Question 15125
 Let T be a linear operator in V. Prove that T^2 = 0 if and only if the range T is a subset of null  T.
 [This question is very easy,you should know how to prove it)
 Proof: (==>)
        If y is in R(T) (range of T) then there is x in V such that
         y = T(x). Hence, {{{T(y) = T(T(x)) = T^2(x) = 0 }}}
        This shows R(T) is a subset of N(T).

        (<==)
        If R(T) < N(T), then 
        for any x in V, T(x) is in R(T) < N(T). 
        Hence , {{{ T^2(x) = T(T(x)) = 0 }}} 
       QED


 You should work hard.


 Kenny