\in F,\ x\in V$
$\exists e_m\in F\forall x\in V:e_m x = x$ - this $e_m$ is denoted $1$ once proved unique.

We denote a vector space as "Let

$(V,F)$ be a vector space" often we will write simply "let

$V$ be a vector space" if it is understood what the field is, because mathematicians are lazy

A normed vector space may be denoted

$(V,\|\cdot\|_V,F)$

Take

$\mathbb{R}^n$ , an entry

$v\in\mathbb{R}^n$ may be denoted

$(v_1,...,v_n)=v$ , scalar multiplication and addition are defined as follows:

$\lambda\in\mathbb{R},v\in\mathbb{R}^n$ we define scalar multiplication $\lambda v=(\lambda v_1,...,\lambda v_n)$
$u,v\in\mathbb{R}^n$ - we define addition as $u+v=(u_1+v_1,...,u_n+v_n)$