Chapter 42: dual space

dual space and linear functional

42.1 linear equations

a11x1+a12x2++a1nxn=b1a21x1+a22x2++a2nxn=b2am1x1+am2x2++amnxn=bm

(a11a12a1na21a22a2nam1am2amn)(x1x2xn)=(b1b2bn)

Ax=b

(ai1ai2ain)(x1x2xn)=(bi)=bi

(ai1aijain)(x1xjxn)=(bi)=bi

(aij)(xj)=bi

aix=aix=bi

aijxj=aix=aix=bi


aix=aix=aijxj=+aijxj+

if finite,

aix=aix=aijxj=ai1x1++aijxj++ainxn

42.2 matrix multiplication

(a11a12a1na21a22a2nam1am2amn)(x11x12x1px21x22x2pxn1xn2xnp)=(b11b12b1pb21b22b2pbm1bm2bmp)

AX=B

(ai1ai2ain)(x1kx2kxnk)=(bik)=bik

(ai1aijain)(x1kxjkxnk)=(bik)=bik

(aij)(xjk)=bik

aixk=aixk=bik

(a11a12a1na21a22a2nam1am2amn)(x11x12x1px21x22x2pxn1xn2xnp)=(b11b12b1pb21b22b2pbm1bm2bmp)=(a1x1a1x2a1xpa2x1a2x2a2xpamx1amx2amxp)=(a1x1a1x2a1xpa2x1a2x2a2xpamx1amx2amxp)

aijxj=aix=aix=bi

aix=aix=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=aixk=aixk

row(A)col(X)=brow,col

aixk=aixk=aijxjk

aixj=aixj=aikxkj


aixj=aixj=aikxkj=+aikxkj+

if finite,

aixj=aixj=aikxkj=ai1x1j++aikxkj++ainxnj

42.3 functional

( innder product or dot product ) or linear equations

aix=aix=aijxj=+aijxj+

if finite,

aix=aix=aijxj=ai1x1++aijxj++ainxn


actually, several rows

aix=aix=aijxj=+aijxj+

if finite,

aix=aix=aijxj=ai1x1++aijxj++ainxn


in functional aspect,

fi(x)=aix=aix=aijxj=+aijxj+

if finite,

fi(x)=aix=aix=aijxj=ai1x1++aijxj++ainxn


fi(,xj,)=fi(xj)=fi(x)=aix=aix=aijxj=+aijxj+

if finite,

fi(x1,,xj,,xn)=fi(xj)=fi(x)=aix=aix=aijxj=ai1x1++aijxj++ainxn


or simply

fi(,xj,)=fi(xj)=fi(x)=aixj=aixj=aikxkj=+aijxj+

if scalar with complex as the field,

fi:CC

if scalar with a field,

fi:FF

or more abstract notation,

fi:FF if scalar with real as the field,

fi:RR


if finite,

fi(x1,,xj,,xn)=fi(xj)=fi(x)=aixj=aixj=aijxj=ai1x1++aijxj++ainxn

fi(x1,x2,,xn)=fi(x1,,xj,,xn)=fi(xj)=fi(x)=aixj=aixj=aijxj=ai1x1++aijxj++ainxn

if scalar with complex as the field,

fi:CnC

if scalar with a field,

fi:FnF

or more abstract notation,

fi:FnF if scalar with real as the field,

fi:RnR


functionals are a set of fuctions mapping n-dimensional vectors to scalars

fi:FnF


fi(,xj,)=fi(xj)=fi(x)=aix=aix=aijxj=+aijxj+

f={fi|fi:FF}={fi(,xj,)=fi(xj)=fi(x)=aix=aix=aijxj,}

if finite,

fi(x1,,xj,,xn)=fi(xj)=fi(x)=aix=aix=aijxj=ai1x1++aijxj++ainxn

f={fi|fi:FnF}={fi(x1,x2,,xn)=fi(x)=ai1x1++aijxj++ainxn,}


linear functionals are a set of fuctions mapping vectors to scalars linearly

f={fi|fi:FnF}={fi(x1,x2,,xn)=fi(x)=ai1x1++aijxj++ainxn,}

42.4 definition of linear functional

functionals generalized to general vector space

f={fi|fi:VF}

linear functionals generalized to general vector space is a linear transformation

f={fi|{fi:VFx,yV2[fi(x+y)=fi(x)+fi(y)]xV,cF[fi(cx)=cfi(x)]}

f={fi|{fi:VFx,yV[fi(x+y)=fi(x)+fi(y)]xV,cF[fi(cx)=cfi(x)]}

if {F=CV=Cn,

f={fi|{fi:CnCx,y(Cn)2[fi(x+y)=fi(x)+fi(y)]xCn,cC[fi(cx)=cfi(x)]}

f={fi|{fi:CnCx,yCn[fi(x+y)=fi(x)+fi(y)]xCn,cC[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=j10,,0,1,0,,0=(00100)=(00100)=(00100)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

vector space[43]

https://math.stackexchange.com/questions/2381942/the-set-of-all-linear-maps-tv-w-is-a-vector-space

T:VWvV,!wW[w=T(v)]

{V,W are vector spacesT:VW{u,vV[T(u+v)=T(u)+T(v)]vV,cF[T(cv)=cT(v)]linearityT is a linear tranformation

{V,W are vector spaces, both over FT,U:VW{T:VWU:VWT,U are both linear tranformationsvVcF

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+TT,UT,ST[S=T+U](va)vector addition:F×TTcF,TT,UT[U=cT=cT](sm)scalar multiplication{S,T,UT[S+(T+U)=(S+T)+U](a)T,UT[T+U=U+T](c)!OT,TT[O+T=T](e)TT,!TT[(T)+T=O](i)(va)vector addition axioms{b,cF,TT[b(cT)=(bc)T](a)!1F,TT[1T=T](e)cF,T,UT[c(T+U)=cT+cU](dv)b,cF,TT[(b+c)T=bT+cT](ds)(sm)scalar multiplication axiomsT=T(F,+,)=(T,F,+,) is a vector space over the field FT 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)=0wW

(O+T)(v)=O(v)+T(v)=0w+T(v)=T(v)

O1(v)O2(v)=0w0w=0wO1(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:VFfunctional mapping vector to field scalar{x,yV[fi(x+y)=fi(x)+fi(y)]xV,cF[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

vector space[43]

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