I am reading Vector and Tensor Analysis by Borisenko and Tarapov.
Early in its definition of tensors, it describes a transformation to a new coordinate system by
$x_i' = \alpha_{i'k}x_k + X_{0i}$, where $\alpha_{i'k}$ is the "cosine of the angle between the i'th axis of the new system and the kth axis of the old system".
My question is: how does the introduction of the kth index work? For example, when finding $x_1'$, does k assume the value of i, so $x_i' = \alpha_{i'1}x_1 + X_{01}$ ? Or is there an implicit sum over all kth elements?