1.7.13 Linear transformations

\( \underline{\mathrm{y}_{\text {word: }}} \) The operator \( \widehat{\mathrm{A}} \) acts in the space of matrices forming a linear subspace \( M \) in the space of all square matrices second order. 1) Prove that \( \widehat{\mathrm{A}} \) is a linear operator. 2) Find the matrix of the operator \( \widehat{\mathrm{A}} \) in some basis of the space \( M \). 3) Find the eigenvalues ​​and eigenvectors of the operator \( \widehat{A} ​​\) (recall that in this case the vectors are matrices). 4) Prove that \( \widehat{A} ​​\) is an operator of simple type, indicate a basis of eigenvectors. \[ \widehat{\mathrm{A}} X=B X-X B, B=\left(\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right), M=\left\{X=\left(\begin{array}{ll} x & y \\ u & v \end{array}\right): x+v=0\right\} \]