文書の過去の版を表示しています。
行列のランク
連立一次方程式
$$ \begin{cases} a_{11}x_{1}+a_{12}x_{2}+\cdots+a_{1n}x_{n}=b_{1}\\ a_{21}x_{1}+a_{22}x_{2}+\cdots+a_{2n}x_{n}=b_{2}\\ \vdots\\ a_{n1}x_{1}+a_{n2}x_{2}+\cdots+a_{nn}x_{n}=b_{n} \end{cases} $$
係数行列(拡大係数行列)
$$ \begin{cases} a_{11}x_{1}+a_{12}x_{2}+\cdots+a_{1n}x_{n}=b_{1}\\ a_{21}x_{1}+a_{22}x_{2}+\cdots+a_{2n}x_{n}=b_{2}\\ \vdots\\ a_{n1}x_{1}+a_{n2}x_{2}+\cdots+a_{nn}x_{n}=b_{n} \end{cases} \Rightarrow \begin{bmatrix} a_{11}&a_{12}&\cdots&a_{1n}&\,&b_{1}\\ a_{21}&a_{22}&\cdots&a_{2n}&\,&b_{2}\\ \vdots&\vdots&\ddots&\vdots&&\vdots\\ a_{n1}&a_{n2}&\cdots&a_{nn}&\,&b_{n} \end{bmatrix} $$
係数行列(拡大係数行列)
$$ \begin{cases} a_{11}x_{1}+a_{12}x_{2}+\cdots+a_{1n}x_{n}=b_{1}\\ a_{21}x_{1}+a_{22}x_{2}+\cdots+a_{2n}x_{n}=b_{2}\\ \vdots\\ a_{n1}x_{1}+a_{n2}x_{2}+\cdots+a_{nn}x_{n}=b_{n} \end{cases} \Rightarrow \begin{bmatrix} a_{11}&a_{12}&\cdots&a_{1n}&\,&b_{1}\\ a_{21}&a_{22}&\cdots&a_{2n}&\,&b_{2}\\ \vdots&\vdots&\ddots&\vdots&&\vdots\\ a_{n1}&a_{n2}&\cdots&a_{nn}&\,&b_{n} \end{bmatrix} $$