im very lost with transition matrix..thought i understand it alr but turns out i haven..

if i define T to be the transition matrix from S to S'

then let (v)_s be the coordinate vector wrt to S...

then its right of me to say that the coord vector wrt to S' is T(v)_s right?

let S= {v_1,...,v_n}, S' = {v'_1,...,v'_n} be the basis for the vector space V

then in my notes, it states that (v_1,...,v_n) = (v'_1,...,v'_n)T. but i thought it should be

T(v_1,...,v_n) = (v'_1,...,v'_n) since T is from S to S'?

i dont get what it means when my prof puts the T behind the vectors.