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To show that the sum of two tensors is a tensor, I am taking the approach that (here for a vector, or first rank tensor)

$V_i' = a_{ij}V_j$

$W_i' = a_{ij}W_j$

then summing them to get

$V_i' + W_i'= a_{ij}V_j + a_{ij}W_j$

then when you sum and contract that lot you get

$V_i' + W_i'= a_{ij}(V_j + W_j)$

which if I am not mistaken fulfils the criteria for a vector, showing that the sum of two vectors is a vector. Now when we get onto higher rank tensors, I am finding this a bit of a pain, since expanding and contracting that lot is tiresome. Is it a general rule that

$V_{\alpha\beta\gamma}' + W_{\alpha\beta\gamma}'= a_{\alpha~i}a_{\beta~j}a_{\gamma~k}W_{ijk} + a_{\alpha~i}a_{\beta~j}a_{\gamma~k}W_{ijk}$

can be factored to

$V_{\alpha\beta\gamma}' + W_{\alpha\beta\gamma}'= a_{\alpha~i}a_{\beta~j}a_{\gamma~k}(V_{ijk}) + W_{ijk}$

and am I even taking the right approach.

(I am working through the Mary L Boas book here, chapter ten problems to section 11, q3 and 5.)

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I have that book and after looking over that section it apperrs to me that you are on the right track.

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