Chapter 42: dual space
dual space and linear functional
42.1 linear equations
a11x1+a12x2+⋯+a1nxn=b1a21x1+a22x2+⋯+a2nxn=b2⋮am1x1+am2x2+⋯+amnxn=bm
(a11a12⋯a1na21a22⋯a2n⋮⋮⋱⋮am1am2⋯amn)(x1x2⋮xn)=(b1b2⋮bn)
Ax=b
(ai1ai2⋯ain)(x1x2⋮xn)=(bi)=bi
(ai1⋯aij⋯ain)(x1⋮xj⋮xn)=(bi)=bi
(⋯aij⋯)(⋮xj⋮)=bi
a⊺ix=ai⋅x=bi
aijxj=a⊺ix=ai⋅x=bi
ai⋅x=a⊺ix=aijxj=⋯+aijxj+⋯
if finite,
ai⋅x=a⊺ix=aijxj=ai1x1+⋯+aijxj+⋯+ainxn
42.2 matrix multiplication
(a11a12⋯a1na21a22⋯a2n⋮⋮⋱⋮am1am2⋯amn)(x11x12⋯x1px21x22⋯x2p⋮⋮⋱⋮xn1xn2⋯xnp)=(b11b12⋯b1pb21b22⋯b2p⋮⋮⋱⋮bm1bm2⋯bmp)
AX=B
(ai1ai2⋯ain)(x1kx2k⋮xnk)=(bik)=bik
(ai1⋯aij⋯ain)(x1k⋮xjk⋮xnk)=(bik)=bik
(⋯aij⋯)(⋮xjk⋮)=bik
a⊺ixk=ai⋅xk=bik
(a11a12⋯a1na21a22⋯a2n⋮⋮⋱⋮am1am2⋯amn)(x11x12⋯x1px21x22⋯x2p⋮⋮⋱⋮xn1xn2⋯xnp)=(b11b12⋯b1pb21b22⋯b2p⋮⋮⋱⋮bm1bm2⋯bmp)=(a1⋅x1a1⋅x2⋯a1⋅xpa2⋅x1a2⋅x2⋯a2⋅xp⋮⋮⋱⋮am⋅x1am⋅x2⋯am⋅xp)=(a⊺1x1a⊺1x2⋯a⊺1xpa⊺2x1a⊺2x2⋯a⊺2xp⋮⋮⋱⋮a⊺mx1a⊺mx2⋯a⊺mxp)
aijxj=a⊺ix=ai⋅x=bi
ai⋅x=a⊺ix=aijxj
\iddots
in MathJax
https://math.meta.stackexchange.com/questions/23273/mathjax-and-iddots-udots-or-reflectbox
\newcommand\iddots{\mathinner{
\kern1mu\raise1pt{.}
\kern2mu\raise4pt{.}
\kern2mu\raise7pt{\Rule{0pt}{7pt}{0pt}.}
\kern1mu
}}
\begin{aligned}
\begin{pmatrix}a_{{\scriptscriptstyle 11}} & \cdots & a_{{\scriptscriptstyle 1j}} & \cdots & a_{{\scriptscriptstyle 1n}}\\
\vdots & \ddots & \vdots & \iddots & \vdots\\
a_{{\scriptscriptstyle i1}} & \cdots & a_{{\scriptscriptstyle ij}} & \cdots & a_{{\scriptscriptstyle in}}\\
\vdots & \iddots & \vdots & \ddots & \vdots\\
a_{{\scriptscriptstyle m1}} & \cdots & a_{{\scriptscriptstyle mj}} & \cdots & a_{{\scriptscriptstyle mn}}
\end{pmatrix}\begin{pmatrix}x_{{\scriptscriptstyle 11}} & \cdots & x_{{\scriptscriptstyle 1k}} & \cdots & x_{{\scriptscriptstyle 1p}}\\
\vdots & \ddots & \vdots & \iddots & \vdots\\
x_{{\scriptscriptstyle j1}} & \cdots & x_{{\scriptscriptstyle jk}} & \cdots & x_{{\scriptscriptstyle jp}}\\
\vdots & \iddots & \vdots & \ddots & \vdots\\
x_{{\scriptscriptstyle n1}} & \cdots & x_{{\scriptscriptstyle nk}} & \cdots & x_{{\scriptscriptstyle np}}
\end{pmatrix}\\
