next up previous contents
Next: 8 鐃緒申鐃緒申鐃術器申 Up: 2 鐃緒申鐃緒申鐃緒申鐃緒申 Previous: 6 鐃緒申鐃緒申鐃緒申鐃緒申鐃塾刻申鐃緒申表鐃緒申 &nbsp Contents

7 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申表鐃緒申鐃緒申鐃緒申

鐃緒申鐃� 2.31 (鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申表鐃緒申鐃緒申鐃緒申)   鐃駿ワ申鐃夙ワ申鐃緒申鐃� $ U$ 鐃塾器申鐃緒申鐃� $ \{\vec{u}_{1},\vec{u}_{2},\cdots,\vec{u}_{n}\}$ 鐃夙わ申鐃緒申 鐃駿ワ申鐃夙ワ申鐃緒申鐃� $ V$ 鐃塾器申鐃緒申鐃� $ \{\vec{v}_{1},\vec{v}_{2},\cdots,\vec{v}_{m}\}$ 鐃夙わ申鐃暑． 鐃緒申鐃塾とわ申鐃緒申 鐃緒申鐃緒申鐃緒申鐃緒申 $ f:U\to V$ 鐃緒申

$\displaystyle \begin{bmatrix}f(\vec{u}_{1}) & f(\vec{u}_{2}) & \cdots & f(\vec{... ...v}_{1} & \vec{v}_{2} & \cdots & \vec{v}_{m} \end{bmatrix} A, \qquad A:m\times n$    

鐃緒申澆鐃緒申鐃緒申箸鐃緒申鐃� 鐃緒申鐃緒申 $ A$ 鐃緒申 $ U$ 鐃塾器申鐃緒申 $ \{\vec{u}_{1},\vec{u}_{2},\cdots,\vec{u}_{n}\}$ 鐃緒申 $ V$ 鐃塾器申鐃緒申 $ \{\vec{v}_{1},\vec{v}_{2},\cdots,\vec{v}_{m}\}$ 鐃祝関わ申鐃緒申 表鐃緒申鐃緒申鐃緒申鐃夙わ申鐃緒申鐃緒申

鐃緒申鐃緒申 2.32 (表鐃緒申鐃緒申鐃緒申)   鐃緒申鐃緒申鐃宿緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申任蓮鐃� 鐃緒申鐃緒申鐃緒申鐃緒申 $ f:\mathbb{R}^{n}\to\mathbb{R}^{m}$ 鐃緒申 $ \vec{y}=f(\vec{x})=\tilde{A}\vec{x}$ 鐃緒申表鐃緒申鐃緒申鐃夙わ申 $ \tilde{A}$ 鐃緒申表鐃緒申鐃緒申鐃緒申噺討屐鐃緒申箸鐃緒申鐃緒申鐃塾でわ申鐃獣わ申鐃緒申 鐃緒申鐃塾刻申鐃緒申 $ \tilde{A}$ 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃祝わ申鐃宿緒申鐃緒申鐃緒申鐃� $ A$ 鐃夙わ申 鐃緒申鐃緒申鐃宿醐申鐃緒申鐃緒申砲箸鐃夙わ申鐃緒申鐃竣わ申鐃暑．

鐃緒申鐃緒申鐃緒申鐃緒申 $ f:\mathbb{R}^{n}\to\mathbb{R}^{m}$ 鐃緒申佑鐃緒申襦� $ \mathbb{R}^{n}$ 鐃緒申標鐃緒申鐃緒申鐃緒申 $ \vec{e}_{1},\vec{e}_{2},\cdots,\vec{e}_{n}$ 鐃夙わ申鐃緒申 $ \mathbb{R}^{m}$ 鐃緒申標鐃緒申鐃緒申鐃緒申 $ \vec{e}'_{1},\vec{e}'_{2},\cdots,\vec{e}'_{m}$ 鐃夙わ申鐃緒申鐃緒申 鐃緒申鐃塾とわ申鐃緒申鐃緒申鐃緒申鐃宿緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申

$\displaystyle \begin{bmatrix}f(\vec{e}_{1}) & f(\vec{e}_{2}) & \cdots & f(\vec{... ...matrix}\vec{e}'_{1} & \vec{e}'_{2} & \cdots & \vec{e}'_{m} \end{bmatrix} A=EA=A$    

鐃緒申鐃緒申鐃緒申立鐃縦￥申 $ \mathbb{R}^{n}$ 鐃緒申任鐃春のベワ申鐃夙ワ申

$\displaystyle \vec{x}= x_{1}\vec{e}_{1}+ x_{2}\vec{e}_{2}+ \cdots+ x_{n}\vec{e}_{n}$    

