There is a lot of stuff to show here. Let be linear transformations . Define to be the sum (as a function sum) of the linear transformations. Then . Futhermore, . Thus, the set of all linear transformations is closed under this definition. There are more things we need to prove. We need to show that this set forms an abelian group under this operation . We know that and and (where is the trivial linear transformation). And if is the negative linear transformation then . Which shows is an abelian group. To show that is a vector space we need to confirm more things. Can you finish that?