I am beginning to self-learn Einstein Notation, and I tried to prove the rotational invariance of a dot product. In matrix notation (which I am more used to) it is quite simple. If $u$ and $v$ are column vectors, $ u \cdot v = u^T v $. If $R$ is a rotation matrix then the vectors transform as $ u' = Ru $, $ v' = Rv $. Taking the dot product of the transformed vectors, $ u' \cdot v' = u'^Tv' = (Ru)^T(Rv) = u^T R^T R V $ and due to the orthogonality of R,the $ R^T R $ cancels out and we are left with $ u^T v $.
However, I am having difficulties using Einstein notation to prove this same result. If $ R_{ij} $ is a rotation matrix such that $ R_{ik} R_{jk} = \delta_{ij} $, how do we prove that $ ( R_{ij} u_j ) ( R_{ij} v_j ) = u_i v_i $?
@Andrei is right. Use$$(Ru)_i(Rv)_i=R_{ij}R_{ik}u_jv_k=\delta_{jk}u_jv_k=u_jv_j.$$