1.1.7 Vector Algebra

Problem: Find the coordinates of the vector \( X \) in the basis \( e^{\prime}=\left\{e_{1}^{\prime} ; e_{2}^{\prime} ; e_{3}^{\prime}\right\} \), if in the basis \( e=\left\{e_{1} ; e_{2} ; e_{3}\right\} \), \( X=(2 ; 6 ;-3) \), and the conversion from \( e \) to \( e^{\prime} \) is: \[ \left\{\begin{array}{l} e_{1}^{\prime}=e_{1}+e_{2}-2 e_{3} \\ e_{2}^{\prime}=\frac{2}{3} e_{1}-e_{2} \\ e_{3}^{\prime}=-e_{1}+e_{2}+e_{3} \end{array}\right. \]