1.7.3 Linear transformations

Problem: Given the coordinates of the vector \( x=\{2,5,10\} \) in the basis \( \left(e_{1}, e_{2}, e_{3}\right) \). Find its coordinates in the basis \( \left(e_{1}{ }^{\prime}, e_{2}{ }^{\prime}, e_{3}{ }^{\prime}\right) \), where \[ \left\{\begin{array}{c} e_{1}{ }^{\prime}=e_{1}+e_{2}+6 e_{3} \\ e_{2}{ }^{\prime}=\frac{6}{5} e_{1}-e_{2} \\ e_{3}{ }^{\prime}=-e_{1}+e_{2}+e_{3} \end{array} .\right. \]