Let V be a vector space and T a linear transformation from V into V. Prove that the following two statements about T are equivalent.

a) The Intersection of the range of T and the null space of T is the zero subspace of V.

b) If T (T ??) =0, then T??=0.

The mathematical notation shown above as a square is alpha, supposed to be read as :

b) IF T ( T alpha ) =0, then T alpha = 0.

am stuck!!!!

jyotsna.

I think to show two statements are equivalent, it suffices to show that a imples b and b imples a.


Pf:
First part
Assume
Intersection of Range(T) and Ker(T)[ which is the null space] ={0}
WTS: if T(T(x))=0, then T(x)=0 [using x instead of alpha]
now, assume T(T(x))=0. This implies T(x) is an element of Ker(T) by the definition of Kernel. But T(x) is also an element of Range of T by the definition of Range.
This implies T(x) is in the intersection of R(T) and Ker(T).
However, by our initial assumption, the intersection has only one element which is 0.
Therefore T(x) must be 0.

Second Part
Other way round is similar,
Assume if T(T(x))=0, then T(x)=0
WTS: Intersection of Range(T) and Ker(T) ={0}

By definition of Range T(x)=0 is an element of Range(T).
Since T(T(x))=0, T(x)=0 is in the Ker(T).
Since T(x)=0 is in both the Range and Ker, it is in the intersection of R(T) and Ker(T).