Chapter 42: dual space

dual space and linear functional

https://ccjou.wordpress.com/2011/06/13/線性泛函與對偶空間/

42.1 linear equations

\[ \begin{aligned} a_{{\scriptscriptstyle 11}}x_{{\scriptscriptstyle 1}}+a_{{\scriptscriptstyle 12}}x_{{\scriptscriptstyle 2}}+\cdots+a_{{\scriptscriptstyle 1n}}x_{{\scriptscriptstyle n}}= & b_{{\scriptscriptstyle 1}}\\ a_{{\scriptscriptstyle 21}}x_{{\scriptscriptstyle 1}}+a_{{\scriptscriptstyle 22}}x_{{\scriptscriptstyle 2}}+\cdots+a_{{\scriptscriptstyle 2n}}x_{{\scriptscriptstyle n}}= & b_{{\scriptscriptstyle 2}}\\ \vdots\\ a_{{\scriptscriptstyle m1}}x_{{\scriptscriptstyle 1}}+a_{{\scriptscriptstyle m2}}x_{{\scriptscriptstyle 2}}+\cdots+a_{{\scriptscriptstyle mn}}x_{{\scriptscriptstyle n}}= & b_{{\scriptscriptstyle m}} \end{aligned} \]

\[ A\boldsymbol{x}=\boldsymbol{b} \]

\[ \begin{pmatrix}a_{{\scriptscriptstyle i1}} & a_{{\scriptscriptstyle i2}} & \cdots & a_{{\scriptscriptstyle in}}\end{pmatrix}\begin{pmatrix}x_{{\scriptscriptstyle 1}}\\ x_{{\scriptscriptstyle 2}}\\ \vdots\\ x_{{\scriptscriptstyle n}} \end{pmatrix}=\begin{pmatrix}b_{{\scriptscriptstyle i}}\end{pmatrix}=b_{{\scriptscriptstyle i}} \]

\[ \begin{pmatrix}a_{{\scriptscriptstyle i1}} & \cdots & a_{{\scriptscriptstyle ij}} & \cdots & a_{{\scriptscriptstyle in}}\end{pmatrix}\begin{pmatrix}x_{{\scriptscriptstyle 1}}\\ \vdots\\ x_{{\scriptscriptstyle j}}\\ \vdots\\ x_{{\scriptscriptstyle n}} \end{pmatrix}=\begin{pmatrix}b_{{\scriptscriptstyle i}}\end{pmatrix}=b_{{\scriptscriptstyle i}} \]

\[ \begin{pmatrix}\cdots & a_{{\scriptscriptstyle ij}} & \cdots\end{pmatrix}\begin{pmatrix}\vdots\\ x_{{\scriptscriptstyle j}}\\ \vdots \end{pmatrix}=b_{{\scriptscriptstyle i}} \]

\[ \boldsymbol{a}_{{\scriptscriptstyle i}}^{\intercal}\boldsymbol{x}=\boldsymbol{a}_{{\scriptscriptstyle i}}\cdot\boldsymbol{x}=b_{i} \]

\[ a_{{\scriptscriptstyle ij}}x_{{\scriptscriptstyle j}}=\boldsymbol{a}_{{\scriptscriptstyle i}}^{\intercal}\boldsymbol{x}=\boldsymbol{a}_{{\scriptscriptstyle i}}\cdot\boldsymbol{x}=b_{i} \]

\[ \boldsymbol{a}_{{\scriptscriptstyle i}}\cdot\boldsymbol{x}=\boldsymbol{a}_{{\scriptscriptstyle i}}^{\intercal}\boldsymbol{x}=a_{{\scriptscriptstyle ij}}x_{{\scriptscriptstyle j}}=\cdots+a_{{\scriptscriptstyle ij}}x_{{\scriptscriptstyle j}}+\cdots \]

if finite,

\[ \boldsymbol{a}_{{\scriptscriptstyle i}}\cdot\boldsymbol{x}=\boldsymbol{a}_{{\scriptscriptstyle i}}^{\intercal}\boldsymbol{x}=a_{{\scriptscriptstyle ij}}x_{{\scriptscriptstyle j}}=a_{{\scriptscriptstyle i1}}x_{{\scriptscriptstyle 1}}+\cdots+a_{{\scriptscriptstyle ij}}x_{{\scriptscriptstyle j}}+\cdots+a_{{\scriptscriptstyle in}}x_{{\scriptscriptstyle n}} \]

42.2 matrix multiplication

\[ \begin{pmatrix}a_{{\scriptscriptstyle 11}} & a_{{\scriptscriptstyle 12}} & \cdots & a_{{\scriptscriptstyle 1n}}\\ a_{{\scriptscriptstyle 21}} & a_{{\scriptscriptstyle 22}} & \cdots & a_{{\scriptscriptstyle 2n}}\\ \vdots & \vdots & \ddots & \vdots\\ a_{{\scriptscriptstyle m1}} & a_{{\scriptscriptstyle m2}} & \cdots & a_{{\scriptscriptstyle mn}} \end{pmatrix}\begin{pmatrix}x_{{\scriptscriptstyle 11}} & x_{{\scriptscriptstyle 12}} & \cdots & x_{{\scriptscriptstyle 1p}}\\ x_{{\scriptscriptstyle 21}} & x_{{\scriptscriptstyle 22}} & \cdots & x_{{\scriptscriptstyle 2p}}\\ \vdots & \vdots & \ddots & \vdots\\ x_{{\scriptscriptstyle n1}} & x_{{\scriptscriptstyle n2}} & \cdots & x_{{\scriptscriptstyle np}} \end{pmatrix}=\begin{pmatrix}b_{{\scriptscriptstyle 11}} & b_{{\scriptscriptstyle 12}} & \cdots & b_{{\scriptscriptstyle 1p}}\\ b_{{\scriptscriptstyle 21}} & b_{{\scriptscriptstyle 22}} & \cdots & b_{{\scriptscriptstyle 2p}}\\ \vdots & \vdots & \ddots & \vdots\\ b_{{\scriptscriptstyle m1}} & b_{{\scriptscriptstyle m2}} & \cdots & b_{{\scriptscriptstyle mp}} \end{pmatrix} \]

\[ AX=B \]

