Vector space

Detaileddefinition

Vectorspaceisalsocalledlinearspace.Itisoneofthecentralcontentandbasicconceptsoflinearalgebra.LetVbeanon-emptysetandPbeadomain.If:

1.AnoperationisdefinedinV,calledaddition,whichmeansthatanytwoelementsαandβinVcorrespondtoauniqueelementαinVaccordingtoacertainrule.+β,calledthesumofαandβ.

2.AnoperationisdefinedbetweentheelementsofPandV,calledscalarmultiplication(alsoknownasquantitymultiplication),thatis,foranyelementαinVandanyelementkinP,pressAcertainrulecorrespondstoauniqueelementkαinV,whichiscalledtheproductofkandα.

3.Additionandscalarmultiplicationmeetthefollowingconditions:

1)α+β=β+α,foranyα,β∈V.

2)α+(β+γ)=(α+β)+γ,foranyα,β,γ∈V.

3)Thereisanelement0∈V,forallα∈Vthereisα+0=α,element0iscalledthenullelementofV.

4)Foranyα∈V,thereisβ∈Vsothatα+β=0,andβiscalledthenegativeelementofα,Denotedas-α.

5)Fortheunitelement1inP,1α=α(α∈V).

6)Foranyk,l∈P,α∈Vhas(kl)α=k(lα).

7)Foranyk,l∈P,α∈Vhas(k+l)α=kα+lα.

8)Foranyk∈P,α,β∈V,thereisk(α+β)=kα+kβ,andthenViscalledalinearspaceorvectorspaceonthedomainP.TheelementsinVarecalledvectors,thezeroelementsofVarecalledzerovectors,andPiscalledthebasefieldoflinearspace.WhenPisarealnumberfield,Viscalledareallinearspace.WhenPisacomplexnumberfield,ViscalledacomplexLinearspace.Forexample,ifVisasetcomposedofallvectors(directedlinesegments)inathree-dimensionalgeometricspace,andPisarealnumberfieldR,thentheadditionofVtovectors(thatis,theparallelogramrule)andthemultiplicationofnumbersandvectorsformalinearityontherealnumberfieldRspace.Foranotherexample,ifVisthesetM_mn(P)composedofallm×nmatricesinthenumberfieldP,theadditionandscalarmultiplicationofVaretheadditionofmatricesandthemultiplicationofnumbersandmatrices,respectively,thenM_mn(P)isalinearspaceonthenumberdomainP.ThevectorinVisanm×nmatrix.Foranotherexample,thesetPcomposedofalln-elementvectors(a₁,a₂,...,a_n)onthedomainPisforaddition:(a₁,a₂,...,a_n)+(b₁,b₂,...,b_n)=(a₁+b₁,a₂+b₂,...,a_n+b_n)andscalarmultiplication:λ(a₁,a₂,…,a_n)=(λa₁,λa₂,…,λa_n)constitutethelinearspaceonthedomainP,whichiscalledthen-elementvectorspaceonthedomainP.

Linearspaceisinvestigatingtheessentialpropertiesofalargenumberofmathematicalobjects(suchasvectorsingeometryandphysics,n-elementvectors,matrices,polynomialsinalgebra,functionsinanalysis,etc.)Themathematicalconceptsabstractedlater,manyresearchobjectsinmodernmathematics,suchasnormedlinearspacesandmodules,arecloselyrelatedtolinearspaces.Itstheoriesandmethodshavepenetratedintomanyfieldsofnaturalscienceandengineeringtechnology.Hamilton(W.R.)firstintroducedthetermvectorandpioneeredvectortheoryandvectorcalculation.Grassmann(H.G.)firstproposedthesystemtheoryofmultidimensionalEuclideanspace.From1844to1847,heandCauchy(A.-L.)respectivelyproposedanabstractn-dimensionalspacethatwasseparatedfromallspaceintuitionandbecameapurelymathematicalconcept.Toeplitz(O.)extendedthemaintheoremsoflinearalgebratogenerallinearspacesonarbitraryfields.

Axiomaticdefinition

LetFbeadomain.AvectorspaceonFistwooperationsofasetV:

Vectoraddition:V+V→V,denotedasv+w,Vv,w∈V

Scalarmultiplication:F×V→V,denotedasa·v,Va∈F,v∈V

Meetsthefollowingaxioms(∀a,b∈Fandu,v,w∈V):

1. Associativelawofvectoraddition:u+(v+w)=(u+v)+w;

2. Commutativelawofvectoraddition:v+w=​​w+v;

3. Theidentityelementofvectoraddition:Vcontainsazerovectorcalled0,∀v∈V,v+0=v;

4. Theinverseelementofvectoraddition:∀v∈V,∃w∈V,suchthatv+w=​​0;

5. Scalarmultiplicationisallocatedtovectoraddition:a(v+w)=av+aw;

6. Scalarmultiplicationisallocatedtofieldaddition:(a+b)v=av+bv;

7. Scalarmultiplicationisconsistentwithscalardomainmultiplication:a(bv)=(ab)v;

8. Scalarmultiplicationhasidentityelement:1v=v,where1referstothemultiplicationidentityelementofdomainF.

Sometextbooksalsoemphasizethefollowingtwoaxioms:

