Suppose that $(\varphi,V)$ is an irreducible finite dimensional representation of a Lie group $G$ and let $\psi=\bigoplus_{i=1}^n{\varphi}$ the representation on $W=\bigoplus_{i=1}^n{V}$. I want to show that $Hom_G(V,W)$ is $n$-dimensional and that $End_G(V\oplus V)$ is $4$-dimensional.
For the first part i got the idea that it may be possible to prove that $Hom_G(V,W)\cong \bigoplus_{i=1}^nHom_G(V,V)$ and from Schurs Lemma we may conclude that $Hom_G(V,V)$ is one-dimensional, thus the sum is n-dimensional. But i don't know how to write this down on a good way. Can someone help me with this? Thanks.