1.7.5 Linear transformations

Problem: Given the matrix of the operator \( A=\left(\begin{array}{lll}2 & 1 & 2 \\ 3 & 0 & 2 \\ 1 & 0 & 1\end{array}\right) \) in basis \( \left(e_{1}, e_{2}, e_{3}\right) \). Find its matrix in basis \( \left(e_{1}^{\prime}, e_{2}^{\prime}, e_{3}{ }^{\prime}\right) \), where \( \left\{\begin{array}{c}e_{1}{ }^{\prime}=e_{1}-e_{2}+e_{3} \\ e_{2}{ }^{\prime}=-e_{1}+e_{2}-2 e_{3} \\ e_{3}{ }^{\prime}=-e_{1}+2 e_{2}+e_{3}\end{array}\right. \).