Let V and W be vector space over a field F, and let T,U: V -> W be linear. Prove that the collection of all linear transformation from V to W is a vector space over F.

Proof.

Let F denote the set of all linear transformations from V to W.

Let T,U,S be in F, and let t,u,s be in V.

Commutativity of addition: (T+U)(t+u)=T(t)+U(u)=U(u)+T(t)=(U+T)(u+t)

Is this the right way to do it? Thanks