\[ \begin{pmatrix}a_{{\scriptscriptstyle i1}} & a_{{\scriptscriptstyle i2}} & \cdots & a_{{\scriptscriptstyle in}}\end{pmatrix}\begin{pmatrix}x_{{\scriptscriptstyle 1k}}\\ x_{{\scriptscriptstyle 2k}}\\ \vdots\\ x_{{\scriptscriptstyle nk}} \end{pmatrix}=\begin{pmatrix}b_{{\scriptscriptstyle ik}}\end{pmatrix}=b_{{\scriptscriptstyle ik}} \]

\[ \begin{pmatrix}a_{{\scriptscriptstyle i1}} & \cdots & a_{{\scriptscriptstyle ij}} & \cdots & a_{{\scriptscriptstyle in}}\end{pmatrix}\begin{pmatrix}x_{{\scriptscriptstyle 1k}}\\ \vdots\\ x_{{\scriptscriptstyle jk}}\\ \vdots\\ x_{{\scriptscriptstyle nk}} \end{pmatrix}=\begin{pmatrix}b_{{\scriptscriptstyle ik}}\end{pmatrix}=b_{{\scriptscriptstyle ik}} \]

\[ \begin{pmatrix}\cdots & a_{{\scriptscriptstyle ij}} & \cdots\end{pmatrix}\begin{pmatrix}\vdots\\ x_{{\scriptscriptstyle jk}}\\ \vdots \end{pmatrix}=b_{{\scriptscriptstyle ik}} \]

\[ \boldsymbol{a}_{{\scriptscriptstyle i}}^{\intercal}\boldsymbol{x}_{{\scriptscriptstyle k}}=\boldsymbol{a}_{{\scriptscriptstyle i}}\cdot\boldsymbol{x}_{{\scriptscriptstyle k}}=b_{ik} \]

\[ \begin{aligned} \begin{pmatrix}a_{{\scriptscriptstyle 11}} & a_{{\scriptscriptstyle 12}} & \cdots & a_{{\scriptscriptstyle 1n}}\\ a_{{\scriptscriptstyle 21}} & a_{{\scriptscriptstyle 22}} & \cdots & a_{{\scriptscriptstyle 2n}}\\ \vdots & \vdots & \ddots & \vdots\\ a_{{\scriptscriptstyle m1}} & a_{{\scriptscriptstyle m2}} & \cdots & a_{{\scriptscriptstyle mn}} \end{pmatrix}\begin{pmatrix}x_{{\scriptscriptstyle 11}} & x_{{\scriptscriptstyle 12}} & \cdots & x_{{\scriptscriptstyle 1p}}\\ x_{{\scriptscriptstyle 21}} & x_{{\scriptscriptstyle 22}} & \cdots & x_{{\scriptscriptstyle 2p}}\\ \vdots & \vdots & \ddots & \vdots\\ x_{{\scriptscriptstyle n1}} & x_{{\scriptscriptstyle n2}} & \cdots & x_{{\scriptscriptstyle np}} \end{pmatrix}\\ =\begin{pmatrix}b_{{\scriptscriptstyle 11}} & b_{{\scriptscriptstyle 12}} & \cdots & b_{{\scriptscriptstyle 1p}}\\ b_{{\scriptscriptstyle 21}} & b_{{\scriptscriptstyle 22}} & \cdots & b_{{\scriptscriptstyle 2p}}\\ \vdots & \vdots & \ddots & \vdots\\ b_{{\scriptscriptstyle m1}} & b_{{\scriptscriptstyle m2}} & \cdots & b_{{\scriptscriptstyle mp}} \end{pmatrix}= & \begin{pmatrix}\boldsymbol{a}_{{\scriptscriptstyle 1}}\cdot\boldsymbol{x}_{{\scriptscriptstyle 1}} & \boldsymbol{a}_{{\scriptscriptstyle 1}}\cdot\boldsymbol{x}_{{\scriptscriptstyle 2}} & \cdots & \boldsymbol{a}_{{\scriptscriptstyle 1}}\cdot\boldsymbol{x}_{{\scriptscriptstyle p}}\\ \boldsymbol{a}_{{\scriptscriptstyle 2}}\cdot\boldsymbol{x}_{{\scriptscriptstyle 1}} & \boldsymbol{a}_{{\scriptscriptstyle 2}}\cdot\boldsymbol{x}_{{\scriptscriptstyle 2}} & \cdots & \boldsymbol{a}_{{\scriptscriptstyle 2}}\cdot\boldsymbol{x}_{{\scriptscriptstyle p}}\\ \vdots & \vdots & \ddots & \vdots\\ \boldsymbol{a}_{{\scriptscriptstyle m}}\cdot\boldsymbol{x}_{{\scriptscriptstyle 1}} & \boldsymbol{a}_{{\scriptscriptstyle m}}\cdot\boldsymbol{x}_{{\scriptscriptstyle 2}} & \cdots & \boldsymbol{a}_{{\scriptscriptstyle m}}\cdot\boldsymbol{x}_{{\scriptscriptstyle p}} \end{pmatrix}\\ = & \begin{pmatrix}\boldsymbol{a}_{{\scriptscriptstyle 1}}^{\intercal}\boldsymbol{x}_{{\scriptscriptstyle 1}} & \boldsymbol{a}_{{\scriptscriptstyle 1}}^{\intercal}\boldsymbol{x}_{{\scriptscriptstyle 2}} & \cdots & \boldsymbol{a}_{{\scriptscriptstyle 1}}^{\intercal}\boldsymbol{x}_{{\scriptscriptstyle p}}\\ \boldsymbol{a}_{{\scriptscriptstyle 2}}^{\intercal}\boldsymbol{x}_{{\scriptscriptstyle 1}} & \boldsymbol{a}_{{\scriptscriptstyle 2}}^{\intercal}\boldsymbol{x}_{{\scriptscriptstyle 2}} & \cdots & \boldsymbol{a}_{{\scriptscriptstyle 2}}^{\intercal}\boldsymbol{x}_{{\scriptscriptstyle p}}\\ \vdots & \vdots & \ddots & \vdots\\ \boldsymbol{a}_{{\scriptscriptstyle m}}^{\intercal}\boldsymbol{x}_{{\scriptscriptstyle 1}} & \boldsymbol{a}_{{\scriptscriptstyle m}}^{\intercal}\boldsymbol{x}_{{\scriptscriptstyle 2}} & \cdots & \boldsymbol{a}_{{\scriptscriptstyle m}}^{\intercal}\boldsymbol{x}_{{\scriptscriptstyle p}} \end{pmatrix} \end{aligned} \]

\[ a_{{\scriptscriptstyle ij}}x_{{\scriptscriptstyle j}}=\boldsymbol{a}_{{\scriptscriptstyle i}}^{\intercal}\boldsymbol{x}=\boldsymbol{a}_{{\scriptscriptstyle i}}\cdot\boldsymbol{x}=b_{i} \]

\[ \boldsymbol{a}_{{\scriptscriptstyle i}}\cdot\boldsymbol{x}=\boldsymbol{a}_{{\scriptscriptstyle i}}^{\intercal}\boldsymbol{x}=a_{{\scriptscriptstyle ij}}x_{{\scriptscriptstyle j}} \]