=\begin{pmatrix}b_{{\scriptscriptstyle 11}} & \cdots & b_{{\scriptscriptstyle 1k}} & \cdots & b_{{\scriptscriptstyle 1p}}\\
\vdots & \ddots & \vdots & \iddots & \vdots\\
b_{{\scriptscriptstyle i1}} & \cdots & b_{{\scriptscriptstyle ik}} & \cdots & b_{{\scriptscriptstyle ip}}\\
\vdots & \iddots & \vdots & \ddots & \vdots\\
b_{{\scriptscriptstyle m1}} & \cdots & b_{{\scriptscriptstyle mk}} & \cdots & b_{{\scriptscriptstyle mp}}
\end{pmatrix}= & \begin{pmatrix}\boldsymbol{a}_{{\scriptscriptstyle 1}}\cdot\boldsymbol{x}_{{\scriptscriptstyle 1}} & \cdots & \boldsymbol{a}_{{\scriptscriptstyle 1}}\cdot\boldsymbol{x}_{{\scriptscriptstyle k}} & \cdots & \boldsymbol{a}_{{\scriptscriptstyle 1}}\cdot\boldsymbol{x}_{{\scriptscriptstyle p}}\\
\vdots & \ddots & \vdots & \iddots & \vdots\\
\boldsymbol{a}_{{\scriptscriptstyle i}}\cdot\boldsymbol{x}_{{\scriptscriptstyle 1}} & \cdots & \boldsymbol{a}_{{\scriptscriptstyle i}}\cdot\boldsymbol{x}_{{\scriptscriptstyle k}} & \cdots & \boldsymbol{a}_{{\scriptscriptstyle i}}\cdot\boldsymbol{x}_{{\scriptscriptstyle p}}\\
\vdots & \iddots & \vdots & \ddots & \vdots\\
\boldsymbol{a}_{{\scriptscriptstyle m}}\cdot\boldsymbol{x}_{{\scriptscriptstyle 1}} & \cdots & \boldsymbol{a}_{{\scriptscriptstyle m}}\cdot\boldsymbol{x}_{{\scriptscriptstyle k}} & \cdots & \boldsymbol{a}_{{\scriptscriptstyle m}}\cdot\boldsymbol{x}_{{\scriptscriptstyle p}}
\end{pmatrix}\\
= & \begin{pmatrix}\boldsymbol{a}_{{\scriptscriptstyle 1}}^{\intercal}\boldsymbol{x}_{{\scriptscriptstyle 1}} & \cdots & \boldsymbol{a}_{{\scriptscriptstyle 1}}^{\intercal}\boldsymbol{x}_{{\scriptscriptstyle k}} & \cdots & \boldsymbol{a}_{{\scriptscriptstyle 1}}^{\intercal}\boldsymbol{x}_{{\scriptscriptstyle p}}\\
\vdots & \ddots & \vdots & \iddots & \vdots\\
\boldsymbol{a}_{{\scriptscriptstyle i}}^{\intercal}\boldsymbol{x}_{{\scriptscriptstyle 1}} & \cdots & \boldsymbol{a}_{{\scriptscriptstyle i}}^{\intercal}\boldsymbol{x}_{{\scriptscriptstyle k}} & \cdots & \boldsymbol{a}_{{\scriptscriptstyle i}}^{\intercal}\boldsymbol{x}_{{\scriptscriptstyle p}}\\
\vdots & \iddots & \vdots & \ddots & \vdots\\