鐃緒申 $ f$ 鐃叔写わ申鐃緒申

$\displaystyle \vec{y}$ $\displaystyle =f(\vec{x})= f(x_{1}\vec{e}_{1}+x_{2}\vec{e}_{2}+\cdots+x_{n}\vec{e}_{n})$    
  $\displaystyle = x_{1}f(\vec{e}_{1})+x_{2}f(\vec{e}_{2})+\cdots+x_{n}f(\vec{e}_{n})$    
  $\displaystyle = \begin{bmatrix}f(\vec{e}_{1}) & f(\vec{e}_{2}) & \cdots & f(\ve... ...}) \end{bmatrix} \begin{bmatrix}x_{1} \\ x_{2} \\ \vdots \\ x_{n} \end{bmatrix}$    
  $\displaystyle =A\vec{x}=\tilde{A}\vec{x}$    

鐃夙なる．$ A$ 鐃緒申 $ \tilde{A}$ 鐃熟逸申鐃竣わ申鐃暑．

鐃緒申 2.33 (表鐃緒申鐃緒申鐃緒申龍鐃緒申鐃緒申鐃�)   鐃緒申鐃緒申鐃緒申鐃緒申 $ f:\mathbb{R}^{2}\to\mathbb{R}^{3}$;

$\displaystyle \vec{y}=f(\vec{x})=A\vec{x}= \begin{bmatrix}2 & 1 \\ 1 & 0 \\ 4 & 3 \end{bmatrix} \vec{x}$    

鐃緒申佑鐃緒申襦� 標鐃緒申鐃緒申鐃緒申

  $\displaystyle \mathbb{R}^{2}\ni \vec{e}_{1}= \begin{bmatrix}1 \\ 0 \end{bmatrix... ...\\ 1 \\ 0 \end{bmatrix}, \vec{e}'_{3}= \begin{bmatrix}0 \\ 0 \\ 1 \end{bmatrix}$    

鐃夙わ申鐃暑． 鐃緒申鐃塾とわ申鐃緒申

  $\displaystyle f(\vec{e}_{1})=A\vec{e}_{1}= \begin{bmatrix}2 & 1 \\ 1 & 0 \\ 4 &... ...n{bmatrix}2 \\ 1 \\ 4 \end{bmatrix} = 2\vec{e}'_{1}+\vec{e}'_{2}+4\vec{e}'_{3},$    
  $\displaystyle f(\vec{e}_{3})=A\vec{e}_{2}= \begin{bmatrix}2 & 1 \\ 1 & 0 \\ 4 &... ...bmatrix} = \begin{bmatrix}1 \\ 0 \\ 3 \end{bmatrix} = \vec{e}'_{1}+\vec{e}'_{3}$    

鐃叔わ申鐃暑．鐃緒申辰鐃�

$\displaystyle \begin{bmatrix}f(\vec{e}_{1}) & f(\vec{e}_{2}) \end{bmatrix} = \b... ...ix} = \begin{bmatrix}\vec{e}'_{1} & \vec{e}'_{2} & \vec{e}'_{3} \end{bmatrix} A$    

鐃緒申鐃緒申立鐃緒申鐃暑． 鐃緒申鐃緒申鐃緒申鐃緒申 $ f$ 鐃緒申標鐃緒申鐃緒申鐃祝わ申鐃緒申鐃緒申表鐃緒申鐃緒申鐃緒申鐃� $ A$ 鐃叔わ申鐃暑．

鐃緒申鐃緒申 2.34 (表鐃緒申鐃緒申鐃緒申)   鐃緒申鐃緒申鐃緒申鐃緒申 $ \vec{y}=f(\vec{x})=A\vec{x}$ 鐃緒申標鐃緒申鐃緒申鐃緒申 鐃緒申鐃緒申鐃緒申表鐃緒申鐃緒申鐃緒申鐃� $ A$ 鐃叔わ申鐃暑．

鐃緒申鐃緒申 2.35 (鐃緒申鐃緒申鐃緒申蟯刻申鐃緒申鐃緒申箸鐃緒申鐃宿緒申鐃緒申鐃緒申鐃�)   鐃緒申鐃緒申鐃緒申鐃緒申 $ f:U\to V$ 鐃緒申 $ U$ 鐃塾器申鐃緒申 $ \{\vec{u}_{1}$, $ \vec{u}_{2}$, $ \cdots$, $ \vec{u}_{n}\}$ 鐃緒申 $ V$ 鐃塾器申鐃緒申 $ \{\vec{v}_{1}$, $ \vec{v}_{2}$, $ \cdots$, $ \vec{v}_{m}\}$ 鐃緒申 鐃舜わ申鐃緒申表鐃緒申鐃緒申鐃緒申鐃� $ A$ 鐃夙わ申鐃暑． 鐃緒申鐃淑わ申鐃緒申鐃�

