# Difference between revisions of "Vector space"

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

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==Definition== | ==Definition== | ||

A vector space {{M|V}} over a [[Field|field]] {{M|F}} is a non empty set {{M|V}} and the binary operations: | A vector space {{M|V}} over a [[Field|field]] {{M|F}} is a non empty set {{M|V}} and the binary operations: | ||

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# <math>\lambda(\mu x)=(\lambda\mu)x\ \forall\lambda,\mu\in F,\ x\in V</math> | # <math>\lambda(\mu x)=(\lambda\mu)x\ \forall\lambda,\mu\in F,\ x\in V</math> | ||

# <math>\exists e_m\in F\forall x\in V:e_m x = x</math> - this <math>e_m</math> is denoted <math>1</math> once proved unique. | # <math>\exists e_m\in F\forall x\in V:e_m x = x</math> - this <math>e_m</math> is denoted <math>1</math> once proved unique. | ||

+ | ===Notation=== | ||

+ | We denote a vector space as "Let <math>(V,F)</math> be a vector space" often we will write simply "let <math>V</math> be a vector space" if it is understood what the field is, because [[Mathematicians are lazy|mathematicians are lazy]] | ||

+ | |||

+ | A [[Norm|normed]] vector space may be denoted <math>(V,\|\cdot\|_V,F)</math> | ||

===Example=== | ===Example=== | ||

Take <math>\mathbb{R}^n</math>, an entry <math>v\in\mathbb{R}^n</math> may be denoted <math>(v_1,...,v_n)=v</math>, scalar multiplication and addition are defined as follows: | Take <math>\mathbb{R}^n</math>, an entry <math>v\in\mathbb{R}^n</math> may be denoted <math>(v_1,...,v_n)=v</math>, scalar multiplication and addition are defined as follows: | ||

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* <math>u,v\in\mathbb{R}^n</math> - we define addition as <math>u+v=(u_1+v_1,...,u_n+v_n)</math> | * <math>u,v\in\mathbb{R}^n</math> - we define addition as <math>u+v=(u_1+v_1,...,u_n+v_n)</math> | ||

− | == | + | ==Important concepts== |

− | + | * [[Linear map|Linear maps]] - the homomorphisms and isomorphisms of vector spaces | |

+ | * [[Span, linear independence, linear dependence, basis and dimension]] | ||

+ | * [[Norm]] | ||

+ | * [[Linear isometry]] | ||

{{Definition|Linear Algebra}} | {{Definition|Linear Algebra}} |

## Latest revision as of 16:30, 23 August 2015

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 [ilmath]V[/ilmath] over a field [ilmath]F[/ilmath] is a non empty set [ilmath]V[/ilmath] and the binary operations:

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

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

### Axioms of a vector space

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

### Notation

We denote a vector space as "Let [math](V,F)[/math] be a vector space" often we will write simply "let [math]V[/math] be a vector space" if it is understood what the field is, because mathematicians are lazy

A normed vector space may be denoted [math](V,\|\cdot\|_V,F)[/math]

### Example

Take [math]\mathbb{R}^n[/math], an entry [math]v\in\mathbb{R}^n[/math] may be denoted [math](v_1,...,v_n)=v[/math], scalar multiplication and addition are defined as follows:

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

## Important concepts

- Linear maps - the homomorphisms and isomorphisms of vector spaces
- Span, linear independence, linear dependence, basis and dimension
- Norm
- Linear isometry