Chapter 43: vector space

https://www.bilibili.com/video/BV1ez4y1n7h8

43.1 What is a vector?

What is a vector? or What is an element in a vector space?

Binary operations defined on a vector space satisfying some properties is more important than what is a vector.

ultimate answer: double dual concept[43.4.1.2]

43.2 vector space definition

https://tex.stackexchange.com/a/141489 multiline node

vector space construction

Fig. 43.1: vector space construction

{F is a field(f)fieldV≠∅(ne)nonempty set+:V×V=V2+→V⇔∀u,v∈V,∃!w∈V[w=u+v](va)vector addition⋅:F×V⋅→V⇔∀s∈F,∀v∈V,∃!u∈V[u=sv=s⋅v](sm)scalar multiplication{∃!0∈V,∀v∈V[0+v=v](e)identity∀v∈V,∃!−v∈V[(−v)+v=0](i)inverse∀u,v,w∈V[u+(v+w)=(u+v)+w](a)associativity∀u,v∈V[u+v=v+u](c)commutativity(va)axioms{∃!1∈F,∀v∈V[1v=v](e)identity∀s,t∈F,v∈V[s(tv)=(st)v](a)associativity∀s,t∈F,v∈V[(s+t)v=sv+tv](ds)scalar distributivity∀s∈F,u,v∈V[s(u+v)=su+sv](dv)vector distributivity(sm)axioms⇔V=V(F,+,⋅)=(V,F,+,⋅) is a vector space over the field F⇔V is a vector space

43.2.1 commutative group structure of vector space

(va) axioms = vector addition axioms

V=(V,+) is a commutative group⇔V=(V,+) is an abelian group⇔{V=(V,+)=(V,+V) is a group(g)group∀u,v∈V[u+v=v+u](c)commutativity⇔{{+:V×V=V2+→V⇔∀u,v∈V,∃!w∈V[w=u+v](cl)closure∃!0∈V,∀v∈V[0+v=v](e)identity∀v∈V,∃!−v∈V[(−v)+v=0](i)inverse∀u,v,w∈V[u+(v+w)=(u+v)+w](a)associativity(g)∀u,v∈V[u+v=v+u](c)

V is a vector space⇔V=V(F,+,⋅)=(V,F,+,⋅) is a vector space over the field F⇔{F is a field(f)fieldV≠∅(ne)nonempty setV=(V,+) is a commutative group⇔V=(V,+) is an abelian group(va)vector addition⋅:F×V⋅→V⇔∀s∈F,∀v∈V,∃!u∈V[u=sv=s⋅v](sm)scalar multiplication{∃!1∈F,∀v∈V[1v=v](e)identity∀s,t∈F,v∈V[s(tv)=(st)v](a)associativity∀s,t∈F,v∈V[(s+t)v=sv+tv](ds)scalar distributivity∀s∈F,u,v∈V[s(u+v)=su+sv](dv)vector distributivity(sm)axioms⇔{F is a field(f)fieldV≠∅(ne)nonempty set{V=(V,+)=(V,+V) is a group(g)group∀u,v∈V[u+v=v+u](c)commutativity(va)vector addition⋅=⋅F×V:F×V⋅→V⇔∀s∈F,∀v∈V,∃!u∈V[u=sv=s⋅v](sm)scalar multiplication{∃!1∈F,∀v∈V[1v=v](e)identity∀s,t∈F,v∈V[s(tv)=(st)v](a)associativity∀s,t∈F,v∈V[(s+t)v=sv+tv](ds)scalar distributivity∀s∈F,u,v∈V[s(u+v)=su+sv](dv)vector distributivity(sm)axioms⇔{F=F(+F,⋅F)=(F,+F,⋅F)=(F,+,⋅) is a field(f)V≠∅(ne){{+:V×V=V2+→V⇔∀u,v∈V,∃!w∈V[w=u+v](cl)closure∃!0∈V,∀v∈V[0+v=v](e)identity∀v∈V,∃!−v∈V[(−v)+v=0](i)inverse∀u,v,w∈V[u+(v+w)=(u+v)+w](a)associativity(g)∀u,v∈V[u+v=v+u](c)(va){⋅:F×V⋅→V⇔∀s∈F,∀v∈V,∃!u∈V[u=sv=s⋅v](cl)closure∃!1∈F,∀v∈V[1v=v](e)identity∀s,t∈F,v∈V[s(tv)=s⋅F×V(t⋅F×Vv)=(s⋅Ft)⋅F×Vv=(st)v](a)associativity∀s,t∈F,v∈V[(s+t)v=(s+Ft)v=sv+Vtv=sv+tv](ds)scalar distributivity∀s∈F,u,v∈V[s(u+v)=su+sv](dv)vector distributivity(sm)