Visclosedundervectoraddition:v+w∈V

Visclosedunderscalarmultiplication:av∈V

Moreabstractly,avectorspaceonFisanF-module.ThemembersofVarecalledvectors,andthemembersofFarecalledscalars.IfFistherealnumberfieldR,Viscalledtherealvectorspace;ifFisthecomplexnumberfieldC,Viscalledthecomplexvectorspace;ifFisthefinitefield,Viscalledthefinitefieldvectorspace;forthegeneralfieldF,ViscalledF-vectorspace.

Thefirst4axiomsexplainthatthevectorVisanAbeliangroupinvectoraddition,andtheremaining4axiomsareusedforscalarmultiplication.

Thefollowingaresomeofthecharacteristicsthatareeasilyderivedfromthevectorspaceaxioms:

• Thezerovector0∈V(Axiom3)isunique

• a0=0,∀a∈F

• 0v=0,∀v∈V,where0istheadditionofFUnitelement

• av=0,youcandeduceeithera=0orv=0

• TheadditiveinverseofvYuan(Axiom4)isunique(writtenas−v).Bothv−wandv+(−w)arestandard.

• (−1)v=−v,∀v∈V

• (−a)v=a(−v)=−(av),∀a∈F,∀v∈V

Linearityindependence

IfVisalinearspace,iftherearecoefficientsc₁,c2,...,c_n∈F,suchthatc₁v₁+c₂v₂+...+c_nv_n=0,thenthereareafinitenumberofvectorsv₁,v₂,...,v_narecalledlinearlyrelated.

Onthecontrary,thissetofvectorsiscalledlinearlyindependent.Moregenerally,ifthereareaninfinitenumberofvectors,wesaythattheseinfinitevectorsarelinearlyindependent,andifanyfinitenumberofthemarelinearlyindependent.

Subspace

LetWbeanon-emptysubsetofvectorspaceV,ifWisclosedundertheadditionofVandscalarmultiplication,andthezerovector0∈W,thenCallWthelinearsubspaceofV.

GivenavectorsetB,thenthesmallestsubspacecontainingitiscalleditsexpansion,denotedasspan(B).Inaddition,theexpansionoftheemptysetcanbespecifiedas{0}.

GivenavectorsetB,ifitsexpansionisthevectorspaceV,thenBiscalledthegeneratingsetofV.

GivenavectorsetB,ifBislinearlyindependent,andBcangenerateV,thenBiscalledabasisofV.IfV={0},theonlybasisistheemptyset.Foranon-zerovectorspaceV,thebasisisthesmallestgeneratingsetofV,whichisalsoamaximallylinearlyindependentgroup.

IfavectorspaceVhasageneratingsetwithafinitenumberofelements,thenVissaidtobeafinite-dimensionalspace.Allthebasisofavectorspacehavethesamecardinality,whichiscalledthedimensionofthespace.Forexample,realnumbervectorspace:R⁰,R¹,R²,R³,…,RThedimensionofⁿisn.

Eachvectorinthespacehasauniquemethodtoexpressasalinearcombinationofvectorsinthebase.Moreover,byarrangingthevectorsinthebasistorepresentanorderedbasis,eachvectorcanberepresentedbyacoordinatesystem.

Linearmapping

IfbothVandWarevectorspacesonthedomainF,alineartransformationfromVtoWor"linearmapping"canbeset".ThesemappingsfromVtoWhavesomethingincommon,thatis,theymaintainthesumandscalarquotient.ThissetcontainsalllinearmappingsfromVtoW,describedbyL(V,W),andisalsoavectorspaceonthedomainF.WhenVandWaredetermined,thelinearmappingcanbeexpressedbyamatrix.

Isomorphismisaone-to-onelinearmapping.IfthereisanisomorphismbetweenVandW,wecallthesetwospacesisomorphism;eachn-dimensionalvectorspaceonthedomainFisthesameasthevectorspaceFIsomorphism.

AvectorspaceintheFfieldplusalinearmappingcanformacategory,thatis,theAbeliancategory.

Extrastructure

Thestudyofvectorspacesnaturallyinvolvessomeextrastructure.Theadditionalstructureisasfollows:

Arealorcomplexvectorspaceplustheconceptoflength.Thenormiscalledthenormedvectorspace.

Theconceptofarealorcomplexvectorspacepluslengthandangleiscalledinnerproductspace.

Avectorspaceplustopologicallyconsistentoperations(additionandscalarmultiplicationarecontinuousmappings)iscalledatopologicalvectorspace.

Avectorspaceplusabilinearoperator(definedasvectormultiplication)isafieldalgebra.