$\displaystyle \begin{bmatrix}f(\vec{u}_{1}) & f(\vec{u}_{2}) & \cdots & f(\vec{... ...\begin{bmatrix}\vec{v}_{1} & \vec{v}_{2} & \cdots & \vec{v}_{m} \end{bmatrix} A$    

鐃夙わ申鐃暑． $ U$ 鐃塾器申鐃緒申 $ \{\vec{u}'_{1}$, $ \vec{u}'_{2}$, $ \cdots$, $ \vec{u}'_{n}\}$ 鐃緒申 $ V$ 鐃塾器申鐃緒申 $ \{\vec{v}'_{1}$, $ \vec{v}'_{2}$, $ \cdots$, $ \vec{v}'_{m}\}$ 鐃緒申 鐃舜わ申鐃緒申表鐃緒申鐃緒申鐃緒申鐃� $ B$ 鐃夙わ申鐃暑． 鐃緒申鐃淑わ申鐃緒申鐃�

$\displaystyle \begin{bmatrix}f(\vec{u}'_{1}) & f(\vec{u}'_{2}) & \cdots & f(\ve... ...gin{bmatrix}\vec{v}'_{1} & \vec{v}'_{2} & \cdots & \vec{v}'_{m} \end{bmatrix} B$    

鐃夙わ申鐃暑． 鐃緒申鐃塾とわ申

$\displaystyle B=Q^{-1}AP$    

鐃緒申鐃緒申鐃緒申立鐃縦￥申 鐃緒申鐃緒申鐃緒申 $ P$, $ Q$ 鐃熟器申鐃緒申鐃緒申儡鐃緒申鐃緒申鐃叔わ申鐃所，

$\displaystyle \begin{bmatrix}\vec{u}'_{1} & \vec{u}'_{2} & \cdots & \vec{u}'_{n} \end{bmatrix}$ $\displaystyle = \begin{bmatrix}\vec{u}_{1} & \vec{u}_{2} & \cdots & \vec{u}_{n} \end{bmatrix}P,$    
$\displaystyle \begin{bmatrix}\vec{v}'_{1} & \vec{v}'_{2} & \cdots & \vec{v}'_{m} \end{bmatrix}$ $\displaystyle = \begin{bmatrix}\vec{v}_{1} & \vec{v}_{2} & \cdots & \vec{v}_{m} \end{bmatrix}Q$    

鐃叔わ申鐃暑．


鐃淑常申鐃緒申鐃緒申    

  $\displaystyle \begin{bmatrix}f(\vec{u}'_1) & f(\vec{u}'_2) & \cdots & f(\vec{u}... ...(\sum_{k}p_{k2}\vec{u}'_k) & \cdots & f(\sum_{k}p_{kn}\vec{u}'_k) \end{bmatrix}$    
  $\displaystyle = \begin{bmatrix}\sum_{k}p_{k1}f(\vec{u}'_k) & \sum_{k}p_{k2}f(\vec{u}'_k) & \cdots & \sum_{k}p_{kn}f(\vec{u}'_k) \end{bmatrix}$    
  $\displaystyle = \begin{bmatrix}f(\vec{u}_1) & f(\vec{u}_2) & \cdots & f(\vec{u}_n) \end{bmatrix} P$    
  $\displaystyle = \begin{bmatrix}\vec{v}_1 & \vec{v}_2 & \cdots & \vec{v}_m \end{bmatrix} AP$    

鐃殉わ申鐃緒申

  $\displaystyle \begin{bmatrix}f(\vec{u}'_1) & f(\vec{u}'_2) & \cdots & f(\vec{u}... ... B = \begin{bmatrix}\vec{v}_1 & \vec{v}_2 & \cdots & \vec{v}_m \end{bmatrix} QB$    

鐃緒申辰董鐃�

  $\displaystyle AP=QB$    
  $\displaystyle B=Q^{-1}AP$    

鐃緒申鐃緒申鐃緒申立鐃縦￥申

鐃緒申 2.36 (鐃緒申鐃緒申鐃緒申蟯刻申鐃緒申鐃緒申箸鐃緒申鐃宿緒申鐃緒申鐃緒申鐃塾駈申鐃緒申鐃緒申)   鐃緒申鐃緒申鐃緒申鐃緒申 $ f:\mathbb{R}^{3}\to\mathbb{R}^{2}$;

$\displaystyle \vec{y}=f(\vec{x})=A\vec{x}= \begin{bmatrix}2 & 4 & 1 \\ 1 & -1 & 0 \end{bmatrix} \vec{x}$    

鐃緒申佑鐃緒申襦� $ \mathbb{R}^{3}$ 鐃塾器申鐃緒申鐃�

$\displaystyle \left\{ \begin{bmatrix}2 \\ 0 \\ 3 \end{bmatrix}, \begin{bmatrix}... ...ix}1 \\ 0 \\ 1 \end{bmatrix} \right\} = \{\vec{u}_{1},\vec{u}_{2},\vec{u}_{3}\}$    

