1.7.8 Linear transformations

condition: 1) Which of the listed transformations is a linear operator in the space \( \mathbb{R}^{3} \) ? 2) Find the matrix of the operator in the canonical basis of the space \( \mathbb{R}^{3} \). 3) Find the eigenvalues ​​and eigenvectors of the operator. Is this operator a simple type operator? 4) Find the kernel of the operator. 5) Is this operator invertible? If yes, find the inverse operator. \[ \begin{array}{l} \widehat{\mathrm{A}}=\left(3 x_{1}-x_{2}-x_{3}, 2 x_{2}+x_{3}, x_{2}+2 x_{3}\right) \\ \widehat{\mathrm{B}}=\left(3 x_{1}-1-x_{3}, 2+x_{3}, x_{2}+2 x_{3}\right) \\ \widehat{\mathrm{C}}=\left(3 x_{1}^{2}-x_{2}-x_{3}, 2 x_{2}+x_{3}^{2}, x_{2}+2 x_{3}\right) . \end{array}\]