= Solution
Use the <Einstein summation convention>. Substitute the given <basis> transformation into the expansion of the <vector>:
$$
v_j e_j=v'_iR_{ij}e_j.
$$
By <linear independence> of the <basis> <vectors>, $v_j=R_{ij}v'_i$. In <matrix> notation $v=R^Tv'$. A <rotation matrix> is <orthogonal>, so multiplication by $R$ gives
$$
\boxed{v'_i=R_{ij}v_j,\qquad v_i=R_{ji}v'_j.}
$$
The placement of the transpose follows from the specified convention $e'_i=R_{ij}e_j$. It must not be guessed from a different convention for an active rotation of a <vector>.
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