Homework SourseLet V be a vector space and ST LV Suppose range S null T Pro
Let V be a vector space, and S,T L(V). Suppose range S null T. Prove that
(ST)2 =0.
Let V be a vector space, and S,T L(V). Suppose range S null T. Prove that
(ST)2 =0.
Let V be a vector space, and S,T L(V). Suppose range S null T. Prove that
(ST)2 =0.
Solution
Given S, T: V-> V such that range (S) is contianed in Null (T).
Range (S)= {Sv: v in V}
and
Null(T)= {w in V: T(w)=0}.
GIven that T(S(v))=0 for all v in V ..................(1)
Consider (ST)2 (v)= (STST)(v) (by definition)
=S(T(S(Tv)))
=S(T(S(u))) where u =Tv
=S(TS(u))
=S(0) (follows from (1))
=0 (as S is linear)
As (ST)2 (v)=0 for all v in V, it follows that (ST)2 is identically 0 as a linear transformation.
Thus (ST)2=0
