1.1.5 Vector Algebra

Problem: Vector \( X=(7 ; 7 ; 2) \) is given in the basis \( e=\left\{e_{1}, e_{2}, e_{3}\right\} \). Find the coordinates of \( X \) in the basis of \( e^{\prime}=\left\{e_{1}^{\prime}, e_{2}^{\prime}, e_{3}^{\prime}\right\} \), where \[ \left\{\begin{array}{l} e_{1}^{\prime}=e_{1}+e_{2}+\frac{6}{7} e_{3} \\ e_{2}^{\prime}=-6 e_{1}-e_{2} \\ e_{3}^{\prime}=-e_{1}+e_{2}+e_{3} \end{array}\right. \]