I'm having some confusion with index notation and how it works with contravariance/covariance.
$(v_{new})^i=\frac{\partial (x_{new})^i}{\partial (x_{old})^j}(v_{old})^j$
$(v_{new})^i=J^i_{\ j}(v_{old})^j$
$(v_{new})_i=\frac{\partial (x_{old})^j}{\partial (x_{new})^i}(v_{old})_j$
$(v_{new})_i=(J^{-1})^j_{\ i}(v_{old})_j$
So these are the standard rules for transforming contra and covariant vectors. Now if we want to convert this into a matrix equation is there an exact set of rules with regards index position?
For example for the covariant transformation I can transpose the matrix which swaps the index order(Not sure how this makes sense) and this gives the right answer if we treat the covariant vectors as columns.
Or I can move the $(v_{old})_j$ to the right of the J inverse and treat it as a row vector and this gives the right answer and I don't need to even consider what a transpose is in this interpretation.
Now both of these interpretations give the correct answers but they seem to have different meanings for upper vs lower and horizontal order.
Is there a best way to think about this, which way makes the most sense in terms of raising/lowering with metric tensors and transforming higher order tensors?