Question

Let U, V, and W be a nontrivial finite dimensional vector space over a scaler field F.
(a) let S∈L(U,V) (this means that S is a linear operator from U to V) be an arbitrary fixed operator. Define R:L(V,W)→L(U,W) by R(T)=TS , T∈L(V,W). (note: TS means T of S)
Prove that R is injective iff S is surjective. Prove that R is surjective iff S is injective.