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Suppose we have a set of numbers $T_{ij}=B_{ij}+a_i$ and another set $A_{ij}$.

Now, according to index notation, $A_{ij}T_{ij}=A_{ij}B_{ij}+A_{ij}a_i$.

The $1^{st}$ term in the RHS is a summation over both indices $i$ and $j$ while the $2^{nd}$ is a summation over $i$ only.

However, when calculating the value without index notation, we get $$A_{ij}T_{ij}=\sum_i\sum_jA_{ij}T_{ij}+A_{ij}a_i$$Both terms are summed up over $i$ and $j$. So where is the problem ?

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  • $\begingroup$ You need to treat $a_i$ as though it also had an index $j$ which it doesn't depend on. Otherwise the first equation doesn't make sense. $\endgroup$ Mar 1, 2017 at 21:09
  • $\begingroup$ Ok now I understand thanks guys ! $\endgroup$
    – Tofi
    Mar 1, 2017 at 21:13

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