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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

C

_{ij}v

_{j}=A

_{ij}v

_{j}+B

_{ij}v

_{j}

which holds for any |v>, so

C=A+B

as this is how we define matrix addition.

However, why couldn't I have written

C

_{ij}v

_{j}=A

_{ik}v

_{k}+B

_{il}v

_{l}

because I have changed only dummy variables, not affecting the sum. This would then not lead to C

_{ij}v

_{j}=A

_{ij}v

_{j}+B

_{ij}v

_{j}. 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.