Difference between revisions of "Vector space"
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| − | {{ | + | ==Definition== |
| + | A vector space {{M|V}} over a [[Field|field]] {{M|F}} is a non empty set {{M|V}} 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. | ||
| + | ===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> | ||
| + | |||
| + | ==Homomorphism between vector spaces== | ||
| + | A homomorphism between vector spaces is a [[Linear map|linear map]] | ||
| + | |||
| + | {{Definition|Linear Algebra}} | ||
Revision as of 15:04, 7 March 2015
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.
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]
Homomorphism between vector spaces
A homomorphism between vector spaces is a linear map
