In an exam I needed to prove that the set of all the endomorphisms of $\mathbb{R}^3$ form an abelian group (I take endomorphisms as linear transformations from a Vector Space to itself).
I tried to answer using the usual sum of functions and seemed correct, but I believe it was too easy that way. To try to make that set a group with the usual function composition, how can I guarantee the existence of inverses? More than that, how can I prove conmutativity?
The set of endomorphisms for $(\mathbb{R}^3,+)$ will form an abelian group under the operation of (pointwise) addition of functions and not with composition, i.e. $$(f+g)(x)=f(x)+g(x).$$ In which case the inverse will be $-f(x)$ and commutativity is the outcome of commutative addition in $\mathbb{R}^3$.
In general, if we have $(G,+)$ an abelian group then the set of endomorphisms of $G$ will form an abelian group under function addition.