基变换与坐标变换

castelu

　　设$\epsilon_1$，$\epsilon_2$，$\cdots$，$\epsilon_n$与$\epsilon'_1$，$\epsilon'_2$，$\cdots$，$\epsilon'_n$是$n$维线性空间$V$中两组基，它们的关系是
$$ \left\{ \begin{array}{l} \epsilon'_1=a_{11}\epsilon_1+a_{21}\epsilon_2+\cdots+a_{n1}\epsilon_n\\ \epsilon'_2=a_{12}\epsilon_1+a_{22}\epsilon_2+\cdots+a_{n2}\epsilon_n\\ \cdots\\ \epsilon'_n=a_{1n}\epsilon_1+a_{2n}\epsilon_2+\cdots+a_{n n}\epsilon_n \end{array} \right. 。$$
　　设向量$\xi$在这两组基下的坐标分别是$(x_1,x_2,\cdots,x_n)$与$(x'_1,x'_2,\cdots,x'_n)$，则有
$$ \left( {\begin{array}{*{20}{c}} x_1\\ x_2\\ \vdots\\ x_n \end{array}} \right) = \left( {\begin{array}{*{20}{c}} a_{11}&a_{12}&\cdots&a_{1n}\\ a_{21}&a_{22}&\cdots&a_{2n}\\ \vdots&\vdots&&\vdots\\ a_{n1}&a_{n2}&\cdots&a_{n n} \end{array}} \right) \cdot \left( {\begin{array}{*{20}{c}} x'_1\\ x'_2\\ \vdots\\ x'_n \end{array}} \right) ，$$
　　或者
$$ \left( {\begin{array}{*{20}{c}} x'_1\\ x'_2\\ \vdots\\ x'_n \end{array}} \right) = \left( {\begin{array}{*{20}{c}} a_{11}&a_{12}&\cdots&a_{1n}\\ a_{21}&a_{22}&\cdots&a_{2n}\\ \vdots&\vdots&&\vdots\\ a_{n1}&a_{n2}&\cdots&a_{n n} \end{array}} \right) ^{-1} \cdot \left( {\begin{array}{*{20}{c}} x_1\\ x_2\\ \vdots\\ x_n \end{array}} \right) ，$$
　　其中矩阵$A= \left( {\begin{array}{*{20}{c}} a_{11}&a_{12}&\cdots&a_{1n}\\ a_{21}&a_{22}&\cdots&a_{2n}\\ \vdots&\vdots&&\vdots\\ a_{n1}&a_{n2}&\cdots&a_{n n} \end{array}} \right) $称为由基$\epsilon_1$，$\epsilon_2$，$\cdots$，$\epsilon_n$到$\epsilon'_1$，$\epsilon'_2$，$\cdots$，$\epsilon'_n$的过渡矩阵。