概念問題 解答

線形代数　行列

問題12の解答

${\displaystyle \overrightarrow{\text{OA}}=\left(\begin{array}{c}1\\ 0\end{array}\right)}$ ， ${\displaystyle \overrightarrow{\text{OB}}=\left(\begin{array}{c}0\\ 1\end{array}\right)}$

${\displaystyle \overrightarrow{\text{O}{\text{A}}^{\prime }}=\left(\begin{array}{c}2\\ 1\end{array}\right)}$ ， ${\displaystyle \overrightarrow{\text{O}{\text{B}}^{\prime }}=\left(\begin{array}{c}2\\ 1\end{array}\right)}$

表現行列を ${\displaystyle \left(\begin{array}{cc}a& c\\ b& d\end{array}\right)}$ とすると

${\displaystyle \left(\begin{array}{c}2\\ 1\end{array}\right)=\left(\begin{array}{cc}a& c\\ b& d\end{array}\right)\left(\begin{array}{c}1\\ 0\end{array}\right)}$　･･････(1)

${\displaystyle \left(\begin{array}{c}1\\ 2\end{array}\right)=\left(\begin{array}{cc}a& c\\ b& d\end{array}\right)\left(\begin{array}{c}0\\ 1\end{array}\right)}$　･･････(2)

の関係がある．

(1)より

${\displaystyle \left(\begin{array}{c}2\\ 1\end{array}\right)=\left(\begin{array}{c}a\\ b\end{array}\right)}$　･･････(3)

(2)より

${\displaystyle \left(\begin{array}{c}1\\ 2\end{array}\right)=\left(\begin{array}{c}c\\ d\end{array}\right)}$　･･････(4)

(3)，(4)より

${\displaystyle \left(\begin{array}{cc}a& c\\ b& d\end{array}\right)=\left(\begin{array}{cc}2& 1\\ 1& 2\end{array}\right)}$

となる．

したがって，答は1である．