鐃夙わ申鐃緒申 $ \mathbb{R}^{2}$ 鐃塾器申鐃緒申鐃�

$\displaystyle \left\{ \begin{bmatrix}1 \\ 1 \end{bmatrix}, \begin{bmatrix}2 \\ 3 \end{bmatrix} \right\} = \{\vec{v}_{1},\vec{v}_{2}\}$    

鐃夙わ申鐃暑． $ f$ 鐃塾器申鐃緒申 $ \{\vec{u}_{1},\vec{u}_{2},\vec{u}_{3}\}$ 鐃緒申 鐃緒申鐃緒申 $ \{\vec{v}_{1},\vec{v}_{2}\}$ 鐃祝関わ申鐃緒申 表鐃緒申鐃緒申鐃緒申 $ B$ 鐃緒申鐃緒申襦� 鐃淑わ申鐃緒申鐃緒申鐃�

$\displaystyle \begin{bmatrix}f(\vec{u}_{1}) & f(\vec{u}_{2}) & f(\vec{u}_{3}) \end{bmatrix} = \begin{bmatrix}\vec{v}_{1} & \vec{v}_{2} \end{bmatrix}B$    

鐃緒申澆鐃緒申鐃� $ B$ 鐃緒申鐃緒申襦� 鐃殉わ申鐃緒申 $ \mathbb{R}^{3}$ 鐃緒申 標鐃緒申鐃緒申鐃� $ \{\vec{e}_{1},\vec{e}_{2},\vec{e}_{3}\}$ 鐃緒申 $ \mathbb{R}^{2}$ 鐃緒申 標鐃緒申鐃緒申鐃� $ \{\vec{e}'_{1},\vec{e}'_{2}\}$ 鐃緒申佑鐃緒申襦� 鐃緒申鐃塾とわ申鐃緒申標鐃緒申鐃緒申鐃祝わ申鐃緒申鐃緒申 $ f$ 鐃緒申表鐃緒申鐃緒申鐃緒申鐃� $ A$ 鐃叔わ申鐃暑か鐃初，

$\displaystyle \begin{bmatrix}f(\vec{e}_{1}) & f(\vec{e}_{2}) & f(\vec{e}_{3}) \end{bmatrix} = \begin{bmatrix}\vec{e}'_{1} & \vec{e}'_{2} \end{bmatrix}A =A$    

鐃緒申鐃緒申鐃緒申立鐃縦￥申 鐃縦わ申鐃祝￥申鐃緒申鐃緒申鐃緒申儡鐃緒申鐃緒申鐃緒申

  $\displaystyle \begin{bmatrix}\vec{u}_{1} & \vec{u}_{2} & \vec{u}_{3} \end{bmatrix} = \begin{bmatrix}\vec{e}_{1} & \vec{e}_{2} & \vec{e}_{3} \end{bmatrix}P=EP=P,$    
  $\displaystyle \begin{bmatrix}\vec{v}_{1} & \vec{v}_{2} \end{bmatrix} = \begin{bmatrix}\vec{e}'_{1} & \vec{e}'_{2} \end{bmatrix}Q=EQ=Q$    

鐃叔わ申鐃暑． 鐃淑常申鐃所，

  $\displaystyle \begin{bmatrix}f(\vec{u}_{1}) & f(\vec{u}_{2}) & f(\vec{u}_{3}) \... ...end{bmatrix}P = \begin{bmatrix}\vec{e}'_{1} & \vec{e}'_{2} \end{bmatrix}AP =AP,$    
  $\displaystyle \begin{bmatrix}f(\vec{u}_{1}) & f(\vec{u}_{2}) & f(\vec{u}_{3}) \... ...\end{bmatrix}B = \begin{bmatrix}\vec{e}'_{1} & \vec{e}'_{2} \end{bmatrix}QB =QB$    

鐃緒申鐃緒申鐃緒申立鐃縦￥申 鐃緒申辰董鐃�

$\displaystyle B=Q^{-1}AP= \begin{bmatrix}1 & 2 \\ 1 & 3 \end{bmatrix}^{-1} \beg... ... 1 & 1 \end{bmatrix} = \begin{bmatrix}17 & 17 & 7 \\ -5 & -6 & -2 \end{bmatrix}$    

鐃緒申鐃緒申鐃暑．

next up previous contents
Next: 8 鐃緒申鐃緒申鐃術器申 Up: 2 鐃緒申鐃緒申鐃緒申鐃緒申 Previous: 6 鐃緒申鐃緒申鐃緒申鐃緒申鐃塾刻申鐃緒申表鐃緒申 &nbsp Contents

Kondo Koichi
Created at 2004/12/13