概念問題 解答

線形代数　行列

問題2の解答

${\displaystyle \left(\begin{array}{c}1\\ 0\end{array}\right)\stackrel{f}{\to }\left(\begin{array}{c}2\\ 2\end{array}\right)}$ ，${\displaystyle \left(\begin{array}{c}0\\ 1\end{array}\right)\stackrel{f}{\to }\left(\begin{array}{c}3\\ 1\end{array}\right)}$

のとき，1列に${\displaystyle \left(\begin{array}{c}2\\ 2\end{array}\right)}$ ，2列に${\displaystyle \left(\begin{array}{c}3\\ 1\end{array}\right)}$ を当てはめた行列

${\displaystyle \left(\begin{array}{cc}2& 3\\ 2& 1\end{array}\right)}$

が${\displaystyle f}$ の表現行列となる．

よって，答は2となる．

与えられた条件より，具体的に表現行列を求めてみる．表現行列を

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

とおく．

${\displaystyle \left(\begin{array}{c}1\\ 0\end{array}\right)\stackrel{f}{\to }\left(\begin{array}{c}2\\ 2\end{array}\right)}$より

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

${\displaystyle \left(\begin{array}{c}0\\ 1\end{array}\right)\stackrel{f}{\to }\left(\begin{array}{c}3\\ 1\end{array}\right)}$より

${\displaystyle \left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\left(\begin{array}{c}0\\ 1\end{array}\right)=\left(\begin{array}{c}3\\ 1\end{array}\right)}$ ，${\displaystyle \left(\begin{array}{c}b\\ c\end{array}\right)=\left(\begin{array}{c}3\\ 1\end{array}\right)}$　･･････(2)

(1)，(2)より

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

となる．