\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}

$$
\begin{pmatrix}\ddots & \vdots & \iddots\\
\cdots & a_{{\scriptscriptstyle ij}} & \cdots\\
\iddots & \vdots & \ddots
\end{pmatrix}\begin{pmatrix}\ddots & \vdots & \iddots\\
\cdots & x_{{\scriptscriptstyle jk}} & \cdots\\
\iddots & \vdots & \ddots
\end{pmatrix}=\begin{pmatrix}\ddots & \vdots & \iddots\\
\cdots & b_{{\scriptscriptstyle ik}} & \cdots\\
\iddots & \vdots & \ddots
\end{pmatrix}=\begin{pmatrix}\ddots & \vdots & \iddots\\
\cdots & \boldsymbol{a}_{{\scriptscriptstyle i}}\cdot\boldsymbol{x}_{{\scriptscriptstyle k}} & \cdots\\
\iddots & \vdots & \ddots
\end{pmatrix}=\begin{pmatrix}\ddots & \vdots & \iddots\\
\cdots & \boldsymbol{a}_{{\scriptscriptstyle i}}^{\intercal}\boldsymbol{x}_{{\scriptscriptstyle k}} & \cdots\\
\iddots & \vdots & \ddots
\end{pmatrix}
$$

\[ b_{{\scriptscriptstyle ik}}=a_{{\scriptscriptstyle ij}}x_{{\scriptscriptstyle jk}}=\boldsymbol{a}_{{\scriptscriptstyle i}}\cdot\boldsymbol{x}_{{\scriptscriptstyle k}}=\boldsymbol{a}_{{\scriptscriptstyle i}}^{\intercal}\boldsymbol{x}_{{\scriptscriptstyle k}} \]

\[ \mathrm{row}\left(A\right)\mathrm{col}\left(X\right)=b_{\mathrm{row},\mathrm{col}} \]

\[ \boldsymbol{a}_{{\scriptscriptstyle i}}\cdot\boldsymbol{x}_{{\scriptscriptstyle k}}=\boldsymbol{a}_{{\scriptscriptstyle i}}^{\intercal}\boldsymbol{x}_{{\scriptscriptstyle k}}=a_{{\scriptscriptstyle ij}}x_{{\scriptscriptstyle jk}} \]

\[ \boldsymbol{a}_{{\scriptscriptstyle i}}\cdot\boldsymbol{x}_{{\scriptscriptstyle j}}=\boldsymbol{a}_{{\scriptscriptstyle i}}^{\intercal}\boldsymbol{x}_{{\scriptscriptstyle j}}=a_{{\scriptscriptstyle ik}}x_{{\scriptscriptstyle kj}}=\cdots+a_{{\scriptscriptstyle ik}}x_{{\scriptscriptstyle kj}}+\cdots \]

if finite,

\[ \boldsymbol{a}_{{\scriptscriptstyle i}}\cdot\boldsymbol{x}_{{\scriptscriptstyle j}}=\boldsymbol{a}_{{\scriptscriptstyle i}}^{\intercal}\boldsymbol{x}_{{\scriptscriptstyle j}}=a_{{\scriptscriptstyle ik}}x_{{\scriptscriptstyle kj}}=a_{{\scriptscriptstyle i1}}x_{{\scriptscriptstyle 1j}}+\cdots+a_{{\scriptscriptstyle ik}}x_{{\scriptscriptstyle kj}}+\cdots+a_{{\scriptscriptstyle in}}x_{{\scriptscriptstyle nj}} \]

42.3 functional

( innder product or dot product ) or linear equations

if finite,

\[ \boldsymbol{a}_{{\scriptscriptstyle i}}\cdot\boldsymbol{x}=\boldsymbol{a}_{{\scriptscriptstyle i}}^{\intercal}\boldsymbol{x}=a_{{\scriptscriptstyle ij}}x_{{\scriptscriptstyle j}}=a_{{\scriptscriptstyle i1}}x_{{\scriptscriptstyle 1}}+\cdots+a_{{\scriptscriptstyle ij}}x_{{\scriptscriptstyle j}}+\cdots+a_{{\scriptscriptstyle in}}x_{{\scriptscriptstyle n}} \]

actually, several rows

\[ \begin{array}{c} \vdots\\ \boldsymbol{a}_{{\scriptscriptstyle i}}\cdot\boldsymbol{x}\\ \vdots \end{array}=\begin{array}{c} \vdots\\ \boldsymbol{a}_{{\scriptscriptstyle i}}^{\intercal}\boldsymbol{x}\\ \vdots \end{array}=\begin{array}{c} \vdots\\ a_{{\scriptscriptstyle ij}}x_{{\scriptscriptstyle j}}\\ \vdots \end{array}=\begin{array}{c} \vdots\\ \cdots+a_{{\scriptscriptstyle ij}}x_{{\scriptscriptstyle j}}+\cdots\\ \vdots \end{array} \]

if finite,

\[ \begin{array}{c} \vdots\\ \boldsymbol{a}_{{\scriptscriptstyle i}}\cdot\boldsymbol{x}\\ \vdots \end{array}=\begin{array}{c} \vdots\\ \boldsymbol{a}_{{\scriptscriptstyle i}}^{\intercal}\boldsymbol{x}\\ \vdots \end{array}=\begin{array}{c} \vdots\\ a_{{\scriptscriptstyle ij}}x_{{\scriptscriptstyle j}}\\ \vdots \end{array}=\begin{array}{c} \vdots\\ a_{{\scriptscriptstyle i1}}x_{{\scriptscriptstyle 1}}+\cdots+a_{{\scriptscriptstyle ij}}x_{{\scriptscriptstyle j}}+\cdots+a_{{\scriptscriptstyle in}}x_{{\scriptscriptstyle n}}\\ \vdots \end{array} \]

