Our lecturer gave us a hard exercice to go further in group theory (we stopped at group actions) :
Let G be a group, V and W complex vector spaces and $\rho_1 : G \mapsto GL(V) $ be a group homomorphism where GL(V) is the general linear group of V (i.e. the invertible linear maps V $\mapsto$ V, group under compositions of maps) and let $\rho_2 : G \mapsto GL(W) $.
$(\rho_1,V)$ and $(\rho_2,V)$ are called representation of G. I have to show that :
-For representations V, W of G, the direct sum $V\oplus W$ is a representation of G. i.e. I have to find a homomorphism $\omega_1 : G \mapsto GL(V\oplus W)$.
-Same again but for the tensor product $V\otimes W$ via $g(v\otimes w):=gv\otimes gw$ i.e. I have to find a homomorphism $\omega_2 : G \mapsto GL(V\otimes W)$.
The problem is : I struggle with the definitions of direct sum and tensor product... (never worked on this before).
Here is what I've done for the first question. I have to show that : let $v\in V$ and $w\in V$ and $(\rho, V\oplus W)$ be a representation of G. Then $\forall g\in G, \rho(g)(v+w)=\rho(g)(v)+\rho(g)(w):=\rho_1(g)(v)+\rho_2(g)(w) \in V\oplus W$. But I don't know how to prove it.
I tried to find lectures and understand as much as I could but there is usually no explanations about those two questions, it is considered trivial apparently !
Thanks for your help