Suppose there exists a nonzero linear transformation 
such that .TS = 0.
Since S is non zero we can find w in V such that S(w) 
0. Let v = S(w)
That is v is a nonzero vector of V.
But T(v) = T(S(w)) =TS (w) = 0. That is v is a nonzero vector such that T(v) = 0.
Conversely assume that there exists a nonzero vector v such that T(v) = 0.
We can find a basis containing v. Let {v,w1,w2, ...} be the basis.
Now consider the projection linear transformation defined as P1(v) = v and P1(wi) = 0 for all the basis elements wi .
Then clearly P belongs to L(V,V)
So for every u in V , P(u) = av for some scalar a and therefore
TP(u) =T(P(u)) = T(av) = aT(v) = 0. That is for every vector u in V, TP(u) = 0. Hence TP =0
So the result is proved.