Question
Please show steps: Let T : R^3 ---> R^3 be defined by Wsub1= x -y Wsub2= y -z Wsub3= z -x A.) Is T a one-to-one operator? If T is one-to-one, find itsinvert. If not justify (prove) why not. B.) Is the range of T all of R^3? If yes, prove it. If no,give an example of a vector in R^3 but not in the range of T. Please show steps: Let T : R^3 ---> R^3 be defined by Wsub1= x -y Wsub2= y -z Wsub3= z -x A.) Is T a one-to-one operator? If T is one-to-one, find itsinvert. If not justify (prove) why not. B.) Is the range of T all of R^3? If yes, prove it. If no,give an example of a vector in R^3 but not in the range of T.
Explanation / Answer
A) we follow that T (x,y,z ) = ( x - y , y - z , z - x ) suppose T(x,y,z) = ( 0,0,0) ==> x - y = 0 , y - z = 0 , z - x =0 . ==> x =y=z= 0. T(x,y,z) = ( 0,0,0) ==> kernel isnull space = 1.