\boldsymbol{a}_{{\scriptscriptstyle m}}^{\intercal}\boldsymbol{x}_{{\scriptscriptstyle 1}} & \cdots & \boldsymbol{a}_{{\scriptscriptstyle m}}^{\intercal}\boldsymbol{x}_{{\scriptscriptstyle k}} & \cdots & \boldsymbol{a}_{{\scriptscriptstyle m}}^{\intercal}\boldsymbol{x}_{{\scriptscriptstyle p}}
\end{pmatrix}
\end{aligned}
$$
\begin{pmatrix}\ddots & \vdots & \iddots\\
\cdots & a_{{\scriptscriptstyle ij}} & \cdots\\
\iddots & \vdots & \ddots
\end{pmatrix}\begin{pmatrix}\ddots & \vdots & \iddots\\
\cdots & x_{{\scriptscriptstyle jk}} & \cdots\\
\iddots & \vdots & \ddots
\end{pmatrix}=\begin{pmatrix}\ddots & \vdots & \iddots\\
\cdots & b_{{\scriptscriptstyle ik}} & \cdots\\
\iddots & \vdots & \ddots
\end{pmatrix}=\begin{pmatrix}\ddots & \vdots & \iddots\\
\cdots & \boldsymbol{a}_{{\scriptscriptstyle i}}\cdot\boldsymbol{x}_{{\scriptscriptstyle k}} & \cdots\\
\iddots & \vdots & \ddots
\end{pmatrix}=\begin{pmatrix}\ddots & \vdots & \iddots\\
\cdots & \boldsymbol{a}_{{\scriptscriptstyle i}}^{\intercal}\boldsymbol{x}_{{\scriptscriptstyle k}} & \cdots\\
\iddots & \vdots & \ddots
\end{pmatrix}
$$
bik=aijxjk=ai⋅xk=a⊺ixk
row(A)col(X)=brow,col
ai⋅xk=a⊺ixk=aijxjk
ai⋅xj=a⊺ixj=aikxkj
ai⋅xj=a⊺ixj=aikxkj=⋯+aikxkj+⋯
if finite,
ai⋅xj=a⊺ixj=aikxkj=ai1x1j+⋯+aikxkj+⋯+ainxnj
42.3 functional
( innder product or dot product ) or linear equations
ai⋅x=a⊺ix=aijxj=⋯+aijxj+⋯
if finite,
ai⋅x=a⊺ix=aijxj=ai1x1+⋯+aijxj+⋯+ainxn
actually, several rows
⋮ai⋅x⋮=⋮a⊺ix⋮=⋮aijxj⋮=⋮⋯+aijxj+⋯⋮
if finite,
⋮ai⋅x⋮=⋮a⊺ix⋮=⋮aijxj⋮=⋮ai1x1+⋯+aijxj+⋯+ainxn⋮
in functional aspect,
⋮fi(x)⋮=⋮ai⋅x⋮=⋮a⊺ix⋮=⋮aijxj⋮=⋮⋯+aijxj+⋯⋮
if finite,
⋮fi(x)⋮=⋮ai⋅x⋮=⋮a⊺ix⋮=⋮aijxj⋮=⋮ai1x1+⋯+aijxj+⋯+ainxn⋮
⋮fi(⋯,xj,⋯)⋮=⋮fi(xj)⋮=⋮fi(x)⋮=⋮ai⋅x⋮=⋮a⊺ix⋮=⋮aijxj⋮=⋮⋯+aijxj+⋯⋮
if finite,
⋮fi(x1,⋯,xj,⋯,xn)⋮=⋮fi(xj)⋮=⋮fi(x)⋮=⋮ai⋅x⋮=⋮a⊺ix⋮=⋮aijxj⋮=⋮ai1x1+⋯+aijxj+⋯+ainxn⋮
or simply
fi(⋯,xj,⋯)=fi(xj)=fi(x)=ai⋅xj=a⊺ixj=aikxkj=⋯+aijxj+⋯
if scalar with complex as the field,
fi:C∞→C
if scalar with a field,
fi:F∞→F
or more abstract notation,
fi:F∞→F if scalar with real as the field,
fi:R∞→R
if finite,
fi(x1,⋯,xj,⋯,xn)=fi(xj)=fi(x)=ai⋅xj=a⊺ixj=aijxj=ai1x1+⋯+aijxj+⋯+ainxn
