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
Contents
[hide]Definition
A vector space V over a field F is a non empty set V and the binary operations:
- +:V×V→V given by +(x,y)=x+y - vector addition
- ×:F×V→V given by ×(λ,x)=λx - scalar multiplication
Such that the following 8 "axioms of a vector space" hold
Axioms of a vector space
- (x+y)+z=x+(y+z) ∀x,y,z∈V
- x+y=y+x ∀x,y∈V
- ∃ea∈V∀x∈V:x+ea=x - this ea is denoted 0 once proved unique.
- ∀x∈V ∃y∈V:x+y=ea - this y is denoted −x once proved unique.
- λ(x+y)=λx+λy ∀λ∈F, x,y∈V
- (λ+μ)x=λx+μx ∀λ,μ∈F, x∈V
- λ(μx)=(λμ)x ∀λ,μ∈F, x∈V
- ∃em∈F∀x∈V:emx=x - this em is denoted 1 once proved unique.
Example
Take Rn, an entry v∈Rn may be denoted (v1,...,vn)=v, scalar multiplication and addition are defined as follows:
- λ∈R,v∈Rn we define scalar multiplication λv=(λv1,...,λvn)
- u,v∈Rn - we define addition as u+v=(u1+v1,...,un+vn)
Homomorphism between vector spaces
A homomorphism between vector spaces is a linear map
