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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.

$$ T'_{kl} = \sum_{i=1}^3 \sum_{j=1}^3 a_{ki}a_{lj}T_{ij}, \qquad k, l = 1,2,3 $$ $$ U'_k V'_l = \sum_{i=1}^3 a_{ki} U_i \sum_{j=1}^3 a_{lj} V_j = \sum_{i, j = 1}^3 a_{ki} a_{lj} U_i V_j $$ $$ T'_{\alpha \beta \gamma \delta} = \sum_{i,j,k,l} a_{\alpha i} a_{\beta j} a_{\gamma k} a_{\delta l} T_{ijkl} $$

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These are all equations relating to the change of basis in a linear space. The simplest is one half of the second one: $$U'_k=\sum_{i=1}^3a_{ki}U_i$$ The $a_{ki}$ form a $3 \times 3$ orthogonal matrix, so $A^T=A^{-1}$. This equation takes a vector $U$ represented in one coordinate frame (unprimed) and expresses it in another (primed) frame. The summation is standard matrix multiplication. $A$ is a $3 \times 3$ matrix, $U$ is a $3 \times 1$ column vector, and $U'$ is a $3 \times 1$ column vector, the same one as $U$ but represented by values in the primed frame. If $A$ is a rotation of $\frac \pi 4$ about the $z$ axis, then $$A=\left[\begin {array}{ccc} \frac {\sqrt 2}{2} & -\frac {\sqrt 2}{2} & 0 \\ \frac {\sqrt 2}{2} & \frac {\sqrt 2}{2} &0 \\ 0 & 0 & 1 \end {array}\right]$$. If $$U=\left[\begin {array}{c}1\\0\\0 \end {array}\right]$$ it is a vector pointing one unit to the east. Then $$U'=\left[\begin {array}{c}\frac {\sqrt 2}{2}\\-\frac {\sqrt 2}{2}\\0 \end {array}\right]$$ indicating you should go $\frac {\sqrt 2}{2}$ northeast and $\frac {\sqrt 2}{2}$ southeast to get to the same point.

The other equations work the same way, but each index needs to be transformed. The first is a two index tensor, so you need two applications of the rotation matrix to transform it to the new frame. The last is a four index tensor, so you get four applications of $A$.

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