1.7.8 Linear Transformations

Condition: 1) Which of the listed transformations is a linear operator in space \ (\ mathb {r}^{3} \)? 2) Find the operator’s matrix in the canonical basis of space \ (\ mathbb {r}^{3} \). 3) Find your own values and own vectors of the operator. Is this operator a simple type operator? 4) Find the nucleus of the operator. 5) Will this operator turn? If so, find the reverse operator. \ [\ begin {array} {l} \ widehat {\ mathrm {a} = \ left (3 x_ {1} -x_ {2} -x_ {3}, 2 x_ {2}+x_ {3}, x_} +2 +2 +2 X_ {3} \ RIGHT) \\ \ WIDEHAT {\ MATHRM {B}} = \ Left (3 x_} -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_^{2}, x_ {2} +2 x_} \ right). \ end {Array} \]