Difference between revisions of "Vector space"
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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