in functional aspect,

\[ \begin{array}{c} \vdots\\ f_{{\scriptscriptstyle i}}\left(\boldsymbol{x}\right)\\ \vdots \end{array}=\begin{array}{c} \vdots\\ \boldsymbol{a}_{{\scriptscriptstyle i}}\cdot\boldsymbol{x}\\ \vdots \end{array}=\begin{array}{c} \vdots\\ \boldsymbol{a}_{{\scriptscriptstyle i}}^{\intercal}\boldsymbol{x}\\ \vdots \end{array}=\begin{array}{c} \vdots\\ a_{{\scriptscriptstyle ij}}x_{{\scriptscriptstyle j}}\\ \vdots \end{array}=\begin{array}{c} \vdots\\ \cdots+a_{{\scriptscriptstyle ij}}x_{{\scriptscriptstyle j}}+\cdots\\ \vdots \end{array} \]

if finite,

\[ \begin{array}{c} \vdots\\ f_{{\scriptscriptstyle i}}\left(\boldsymbol{x}\right)\\ \vdots \end{array}=\begin{array}{c} \vdots\\ \boldsymbol{a}_{{\scriptscriptstyle i}}\cdot\boldsymbol{x}\\ \vdots \end{array}=\begin{array}{c} \vdots\\ \boldsymbol{a}_{{\scriptscriptstyle i}}^{\intercal}\boldsymbol{x}\\ \vdots \end{array}=\begin{array}{c} \vdots\\ a_{{\scriptscriptstyle ij}}x_{{\scriptscriptstyle j}}\\ \vdots \end{array}=\begin{array}{c} \vdots\\ a_{{\scriptscriptstyle i1}}x_{{\scriptscriptstyle 1}}+\cdots+a_{{\scriptscriptstyle ij}}x_{{\scriptscriptstyle j}}+\cdots+a_{{\scriptscriptstyle in}}x_{{\scriptscriptstyle n}}\\ \vdots \end{array} \]

\[ \begin{array}{c} \vdots\\ f_{{\scriptscriptstyle i}}\left(\cdots,x_{{\scriptscriptstyle j}},\cdots\right)\\ \vdots \end{array}=\begin{array}{c} \vdots\\ f_{{\scriptscriptstyle i}}\left(x_{{\scriptscriptstyle j}}\right)\\ \vdots \end{array}=\begin{array}{c} \vdots\\ f_{{\scriptscriptstyle i}}\left(\boldsymbol{x}\right)\\ \vdots \end{array}=\begin{array}{c} \vdots\\ \boldsymbol{a}_{{\scriptscriptstyle i}}\cdot\boldsymbol{x}\\ \vdots \end{array}=\begin{array}{c} \vdots\\ \boldsymbol{a}_{{\scriptscriptstyle i}}^{\intercal}\boldsymbol{x}\\ \vdots \end{array}=\begin{array}{c} \vdots\\ a_{{\scriptscriptstyle ij}}x_{{\scriptscriptstyle j}}\\ \vdots \end{array}=\begin{array}{c} \vdots\\ \cdots+a_{{\scriptscriptstyle ij}}x_{{\scriptscriptstyle j}}+\cdots\\ \vdots \end{array} \]

if finite,

\[ \begin{array}{c} \vdots\\ f_{{\scriptscriptstyle i}}\left(x_{{\scriptscriptstyle 1}},\cdots,x_{{\scriptscriptstyle j}},\cdots,x_{{\scriptscriptstyle n}}\right)\\ \vdots \end{array}=\begin{array}{c} \vdots\\ f_{{\scriptscriptstyle i}}\left(x_{{\scriptscriptstyle j}}\right)\\ \vdots \end{array}=\begin{array}{c} \vdots\\ f_{{\scriptscriptstyle i}}\left(\boldsymbol{x}\right)\\ \vdots \end{array}=\begin{array}{c} \vdots\\ \boldsymbol{a}_{{\scriptscriptstyle i}}\cdot\boldsymbol{x}\\ \vdots \end{array}=\begin{array}{c} \vdots\\ \boldsymbol{a}_{{\scriptscriptstyle i}}^{\intercal}\boldsymbol{x}\\ \vdots \end{array}=\begin{array}{c} \vdots\\ a_{{\scriptscriptstyle ij}}x_{{\scriptscriptstyle j}}\\ \vdots \end{array}=\begin{array}{c} \vdots\\ a_{{\scriptscriptstyle i1}}x_{{\scriptscriptstyle 1}}+\cdots+a_{{\scriptscriptstyle ij}}x_{{\scriptscriptstyle j}}+\cdots+a_{{\scriptscriptstyle in}}x_{{\scriptscriptstyle n}}\\ \vdots \end{array} \]

or simply

\[ f_{{\scriptscriptstyle i}}\left(\cdots,x_{{\scriptscriptstyle j}},\cdots\right)=f_{{\scriptscriptstyle i}}\left(x_{{\scriptscriptstyle j}}\right)=f_{{\scriptscriptstyle i}}\left(\boldsymbol{x}\right)=\boldsymbol{a}_{{\scriptscriptstyle i}}\cdot\boldsymbol{x}_{{\scriptscriptstyle j}}=\boldsymbol{a}_{{\scriptscriptstyle i}}^{\intercal}\boldsymbol{x}_{{\scriptscriptstyle j}}=a_{{\scriptscriptstyle ik}}x_{{\scriptscriptstyle kj}}=\cdots+a_{{\scriptscriptstyle ij}}x_{{\scriptscriptstyle j}}+\cdots \]

if scalar with complex as the field,

\[ f_{{\scriptscriptstyle i}}:\mathbb{C}^{\infty}\rightarrow\mathbb{C} \]

if scalar with a field,

\[ f_{{\scriptscriptstyle i}}:\mathbb{F}^{\infty}\rightarrow\mathbb{F} \]

or more abstract notation,

\[ f_{{\scriptscriptstyle i}}:F^{\infty}\rightarrow F \] if scalar with real as the field,

\[ f_{{\scriptscriptstyle i}}:\mathbb{R}^{\infty}\rightarrow\mathbb{R} \]

