1.7.13 Linear Transformations

\ (\ underline {\ mathrm {y} _ {\ text {Sliding:}} \) Operator \ (\ Widehat {\ Mathrm {a}} \) acts in the space of matrices \ (m \) in the space of all square matrixes second order. 1) to prove that \ (\ widehat {\ mathrm {a}} \) is a linear operator. 2) Find the matrix of the operator \ (\ widehat {\ mathrm {a}} \) in some kind of space \ (m \). 3) Find your own values and own vectors of the operator \ (\ widehat {a} \) (recall that in this case the vectors are matrices). 4) to prove that \ (\ widehat {a} \) is a simple -type operator, indicate the basis from its own vectors. \ [\ widehat {\ mathrm {a}} x = b x-x b, b = \ left (\ begin {array} {ll} 0 & 1 \\ 1 & 0 {array} \ right), M = \ left \ {x = \ left (\ begin {array} {ll} x & y \\ u & v \ end {array} \ right): x+v = 0 \ right \} \]