Suppose U is a subspace of V with U=V. Suppose S∈L(U,W) and S=0 (Which means that Su=0 for some u∈U). Define T:V→W by
Tv={Sv0if v∈Uif v∈V and v∈/U
Prove that T is not a linear map on V.
Let u∈U such that Su=0. Let v∈V,v∈/U and consider how u+v∈/U because if it were in U adding −u would imply v∈U which it isn't. Since T(u+v)=0=Tu+Tv we conclude T is not linear.
In essence this exercise was to show you can't naively extend a linear map on a subspace to the full space by setting it to zero on the full space.