Suppose we have linear operators A' and B'. We define their sum C'=A'+B' such that C'|v>=(A'+B')|v>=A'|v>+B'|v>. Now we can represent A',B',C' by matrices A,B,C respectively. I have a question about proving that if C'=A'+B', C=A+B holds. The proof is Using the above with Einstein summation convention, C|v>=A|v>+B|v> and so component i on each side matches. Then Cijvj=Aijvj+Bijvj which holds for any |v>, so C=A+B as this is how we define matrix addition. However, why couldn't I have written Cijvj=Aikvk+Bilvl because I have changed only dummy variables, not affecting the sum. This would then not lead to Cijvj=Aijvj+Bijvj. I'm assuming it's something to do with the fact the next step sort of stops this sum from happening anyway, but I'm not sure.