if finite,

\[ f_{{\scriptscriptstyle i}}\left(x_{{\scriptscriptstyle 1}},\cdots,x_{{\scriptscriptstyle j}},\cdots,x_{{\scriptscriptstyle n}}\right)=f_{{\scriptscriptstyle i}}\left(x_{{\scriptscriptstyle j}}\right)=f_{{\scriptscriptstyle i}}\left(\boldsymbol{x}\right)=\boldsymbol{a}_{{\scriptscriptstyle i}}\cdot\boldsymbol{x}_{{\scriptscriptstyle j}}=\boldsymbol{a}_{{\scriptscriptstyle i}}^{\intercal}\boldsymbol{x}_{{\scriptscriptstyle j}}=a_{{\scriptscriptstyle ij}}x_{{\scriptscriptstyle j}}=a_{{\scriptscriptstyle i1}}x_{{\scriptscriptstyle 1}}+\cdots+a_{{\scriptscriptstyle ij}}x_{{\scriptscriptstyle j}}+\cdots+a_{{\scriptscriptstyle in}}x_{{\scriptscriptstyle n}} \]

\[ \begin{aligned} f_{{\scriptscriptstyle i}}\left(x_{{\scriptscriptstyle 1}},x_{{\scriptscriptstyle 2}},\cdots,x_{{\scriptscriptstyle n}}\right)= & f_{{\scriptscriptstyle i}}\left(x_{{\scriptscriptstyle 1}},\cdots,x_{{\scriptscriptstyle j}},\cdots,x_{{\scriptscriptstyle n}}\right)=f_{{\scriptscriptstyle i}}\left(x_{{\scriptscriptstyle j}}\right)=f_{{\scriptscriptstyle i}}\left(\boldsymbol{x}\right)\\ = & \boldsymbol{a}_{{\scriptscriptstyle i}}\cdot\boldsymbol{x}_{{\scriptscriptstyle j}}=\boldsymbol{a}_{{\scriptscriptstyle i}}^{\intercal}\boldsymbol{x}_{{\scriptscriptstyle j}}=a_{{\scriptscriptstyle ij}}x_{{\scriptscriptstyle j}}=a_{{\scriptscriptstyle i1}}x_{{\scriptscriptstyle 1}}+\cdots+a_{{\scriptscriptstyle ij}}x_{{\scriptscriptstyle j}}+\cdots+a_{{\scriptscriptstyle in}}x_{{\scriptscriptstyle n}} \end{aligned} \]

if scalar with complex as the field,

\[ f_{{\scriptscriptstyle i}}:\mathbb{C}^{n}\rightarrow\mathbb{C} \]

if scalar with a field,

\[ f_{{\scriptscriptstyle i}}:\mathbb{F}^{n}\rightarrow\mathbb{F} \]

or more abstract notation,

\[ f_{{\scriptscriptstyle i}}:F^{n}\rightarrow F \] if scalar with real as the field,

\[ f_{{\scriptscriptstyle i}}:\mathbb{R}^{n}\rightarrow\mathbb{R} \]

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

\[ f_{{\scriptscriptstyle i}}:F^{n}\rightarrow F \]

\[ f=\left\{ f_{{\scriptscriptstyle i}}\middle|f_{{\scriptscriptstyle i}}:F^{\infty}\rightarrow F\right\} =\left\{ \begin{array}{c} \vdots\\ f_{{\scriptscriptstyle i}}\left(\cdots,x_{{\scriptscriptstyle j}},\cdots\right)=f_{{\scriptscriptstyle i}}\left(x_{{\scriptscriptstyle j}}\right)=f_{{\scriptscriptstyle i}}\left(\boldsymbol{x}\right)=\boldsymbol{a}_{{\scriptscriptstyle i}}\cdot\boldsymbol{x}=\boldsymbol{a}_{{\scriptscriptstyle i}}^{\intercal}\boldsymbol{x}=a_{{\scriptscriptstyle ij}}x_{{\scriptscriptstyle j}},\\ \vdots \end{array}\right\} \]

if finite,

\[ f=\left\{ f_{{\scriptscriptstyle i}}\middle|f_{{\scriptscriptstyle i}}:F^{n}\rightarrow F\right\} =\left\{ \begin{array}{c} \vdots\\ f_{{\scriptscriptstyle i}}\left(x_{{\scriptscriptstyle 1}},x_{{\scriptscriptstyle 2}},\cdots,x_{{\scriptscriptstyle n}}\right)=f_{{\scriptscriptstyle i}}\left(\boldsymbol{x}\right)=a_{{\scriptscriptstyle i1}}x_{{\scriptscriptstyle 1}}+\cdots+a_{{\scriptscriptstyle ij}}x_{{\scriptscriptstyle j}}+\cdots+a_{{\scriptscriptstyle in}}x_{{\scriptscriptstyle n}},\\ \vdots \end{array}\right\} \]

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

42.4 definition of linear functional

functionals generalized to general vector space

\[ f=\left\{ f_{{\scriptscriptstyle i}}\middle|f_{{\scriptscriptstyle i}}:V\rightarrow F\right\} \]

linear functionals generalized to general vector space is a linear transformation

\[ f=\left\{ f_{{\scriptscriptstyle i}}\middle|\begin{cases} f_{{\scriptscriptstyle i}}:V\rightarrow F\\ \forall\left\langle \boldsymbol{x},\boldsymbol{y}\right\rangle \in V^{2}\left[f_{{\scriptscriptstyle i}}\left(\boldsymbol{x}+\boldsymbol{y}\right)=f_{{\scriptscriptstyle i}}\left(\boldsymbol{x}\right)+f_{{\scriptscriptstyle i}}\left(\boldsymbol{y}\right)\right]\\ \forall\boldsymbol{x}\in V,c\in F\left[f_{{\scriptscriptstyle i}}\left(c\boldsymbol{x}\right)=cf_{{\scriptscriptstyle i}}\left(\boldsymbol{x}\right)\right] \end{cases}\right\} \]

\[ f=\left\{ f_{{\scriptscriptstyle i}}\middle|\begin{cases} f_{{\scriptscriptstyle i}}:V\rightarrow F\\ \forall\boldsymbol{x},\boldsymbol{y}\in V\left[f_{{\scriptscriptstyle i}}\left(\boldsymbol{x}+\boldsymbol{y}\right)=f_{{\scriptscriptstyle i}}\left(\boldsymbol{x}\right)+f_{{\scriptscriptstyle i}}\left(\boldsymbol{y}\right)\right]\\ \forall\boldsymbol{x}\in V,c\in F\left[f_{{\scriptscriptstyle i}}\left(c\boldsymbol{x}\right)=cf_{{\scriptscriptstyle i}}\left(\boldsymbol{x}\right)\right] \end{cases}\right\} \]

