Consider the equation $$u_k = t_i U_i^k$$
I am told that subscripts represent covectors (row vectors) and superscripts represent column vectors. My intuitive interpretation of the above equation as a matrix multiplication fails since $(1 \times n) \ne (1 \times m) \times (n \times m)$, where $n$ is the range of $k$ and $m$ is the range of $i$.
I am assuming that $U_i^k$ must instead represent an $m \times n$ matrix, but would this not contradict the rule of superscripts representing the column index? Is there a rule in tensor notation which covers this, or did the author of equation make a mistake and should write $U_k^i$?