fi(x1,x2,⋯,xn)=fi(x1,⋯,xj,⋯,xn)=fi(xj)=fi(x)=ai⋅xj=a⊺ixj=aijxj=ai1x1+⋯+aijxj+⋯+ainxn
if scalar with complex as the field,
fi:Cn→C
if scalar with a field,
fi:Fn→F
or more abstract notation,
fi:Fn→F if scalar with real as the field,
fi:Rn→R
functionals are a set of fuctions mapping n-dimensional vectors to scalars
fi:Fn→F
⋮fi(⋯,xj,⋯)⋮=⋮fi(xj)⋮=⋮fi(x)⋮=⋮ai⋅x⋮=⋮a⊺ix⋮=⋮aijxj⋮=⋮⋯+aijxj+⋯⋮
f={fi|fi:F∞→F}={⋮fi(⋯,xj,⋯)=fi(xj)=fi(x)=ai⋅x=a⊺ix=aijxj,⋮}
if finite,
⋮fi(x1,⋯,xj,⋯,xn)⋮=⋮fi(xj)⋮=⋮fi(x)⋮=⋮ai⋅x⋮=⋮a⊺ix⋮=⋮aijxj⋮=⋮ai1x1+⋯+aijxj+⋯+ainxn⋮
f={fi|fi:Fn→F}={⋮fi(x1,x2,⋯,xn)=fi(x)=ai1x1+⋯+aijxj+⋯+ainxn,⋮}
linear functionals are a set of fuctions mapping vectors to scalars linearly
f={fi|fi:Fn→F}={⋮fi(x1,x2,⋯,xn)=fi(x)=ai1x1+⋯+aijxj+⋯+ainxn,⋮}
42.4 definition of linear functional
functionals generalized to general vector space
f={fi|fi:V→F}
linear functionals generalized to general vector space is a linear transformation
f={fi|{fi:V→F∀⟨x,y⟩∈V2[fi(x+y)=fi(x)+fi(y)]∀x∈V,c∈F[fi(cx)=cfi(x)]}
f={fi|{fi:V→F∀x,y∈V[fi(x+y)=fi(x)+fi(y)]∀x∈V,c∈F[fi(cx)=cfi(x)]}
if {F=CV=Cn,
f={fi|{fi:Cn→C∀⟨x,y⟩∈(Cn)2[fi(x+y)=fi(x)+fi(y)]∀x∈Cn,c∈C[fi(cx)=cfi(x)]}
f={fi|{fi:Cn→C∀x,y∈Cn[fi(x+y)=fi(x)+fi(y)]∀x∈Cn,c∈C[fi(cx)=cfi(x)]}
then
fi(x)=ai1x1+⋯+aijxj+⋯+ainxn
satisfying
fi(x+y)=ai1(x+y)1+⋯+aij(x+y)j+⋯+ain(x+y)n=ai1(x1+y1)+⋯+aij(xj+yj)+⋯+ain(xn+yn)=(ai1x1+ai1y1)+⋯+(aijxj+aijyj)+⋯+(ainxn+ainyn)=(ai1x1+⋯+aijxj+⋯+ainxn)+(ai1y1+⋯+aijyj+⋯+ainyn)=fi(x)+fi(y)
fi(cx)=ai1(cx)1+⋯+aij(cx)j+⋯+ain(cx)n=ai1(cx1)+⋯+aij(cxj)+⋯+ain(cxn)=c(ai1x1)+⋯+c(aijxj)+⋯+c(ainxn)=c(ai1x1+⋯+aijxj+⋯+ainxn)=cfi(x)
different functional has different aij
let
aij=fi(ej),ej=⟨j−1⏞0,⋯,0,1,0,⋯,0⟩=(0⋯010⋯0)⊺=(0⋮010⋮0)=(0⋮010⋮0)n×1
if fi is a linear functional, then
fi(x)=fi(x1e1+⋯+xjej+⋯+xnen)=fi(x1e1)+⋯+fi(xjej)+⋯+fi(xnen)=x1fi(e1)+⋯+xjfi(ej)+⋯+xnfi(en)=x1ai1+⋯+xjaij+⋯+xnain=ai1x1+⋯+aijxj+⋯+ainxn=fi(x)
42.5 set of all linear transformations is a vector space
https://math.stackexchange.com/questions/2381942/the-set-of-all-linear-maps-tv-w-is-a-vector-space
T:V→W⇔∀v∈V,∃!w∈W[w=T(v)]
