this is another notation problem which i cant figure out:

let T be the transition matrix from S to S' and T' be the transition matrix from S' to S"and S={v_1,...,v_n} and S'= {v'_1,...,v'_n}

then my notes define that

let T= (t_kj)k,j and T'=(t'_ik)i,k...i was wondering, how do you know what to label the subscript at?

also, why is it then that v_j = summation from k=1 to n of [(t_kj)v'_k] for j=1,..,n

i was thinking that it should be v'_j = summation from k=1 to n of [(t_kj)v_k] since T is from S to S'