ベクトルの1次結合の計算

$3$\displaystyle{\hspace{1}{x}_{1} $ \begin{array} \hspace{1}{a}_{ 11 } & \vspace{6}\\ \hspace{1}{a}_{ 21 } & \vspace{6}\\ \vdots & \vspace{6}\\ \hspace{1}{a}_{ n1 } & \vspace{6}\\ \end{array} $ +\hspace{1}{x}_{2} $ \begin{array} \hspace{1}{a}_{ 12 } & \vspace{6}\\ \hspace{1}{a}_{ 22 } & \vspace{6}\\ \vdots & \vspace{6}\\ \hspace{1}{a}_{ n2 } & \vspace{6}\\ \end{array} $ +\cdots +\hspace{1}{x}_{m} $ \begin{array} \hspace{1}{a}_{ 1m } & \vspace{6}\\ \hspace{1}{a}_{ 2m } & \vspace{6}\\ \vdots & \vspace{6}\\ \hspace{1}{a}_{ nm } & \vspace{6}\\ \end{array} $ }$

(∵ 行列のスカラー倍，列ベクトルは行列に含まれる)

$3$\displaystyle{= $ \begin{array} \hspace{1}{a}_{ 11 }\hspace{1}{x}_{1} & \vspace{6}\\ \hspace{1}{a}_{ 21 }\hspace{1}{x}_{1} & \vspace{6}\\ \vdots & \vspace{6}\\ \hspace{1}{a}_{ n1 }\hspace{1}{x}_{1} & \vspace{6}\\ \end{array} $ + $ \begin{array} \hspace{1}{a}_{ 12 }\hspace{1}{x}_{2} & \vspace{6}\\ \hspace{1}{a}_{ 22 }\hspace{1}{x}_{2} & \vspace{6}\\ \vdots & \vspace{6}\\ \hspace{1}{a}_{ n2 }\hspace{1}{x}_{2} & \vspace{6}\\ \end{array} $ +\cdots + $ \begin{array} \hspace{1}{a}_{ 1m }\hspace{1}{x}_{m} & \vspace{6}\\ \hspace{1}{a}_{ 2m }\hspace{1}{x}_{m} & \vspace{6}\\ \vdots & \vspace{6}\\ \hspace{1}{a}_{ nm }\hspace{1}{x}_{m} & \vspace{6}\\ \end{array} $ }$

(∵ 行列の和)

$3$\displaystyle{= $ \begin{array} \hspace{1}{a}_{ 11 }\hspace{1}{x}_{1}+\hspace{1}{a}_{ 12 }\hspace{1}{x}_{2}+\cdots +\hspace{1}{a}_{ 1m }\hspace{1}{x}_{m} & \vspace{6}\\ \hspace{1}{a}_{ 21 }\hspace{1}{x}_{1}+\hspace{1}{a}_{ 22 }\hspace{1}{x}_{2}+\cdots +\hspace{1}{a}_{ 2m }\hspace{1}{x}_{m} & \vspace{6}\\ \vdots & \vspace{6}\\ \hspace{1}{a}_{ n1 }\hspace{1}{x}_{1}+\hspace{1}{a}_{ n2 }\hspace{1}{x}_{2}+\cdots +\hspace{1}{a}_{ nm }\hspace{1}{x}_{m} & \vspace{6}\\ \end{array} $ }$

行列の積の計算を利用して，行列を行列の積に変換する

$3$\displaystyle{= $ \begin{array} \hspace{1}{a}_{ 11 } & \hspace{1}{a}_{ 12 } &\cdots & \hspace{1}{a}_{ 1m } & \vspace{6}\\ \hspace{1}{a}_{ 21 } & \hspace{1}{a}_{ 22 } &\cdots & \hspace{1}{a}_{ 2m } & \vspace{6}\\ \vdots &\vdots &\ddots &\vdots & \vspace{6}\\ \hspace{1}{a}_{ n1 } & \hspace{1}{a}_{ n2 } &\cdots & \hspace{1}{a}_{ nm } & \vspace{6}\\ \end{array} $ $ \begin{array} \hspace{1}{x}_{1} & \vspace{6}\\ \hspace{1}{x}_{2} & \vspace{6}\\ \vdots & \vspace{6}\\ \hspace{1}{x}_{m} & \vspace{6}\\ \end{array} $ }$

$3$\displaystyle{= $ \hspace{1}{\mathbb{a}}_{1},\hspace{1}{\mathbb{a}}_{2},\cdots ,\hspace{1}{\mathbb{a}}_{m} $ $ \begin{array} \hspace{1}{x}_{1} & \vspace{6}\\ \hspace{1}{x}_{2} & \vspace{6}\\ \vdots & \vspace{6}\\ \hspace{1}{x}_{m} & \vspace{6}\\ \end{array} $ }$

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最終更新日： 2022年6月9日