43.2.2 scalar distributivity

(sm)(ds)

∀s,t∈F,v∈V[(s+t)v=sv+tv]

∀s,t∈F,v∈V[(s+Ft)v=sv+tv]

∀s,t∈F,v∈V[(s+Ft)v=sv+Vtv]

43.3 linearity

{f(x+y)=f(x)+f(y)additivityf(λx)=λf(x)homogeneity⇔f(λx+y)=λf(x)+f(y)⇔f is linear

43.3.1 linear structure of vector space

∀s∈F,u,v∈V[u+sv∈V]

∀s∈F,∀u,v∈V[u+sv∈V]

∀s∈F,⟨u,v⟩∈V2[u+sv∈V]

{∀u,v∈V[u+v∈V]vector addition closure∀s∈F,v∈V[sv∈V]scalar multiplication closure⇔{∀u,v∈V[u+v∈V](a)additivity∀s∈F,v∈V[sv∈V](h)homogeneity⇔∀s∈F,u,v∈V[u+sv∈V] (l)linearity

43.3.2 linear transformation or linear map

{V,W are vector spacesT:V→W{∀u,v∈V[T(u+v)=T(u)+T(v)](a)additivity∀v∈V,c∈F[T(cv)=cT(v)](h)homogeneity(L)⇔{V,W are vector spacesT:V→W∀u,v∈V,c∈F[T(u+cv)=T(u)+cT(v)](l)linearity⇔T is a linear map from V to W⇔T is a linear tranformation

43.4 vector space example

  • arrow vector
  • number
    • integer
    • real
    • complex
    • quaternion
  • function
    • polynomial function
    • continuous function
  • matrix
    • real matrix
    • complex matrix
  • reciprocal space

https://www.bilibili.com/video/BV1NC4y1J7UL

applications in different disciplines

43.4.1 reciprocal space

reciprocal space = 倒易空間

{e1=aa×b≠0e2=bb×c≠0e3=cc×a≠0⇒{e′1=b×cΩe′2=c×aΩe′3=a×bΩ,

Ω=a⋅(b×c)=b⋅(c×a)=c⋅(a×b)

reciprocal space as dual space and contravariant vector

span{e1,e2,e3}=span{a,b,c}=V=R3=span{e′1,e′2,e′3}=span{b×cΩ,c×aΩ,a×bΩ}=span{e1,e2,e3}=span{e∗}∗∈{1,2,3}=V∗

43.4.1.1 Kronecker delta

(e1⋅e′1e1⋅e′2e1⋅e′3e2⋅e′1e2⋅e′2e2⋅e′3e3⋅e′1e3⋅e′2e3⋅e′3)=[δij]=(100010001)=(e1⋅e1e1⋅e2e1⋅e3e2⋅e1e2⋅e2e2⋅e3e3⋅e1e3⋅e2e3⋅e3)

Kronecker delta

ei⋅e′j=δij={1i=j0i≠j

Kronecker delta tensor = Kronecker tensor

ei(ej)=ei⋅ej=δij=δij={1i=j0i≠j

v=vaa+vbb+vcc=v1e1+v2e2+v3e3

e1⋅v=v⋅e′1=v1e1⋅e′1+v2e2⋅e′1+v3e3⋅e′1=v1

e2⋅v=v⋅e′2=v1e1⋅e′2+v2e2⋅e′2+v3e3⋅e′2=v2

e3⋅v=v⋅e′3=v1e1⋅e′3+v2e2⋅e′3+v3e3⋅e′3=v3

v=v1e1+v2e2+v3e3=(v⋅e′1)e1+(v⋅e′2)e2+(v⋅e′3)e3=(e1⋅v)e1+(e2⋅v)e2+(e3⋅v)e3=e1(v)e1+e2(v)e2+e3(v)e3