{V,W are vector spacesT:V→W{∀u,v∈V[T(u+v)=T(u)+T(v)]∀v∈V,c∈F[T(cv)=cT(v)]linearity⇔T is a linear tranformation
{V,W are vector spaces, both over FT,U:V→W{T:V→WU:V→WT,U are both linear tranformationsv∈Vc∈F
There is still linearity over linear transformations
(va)
(T+U)(u+v)=T(u+v)+U(u+v)=[T(u)+T(v)]+[U(u)+U(v)]=[T(u)+U(u)]+[T(v)+U(v)]=(T+U)(u)+(T+U)(v)
(sm)
(T+U)(cv)=T(cv)+U(cv)=cT(v)+cU(v)=c[T(v)+U(v)]=c(T+U)(v)
so we can define
{(T+U)(v)=T(v)+U(v)linear transformation addition(cT)(v)=cT(v)scalar linear transformation multiplication
the set of all linear tranformations is a vector space
T is the set of all linear tranformations
{F(f)F is a fieldT≠∅(ne)nonempty set+:T×T=T2+→T⇔∀T,U∈T,∃S∈T[S=T+U](va)vector addition⋅:F×T⋅→T⇔∀c∈F,∀T∈T,∃U∈T[U=cT=c⋅T](sm)scalar multiplication{∀S,T,U∈T[S+(T+U)=(S+T)+U](a)∀T,U∈T[T+U=U+T](c)∃!O∈T,∀T∈T[O+T=T](e)∀T∈T,∃!−T∈T[(−T)+T=O](i)(va)vector addition axioms{∀b,c∈F,T∈T[b(cT)=(bc)T](a)∃!1∈F,∀T∈T[1T=T](e)∀c∈F,T,U∈T[c(T+U)=cT+cU](dv)∀b,c∈F,T∈T[(b+c)T=bT+cT](ds)(sm)scalar multiplication axioms⇔T=T(F,+,⋅)=(T,F,+,⋅) is a vector space over the field F⇔T is a vector space
Selected proofs of 8 vector space axioms due to some trivial field and vector space properties:
(va)(a)
(S+(T+U))(v)=S(v)+(T+U)(v)=S(v)+T(v)+U(v)=(S+T)(v)+U(v)=((S+T)+U)(v)
(va)(c)
(T+U)(v)=T(v)+U(v)=U(v)+T(v)=(U+T)(v)
(va)(e)
O(v)=0w∈W
(O+T)(v)=O(v)+T(v)=0w+T(v)=T(v)
O1(v)−O2(v)=0w−0w=0w⇒O1(v)=O2(v)
(sm)(dv)
(c(T+U))(v)=c(T+U)(v)=c[T(v)+U(v)]=cT(v)+cU(v)=(cT+cU)(v)
The set of all linear tranformations T is a vector space.
42.6 definition of dual space
V∗=L(V,F)=f={fi|{fi:V→Ffunctional mapping vector to field scalar{∀x,y∈V[fi(x+y)=fi(x)+fi(y)]∀x∈V,c∈F[fi(cx)=cfi(x)](L)linearity}⇔V∗ is a dual space, a set of linear functionals fi mapping vectors in the vector space V to scalars in the field F
https://web.math.sinica.edu.tw/mathmedia/HTMLarticle18.jsp?mID=31304
https://web.math.sinica.edu.tw/mathmedia/author18.jsp?query_filter=%E9%BE%94%E6%98%87
42.7 double dual
double dual = second dual
https://www.zhihu.com/question/444079322/answer/1749490720
Hough Transform