Suppose T LV Prove that Trange T 0 Suppose T LV Prove that

Suppose T L(V). Prove that T/(range T) = 0. Suppose T L(V). Prove that T/(null T) is injective if and only if

Solution

By definition, for any x+rangeT V/(rangeT), we have

                              T/(rangeT)(x+rangeT)=Tx+rangeT.

Note that Tx rangeT,

it follows that T/(rangeT)(x+rangeT)=0.

Since x+rangeT is choosed arbitrarily, we conclude that T/(rangeT)=0.