if $\begin{cases} F=\mathbb{C}\\ V=\mathbb{C}^{n} \end{cases}$,

\[ f=\left\{ f_{{\scriptscriptstyle i}}\middle|\begin{cases} f_{{\scriptscriptstyle i}}:\mathbb{C}^{n}\rightarrow\mathbb{C}\\ \forall\left\langle \boldsymbol{x},\boldsymbol{y}\right\rangle \in\left(\mathbb{C}^{n}\right)^{2}\left[f_{{\scriptscriptstyle i}}\left(\boldsymbol{x}+\boldsymbol{y}\right)=f_{{\scriptscriptstyle i}}\left(\boldsymbol{x}\right)+f_{{\scriptscriptstyle i}}\left(\boldsymbol{y}\right)\right]\\ \forall\boldsymbol{x}\in\mathbb{C}^{n},c\in\mathbb{C}\left[f_{{\scriptscriptstyle i}}\left(c\boldsymbol{x}\right)=cf_{{\scriptscriptstyle i}}\left(\boldsymbol{x}\right)\right] \end{cases}\right\} \]

\[ f=\left\{ f_{{\scriptscriptstyle i}}\middle|\begin{cases} f_{{\scriptscriptstyle i}}:\mathbb{C}^{n}\rightarrow\mathbb{C}\\ \forall\boldsymbol{x},\boldsymbol{y}\in\mathbb{C}^{n}\left[f_{{\scriptscriptstyle i}}\left(\boldsymbol{x}+\boldsymbol{y}\right)=f_{{\scriptscriptstyle i}}\left(\boldsymbol{x}\right)+f_{{\scriptscriptstyle i}}\left(\boldsymbol{y}\right)\right]\\ \forall\boldsymbol{x}\in\mathbb{C}^{n},c\in\mathbb{C}\left[f_{{\scriptscriptstyle i}}\left(c\boldsymbol{x}\right)=cf_{{\scriptscriptstyle i}}\left(\boldsymbol{x}\right)\right] \end{cases}\right\} \]

then

\[ f_{{\scriptscriptstyle i}}\left(\boldsymbol{x}\right)=a_{{\scriptscriptstyle i1}}x_{{\scriptscriptstyle 1}}+\cdots+a_{{\scriptscriptstyle ij}}x_{{\scriptscriptstyle j}}+\cdots+a_{{\scriptscriptstyle in}}x_{{\scriptscriptstyle n}} \]

satisfying

\[ \begin{aligned} f_{{\scriptscriptstyle i}}\left(\boldsymbol{x}+\boldsymbol{y}\right)= & a_{{\scriptscriptstyle i1}}\left(x+y\right)_{{\scriptscriptstyle 1}}+\cdots+a_{{\scriptscriptstyle ij}}\left(x+y\right)_{{\scriptscriptstyle j}}+\cdots+a_{{\scriptscriptstyle in}}\left(x+y\right)_{{\scriptscriptstyle n}}\\ = & a_{{\scriptscriptstyle i1}}\left(x_{{\scriptscriptstyle 1}}+y_{{\scriptscriptstyle 1}}\right)+\cdots+a_{{\scriptscriptstyle ij}}\left(x_{{\scriptscriptstyle j}}+y_{{\scriptscriptstyle j}}\right)+\cdots+a_{{\scriptscriptstyle in}}\left(x_{{\scriptscriptstyle n}}+y_{{\scriptscriptstyle n}}\right)\\ = & \left(a_{{\scriptscriptstyle i1}}x_{{\scriptscriptstyle 1}}+a_{{\scriptscriptstyle i1}}y_{{\scriptscriptstyle 1}}\right)+\cdots+\left(a_{{\scriptscriptstyle ij}}x_{{\scriptscriptstyle j}}+a_{{\scriptscriptstyle ij}}y_{{\scriptscriptstyle j}}\right)+\cdots+\left(a_{{\scriptscriptstyle in}}x_{{\scriptscriptstyle n}}+a_{{\scriptscriptstyle in}}y_{{\scriptscriptstyle n}}\right)\\ = & \left(a_{{\scriptscriptstyle i1}}x_{{\scriptscriptstyle 1}}+\cdots+a_{{\scriptscriptstyle ij}}x_{{\scriptscriptstyle j}}+\cdots+a_{{\scriptscriptstyle in}}x_{{\scriptscriptstyle n}}\right)+\left(a_{{\scriptscriptstyle i1}}y_{{\scriptscriptstyle 1}}+\cdots+a_{{\scriptscriptstyle ij}}y_{{\scriptscriptstyle j}}+\cdots+a_{{\scriptscriptstyle in}}y_{{\scriptscriptstyle n}}\right)\\ = & f_{{\scriptscriptstyle i}}\left(\boldsymbol{x}\right)+f_{{\scriptscriptstyle i}}\left(\boldsymbol{y}\right) \end{aligned} \]

\[ \begin{aligned} f_{{\scriptscriptstyle i}}\left(c\boldsymbol{x}\right)= & a_{{\scriptscriptstyle i1}}\left(cx\right)_{{\scriptscriptstyle 1}}+\cdots+a_{{\scriptscriptstyle ij}}\left(cx\right)_{{\scriptscriptstyle j}}+\cdots+a_{{\scriptscriptstyle in}}\left(cx\right)_{{\scriptscriptstyle n}}\\ = & a_{{\scriptscriptstyle i1}}\left(cx_{{\scriptscriptstyle 1}}\right)+\cdots+a_{{\scriptscriptstyle ij}}\left(cx_{{\scriptscriptstyle j}}\right)+\cdots+a_{{\scriptscriptstyle in}}\left(cx_{{\scriptscriptstyle n}}\right)\\ = & c\left(a_{{\scriptscriptstyle i1}}x_{{\scriptscriptstyle 1}}\right)+\cdots+c\left(a_{{\scriptscriptstyle ij}}x_{{\scriptscriptstyle j}}\right)+\cdots+c\left(a_{{\scriptscriptstyle in}}x_{{\scriptscriptstyle n}}\right)\\ = & c\left(a_{{\scriptscriptstyle i1}}x_{{\scriptscriptstyle 1}}+\cdots+a_{{\scriptscriptstyle ij}}x_{{\scriptscriptstyle j}}+\cdots+a_{{\scriptscriptstyle in}}x_{{\scriptscriptstyle n}}\right)\\ = & cf_{{\scriptscriptstyle i}}\left(\boldsymbol{x}\right) \end{aligned} \]

