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Indeed it does depend, some languages are row major (C, C++) while others are column major (Fortran, and some common HPC libraries are in Fortran).

https://en.wikipedia.org/wiki/Row-_and_column-major_order

I'm not alone!!!!!! :) I was having brain melting with today's http://adventofcode.com/2016/day/8 I'm attempting everything in Elixir and it's quite different from what I'm used to!

>Are there things that transpose memory organization?

Not exactly what you asked, but if you look at most subroutines in linear algebra libraries like LAPACK, they tend to have extra arguments so you can tell if your inputs are already transposed and/or conjugate in order to have more efficient memory access.