lectures:一次変換
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lectures:一次変換 [2019/05/23 22:32] – kimi | lectures:一次変換 [2019/05/23 23:47] – [証明] kimi | ||
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====== 一次変換 ====== | ====== 一次変換 ====== | ||
+ | ===== 定義 ===== | ||
$$ | $$ | ||
\left\{ | \left\{ | ||
\begin{array}{ll} | \begin{array}{ll} | ||
\vec{x}, \vec{y} \in \mathbb{R}^n, | \vec{x}, \vec{y} \in \mathbb{R}^n, | ||
- | | + | |
\end{array} | \end{array} | ||
- | \right. \Leftrightarrow | + | \right. \Leftrightarrow |
$$ | $$ | ||
+ | ==== 証明 ==== | ||
+ | |||
+ | $$\vec{y}=f(\vec{x})$$ | ||
+ | |||
+ | $$\vec{x}=x_1\vec{e}_1+x_2\vec{e}_2+\cdots$$ | ||
+ | |||
+ | $$\vec{y}=y_1\vec{e}_1+y_2\vec{e}_2+\cdots$$ | ||
$$ | $$ | ||
- | \vec{x}\in R\mathbb{R}\left\{ | + | \begin{array}{rcl} |
- | | + | f(\vec{x})& |
- | | + | &=&f(x_1\vec{e}_1)+f(x_2\vec{e}_2)+\cdots |
- | | + | &=&x_1f(\vec{e}_1)+x_2f(\vec{e}_2)+\cdots |
- | \end{array} | + | \end{array} |
- | \right. \Leftrightarrow | + | |
$$ | $$ | ||
+ | |||
+ | $$ | ||
+ | \begin{array}{rcl} | ||
+ | f(\vec{e}_1)& | ||
+ | f(\vec{e}_2)& | ||
+ | & | ||
+ | \end{array} | ||
+ | $$ | ||
+ | |||
+ | $$ | ||
+ | \begin{array}{rcl} | ||
+ | f(\vec{x})& | ||
+ | & | ||
+ | & & | ||
+ | & & | ||
+ | & | ||
+ | & & | ||
+ | & & | ||
+ | \end{array} | ||
+ | $$ | ||
+ | |||
+ | $$ | ||
+ | \begin{array}{rcl} | ||
+ | y_1\vec{e}_1+y_2\vec{e}_2+\cdots & | ||
+ | & & | ||
+ | & & | ||
+ | \end{array} | ||
+ | $$ | ||
+ | |||
+ | $$ | ||
+ | \left\{\begin{array}{rcl} | ||
+ | y_1& | ||
+ | y_2& | ||
+ | & | ||
+ | \end{array} | ||
+ | \right. | ||
+ | $$ | ||
+ | |||
+ | $$ | ||
+ | \left[\begin{array}{c} | ||
+ | y_1\\ | ||
+ | y_2\\ | ||
+ | \vdots\\ | ||
+ | \end{array} | ||
+ | \right] | ||
+ | =\left[\begin{array}{ccc} | ||
+ | a_{11}& | ||
+ | a_{21}& | ||
+ | \vdots & & \\ | ||
+ | \end{array} | ||
+ | \right] | ||
+ | \left[\begin{array}{c} | ||
+ | x_{1}\\ | ||
+ | x_{2}\\ | ||
+ | \vdots\\ | ||
+ | \end{array} | ||
+ | \right] | ||
+ | $$ | ||
+ |
lectures/一次変換.txt · 最終更新: 2022/08/23 13:34 by 127.0.0.1