different functional has different $a_{\scriptscriptstyle{ij}}$

let

\[ a_{{\scriptscriptstyle ij}}=f_{{\scriptscriptstyle i}}\left(\boldsymbol{e}_{{\scriptscriptstyle j}}\right),\boldsymbol{e}_{{\scriptscriptstyle j}}=\left\langle \overset{j-1}{\overbrace{0,\cdots,0}},1,0,\cdots,0\right\rangle =\begin{pmatrix}0 & \cdots & 0 & 1 & 0 & \cdots & 0\end{pmatrix}^{\intercal}=\begin{pmatrix}0\\ \vdots\\ 0\\ 1\\ 0\\ \vdots\\ 0 \end{pmatrix}=\begin{pmatrix}0\\ \vdots\\ 0\\ 1\\ 0\\ \vdots\\ 0 \end{pmatrix}_{n\times1} \]

if $f_{{\scriptscriptstyle i}}$ is a linear functional, then

\[ \begin{aligned} f_{{\scriptscriptstyle i}}\left(\boldsymbol{x}\right)= & f_{{\scriptscriptstyle i}}\left(x_{{\scriptscriptstyle 1}}\boldsymbol{e}_{{\scriptscriptstyle 1}}+\cdots+x_{{\scriptscriptstyle j}}\boldsymbol{e}_{{\scriptscriptstyle j}}+\cdots+x_{{\scriptscriptstyle n}}\boldsymbol{e}_{{\scriptscriptstyle n}}\right)\\ = & f_{{\scriptscriptstyle i}}\left(x_{{\scriptscriptstyle 1}}\boldsymbol{e}_{{\scriptscriptstyle 1}}\right)+\cdots+f_{{\scriptscriptstyle i}}\left(x_{{\scriptscriptstyle j}}\boldsymbol{e}_{{\scriptscriptstyle j}}\right)+\cdots+f_{{\scriptscriptstyle i}}\left(x_{{\scriptscriptstyle n}}\boldsymbol{e}_{{\scriptscriptstyle n}}\right)\\ = & x_{{\scriptscriptstyle 1}}f_{{\scriptscriptstyle i}}\left(\boldsymbol{e}_{{\scriptscriptstyle 1}}\right)+\cdots+x_{{\scriptscriptstyle j}}f_{{\scriptscriptstyle i}}\left(\boldsymbol{e}_{{\scriptscriptstyle j}}\right)+\cdots+x_{{\scriptscriptstyle n}}f_{{\scriptscriptstyle i}}\left(\boldsymbol{e}_{{\scriptscriptstyle n}}\right)\\ = & x_{{\scriptscriptstyle 1}}a_{{\scriptscriptstyle i1}}+\cdots+x_{{\scriptscriptstyle j}}a_{{\scriptscriptstyle ij}}+\cdots+x_{{\scriptscriptstyle n}}a_{{\scriptscriptstyle in}}\\ = & a_{{\scriptscriptstyle i1}}x_{{\scriptscriptstyle 1}}+\cdots+a_{{\scriptscriptstyle ij}}x_{{\scriptscriptstyle j}}+\cdots+a_{{\scriptscriptstyle in}}x_{{\scriptscriptstyle n}}\\ = & f_{{\scriptscriptstyle i}}\left(\boldsymbol{x}\right) \end{aligned} \]

42.5 set of all linear transformations is a vector space

https://ccjou.wordpress.com/2011/04/08/線性變換集合構成向量空間/

vector space^[43]

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

\[ T:V\rightarrow W \Leftrightarrow \forall\boldsymbol{v}\in V,\exists!\boldsymbol{w}\in W\left[\boldsymbol{w}=T\left(\boldsymbol{v}\right)\right] \]

\[ \begin{aligned} & \begin{cases} V,W\text{ are vector spaces}\\ T:V\rightarrow W\\ \begin{cases} \forall\boldsymbol{u},\boldsymbol{v}\in V\left[T\left(\boldsymbol{u}+\boldsymbol{v}\right)=T\left(\boldsymbol{u}\right)+T\left(\boldsymbol{v}\right)\right]\\ \forall\boldsymbol{v}\in V,c\in F\left[T\left(c\boldsymbol{v}\right)=cT\left(\boldsymbol{v}\right)\right] \end{cases} & \text{linearity} \end{cases}\\ \Leftrightarrow & T\text{ is a linear tranformation} \end{aligned} \]

\[ \begin{cases} V,W\text{ are vector spaces, both over }F\\ T,U:V\rightarrow W\begin{cases} T:V\rightarrow W\\ U:V\rightarrow W \end{cases}\\ T,U\text{ are both linear tranformations}\\ \boldsymbol{v}\in V\\ c\in F \end{cases} \]

There is still linearity over linear transformations

$\left(va\right)$

\[ \begin{aligned} \left(T+U\right)\left(\boldsymbol{u}+\boldsymbol{v}\right)= & T\left(\boldsymbol{u}+\boldsymbol{v}\right)+U\left(\boldsymbol{u}+\boldsymbol{v}\right)\\ = & \left[T\left(\boldsymbol{u}\right)+T\left(\boldsymbol{v}\right)\right]+\left[U\left(\boldsymbol{u}\right)+U\left(\boldsymbol{v}\right)\right]\\ = & \left[T\left(\boldsymbol{u}\right)+U\left(\boldsymbol{u}\right)\right]+\left[T\left(\boldsymbol{v}\right)+U\left(\boldsymbol{v}\right)\right]\\ = & \left(T+U\right)\left(\boldsymbol{u}\right)+\left(T+U\right)\left(\boldsymbol{v}\right) \end{aligned} \]

$\left(sm\right)$

\[ \begin{aligned} \left(T+U\right)\left(c\boldsymbol{v}\right)= & T\left(c\boldsymbol{v}\right)+U\left(c\boldsymbol{v}\right)\\ = & cT\left(\boldsymbol{v}\right)+cU\left(\boldsymbol{v}\right)\\ = & c\left[T\left(\boldsymbol{v}\right)+U\left(\boldsymbol{v}\right)\right]\\ = & c\left(T+U\right)\left(\boldsymbol{v}\right) \end{aligned} \]

so we can define

\[ \begin{cases} \left(T+U\right)\left(\boldsymbol{v}\right)=T\left(\boldsymbol{v}\right)+U\left(\boldsymbol{v}\right) & \text{linear transformation addition}\\ \left(cT\right)\left(\boldsymbol{v}\right)=cT\left(\boldsymbol{v}\right) & \text{scalar linear transformation multiplication} \end{cases} \]