{e1(v)=e1⋅v=v⋅e′1=v1e2(v)=e2⋅v=v⋅e′2=v2e3(v)=e3⋅v=v⋅e′3=v3

ei(ej)=ei⋅ej=δij=δij={1i=j0i≠j

reciprocal space is a dual space of its original vector space

V=span{e1,e2,e3}={v1e1+v2e2+v3e3}={3∑j=1vjej}={vjej|{vj∈Fej∈F3}={v|v∈V}V∗=span{e1,e2,e3}={v∗1e1+v∗2e2+v∗3e3}={3∑i=1v∗iei}={v∗iei|{v∗i∈Fei∈F3}={v∗|v∗∈V∗}

v∗(v)=(v∗iei)(v),v∈V=(v∗1e1+v∗2e2+v∗3e3)(v)=v∗1e1(v)+v∗2e2(v)+v∗3e3(v)=v∗1v1+v∗2v2+v∗3v3∈F

element in dual space is a functional or mapping from its original vector space to the field

v∗:V→F

Vv∗→F

V∗={v∗|v∗:V→F}

V={e1e2e3v⋯}e1:↓↓↓↓↓V∗={e2}F⊇{100v1⋯}e3V={e1e2e3v⋯}v∗:↓↓↓↓↓⋮F⊇{v∗1v∗2v∗3v∗ivi⋯}

V∗={v∗|v∗∈V∗}={v∗|v∗:V→F}={v∗|Vv∗→F}={ω|ω:V→F}={ωiei|{ωi∈Fei∈F3}

By defining vector addition and scalar multiplication on the dual space

{+:V∗×V∗→V∗⇔∀ω1,ω2∈V∗,∃!(ω1+ω2)∈V∗[(ω1+ω2)(v)=ω1(v)+ω2(v)]⋅:F×V∗→V∗⇔∀k∈F,∀ω∈V∗,∃!(kω)∈V∗[(kω)(v)=k⋅ω(v)]∀ω∈V∗,∃!0∈V∗[(ω+0)(v)=ω(v)+0(v)=ω(v)]

the dual space also becomes a vector space.

43.4.1.2 double dual concept

double dual space = second dual space

V∗∗=(V∗)∗={ω∗|ω∗:V∗→F}={ω∗|ω∗∈V∗∗}

V∗∗=(V∗)∗=span{eμ}∗μ∈{1,2,3}=span{e1,e2,e3}∗=span{e1∗,e2∗,e3∗}=span{eν∗}ν∈{1,2,3}

ω∗(ω)=(ω∗νeν∗)(ω),ω∈V∗=(ω∗1e1∗+ω∗2e2∗+ω∗3e3∗)(ω)=ω∗1e1∗(ω)+ω∗2e2∗(ω)+ω∗3e3∗(ω)=ω∗1ω1+ω∗2ω2+ω∗3ω3∈F

V∗∗={ω∗|ω∗:V∗→F}

V∗={e1e2e3ω⋯}e1∗:↓↓↓↓↓V∗∗={e2∗}F⊇{100ω1⋯}e3∗V∗={e1e2e3v⋯}ω∗:↓↓↓↓↓⋮F⊇{ω1∗ω2∗ω3∗ωμ∗ωμ⋯}

{e1∗(ω)=e1∗⋅ω=ω⋅e1∗=ω(e1∗)e2∗(ω)=e2∗⋅ω=ω⋅e2∗=ω(e2∗)e3∗(ω)=e3∗⋅ω=ω⋅e3∗=ω(e3∗)

ω∗(ω)=ω∗⋅ω=ω⋅ω∗=ω(ω∗) i.e. f acts on x equivalent to x acts on f

x(f)=x⋅f=f⋅x=f(x)

eμ(eν)=eμ⋅eν=δμν=δμν={1μ=ν0μ≠ν

eν∗(eμ)def.=eν⋅eμ=eμ⋅eν=eμ(eν)⇓V∗∗=span{eν∗}ν∈{1,2,3}≅span{eν}ν∈{1,2,3}=VV∗∗≅V⇓{V∗∗≅VV,V∗∗ are isomorphismindependent of choice of basesV,V∗∗ are naturally isomorphism

V∗∗={ω∗|ω∗:V∗→F}≅V={v|v:V∗→F}

V∗={e1e2e3ω⋯}e1∗:↓↓↓↓↓V∗∗={e2∗}F⊇{100ω1⋯}e3∗V∗={e1e2e3ω⋯}ω∗:↓↓↓↓↓⋮F⊇{ω1∗ω2∗ω3∗ωμ∗ωμ⋯}V∗={e1e2e3v∗⋯}e1:↓↓↓↓↓≅V={e2}F⊇{100v∗1⋯}e3V∗={e1e2e3v∗⋯}v:↓↓↓↓↓⋮F⊇{v1v2v3vμv∗μ⋯}

V≅V∗∗

V≅V∗∗={ω∗|ω∗:V∗→F}

V={v|v:V∗→F}

i.e. vector space is a set of functionals or mappings from its dual space to the field, answering What is a vector?[43.1], and satifying Fig: 43.1.

43.8 subspace

  1. https://www.linkedin.com/in/yiwen-pan-16a90076↩︎