I was trying to prove this question:
Prove that $(T_{1} + T_{2}) S \simeq T_{1}S + T_{2}S $ for any representations $T_{1},T_{2},$ and $S$ of $G$.
So I was reading the Sums of representations and I found that it is description in terms of matrices is as shown below:
Can I say by this that addition of representations can be considered as a diagonal matrix, but why? this idea is so hard for me to believe, can anyone give me a numeric example please?