the set of all linear tranformations is a vector space

$\mathcal{T}$ is the set of all linear tranformations

\[ \begin{aligned} & \begin{cases} F & \left(f\right)F\text{ is a field}\\ \mathcal{T}\ne\emptyset & \left(ne\right)\text{nonempty set}\\ +:\mathcal{T}\times\mathcal{T}=\mathcal{T}^{2}\overset{+}{\rightarrow}\mathcal{T}\Leftrightarrow\forall T,U\in\mathcal{T},\exists S\in\mathcal{T}\left[S=T+U\right] & \left(va\right)\text{vector addition}\\ \cdot:F\times\mathcal{T}\overset{\cdot}{\rightarrow}\mathcal{T}\Leftrightarrow\forall c\in F,\forall T\in\mathcal{T},\exists U\in\mathcal{T}\left[U=cT=c\cdot T\right] & \left(sm\right)\text{scalar multiplication}\\ \begin{cases} \forall S,T,U\in\mathcal{T}\left[S+\left(T+U\right)=\left(S+T\right)+U\right] & \left(a\right)\\ \forall T,U\in\mathcal{T}\left[T+U=U+T\right] & \left(c\right)\\ \exists!O\in\mathcal{T},\forall T\in\mathcal{T}\left[O+T=T\right] & \left(e\right)\\ \forall T\in\mathcal{T},\exists!-T\in\mathcal{T}\left[\left(-T\right)+T=O\right] & \left(i\right) \end{cases} & \left(va\right)\text{vector addition axioms}\\ \begin{cases} \forall b,c\in F,T\in\mathcal{T}\left[b\left(cT\right)=\left(bc\right)T\right] & \left(a\right)\\ \exists!1\in F,\forall T\in\mathcal{T}\left[1T=T\right] & \left(e\right)\\ \forall c\in F,T,U\in\mathcal{T}\left[c\left(T+U\right)=cT+cU\right] & \left(dv\right)\\ \forall b,c\in F,T\in\mathcal{T}\left[\left(b+c\right)T=bT+cT\right] & \left(ds\right) \end{cases} & \left(sm\right)\text{scalar multiplication axioms} \end{cases}\\ \Leftrightarrow & \mathcal{T}=\mathcal{T}\left(F,+,\cdot\right)=\left(\mathcal{T},F,+,\cdot\right)\text{ is a vector space over the field }F\\ \Leftrightarrow & \mathcal{T}\text{ is a vector space} \end{aligned} \]

Selected proofs of 8 vector space axioms due to some trivial field and vector space properties:

$\left(va\right)\left(a\right)$

\[ \begin{aligned} \left(S+\left(T+U\right)\right)\left(\boldsymbol{v}\right)= & S\left(\boldsymbol{v}\right)+\left(T+U\right)\left(\boldsymbol{v}\right)\\ = & S\left(\boldsymbol{v}\right)+T\left(\boldsymbol{v}\right)+U\left(\boldsymbol{v}\right)\\ = & \left(S+T\right)\left(\boldsymbol{v}\right)+U\left(\boldsymbol{v}\right)\\ = & \left(\left(S+T\right)+U\right)\left(\boldsymbol{v}\right) \end{aligned} \]

$\left(va\right)\left(c\right)$

\[ \begin{aligned} \left(T+U\right)\left(\boldsymbol{v}\right)= & T\left(\boldsymbol{v}\right)+U\left(\boldsymbol{v}\right)\\ = & U\left(\boldsymbol{v}\right)+T\left(\boldsymbol{v}\right)\\ = & \left(U+T\right)\left(\boldsymbol{v}\right) \end{aligned} \]

$\left(va\right)\left(e\right)$

\[ O\left(\boldsymbol{v}\right)=0\boldsymbol{w}\in W \]

\[ \begin{aligned} \left(O+T\right)\left(\boldsymbol{v}\right)= & O\left(\boldsymbol{v}\right)+T\left(\boldsymbol{v}\right)\\ = & 0\boldsymbol{w}+T\left(\boldsymbol{v}\right)\\ = & T\left(\boldsymbol{v}\right) \end{aligned} \]

\[ O_{1}\left(\boldsymbol{v}\right)-O_{2}\left(\boldsymbol{v}\right)=0\boldsymbol{w}-0\boldsymbol{w}=0\boldsymbol{w}\Rightarrow O_{1}\left(\boldsymbol{v}\right)=O_{2}\left(\boldsymbol{v}\right) \]

$\left(sm\right)\left(dv\right)$

\[ \begin{aligned} \left(c\left(T+U\right)\right)\left(\boldsymbol{v}\right)= & c\left(T+U\right)\left(\boldsymbol{v}\right)\\ = & c\left[T\left(\boldsymbol{v}\right)+U\left(\boldsymbol{v}\right)\right]\\ = & cT\left(\boldsymbol{v}\right)+cU\left(\boldsymbol{v}\right)\\ = & \left(cT+cU\right)\left(\boldsymbol{v}\right) \end{aligned} \]

\[ \text{The set of all linear tranformations } \mathcal{T} \text{ is a vector space.} \]

\[ \ \tag*{$\Box$} \]

42.6 definition of dual space

\[ \begin{aligned} & V^{*}=L\left(V,F\right)\\ = & f=\left\{ f_{{\scriptscriptstyle i}}\middle|\begin{cases} f_{{\scriptscriptstyle i}}:V\rightarrow F & \text{functional mapping vector to field scalar}\\ \begin{cases} \forall\boldsymbol{x},\boldsymbol{y}\in V\left[f_{{\scriptscriptstyle i}}\left(\boldsymbol{x}+\boldsymbol{y}\right)=f_{{\scriptscriptstyle i}}\left(\boldsymbol{x}\right)+f_{{\scriptscriptstyle i}}\left(\boldsymbol{y}\right)\right]\\ \forall\boldsymbol{x}\in V,c\in F\left[f_{{\scriptscriptstyle i}}\left(c\boldsymbol{x}\right)=cf_{{\scriptscriptstyle i}}\left(\boldsymbol{x}\right)\right] \end{cases} & \left(L\right)\text{linearity} \end{cases}\right\} \\ \Leftrightarrow & V^{*}\text{ is a dual space, a set of linear functionals }f_{{\scriptscriptstyle i}}\text{ mapping vectors in the vector space }V\text{ to scalars in the field }F \end{aligned} \]

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

42.7 double dual

double dual = second dual

https://ccjou.wordpress.com/2014/04/10/雙重對偶/

https://www.zhihu.com/question/444079322/answer/1749490720

Hough Transform

https://www.cnblogs.com/php-rearch/p/6760683.html