I am trying to follow along in a textbook (Boas: Mathematical Methods in the Physical Sciences) and I have a problem understanding how to expand the following summation signs. (They are unrelated to each other. I just need help with understanding how each is expanded out.) I'd appreciate any assistance. I'd also like some links to relevant webpages that show you how to work these out, if available.
$ T'_{kl} = \sum_{i=1}^3 \sum_{j=1}^3 a_{ki}a_{lj}T_{ij}, \qquad k, l = 1,2,3 $ $ U'_k V'_l = \sum_{i=1}^3 a_{ki} U_i \sum_{j=1}^3 a_{lj} V_j = \sum_{i, j = 1}^3 a_{ki} a_{lj} U_i V_j $ $ T'_{\alpha \beta \gamma \delta} = \sum_{i,j,k,l} a_{\alpha i} a_{\beta j} a_{\gamma k} a_{\delta l} T_{ijkl} $