In input-output analysis in economics, for example, the elements of a vector represent the amounts of a commodity (coal, steel, electricity, etc.), and the columns of the demand matrix represent how much of each commodity is required to produce one unit of that commodity as output. So the type of row 1 is "kg of coal", the type of row 2 is "kg of steel", the type of row 3 is "kJ of energy", etc. Given this matrix and a vector representing the starting quantities, you can do some linear algebra to get a matrix representing how much of each quantity to allocate to each sector, and a vector representing the resulting output. The type of these vectors are "[kg Coal, kg Steel, kJ electricity, ...]"

I don't now how you can "have an implicit 1 with the right units in them in order to make the vector have a consistent unit" given this setup.

maybe i don't understand what you're saying but it sounds like you don't understand the problem

it's not that the units are inconsistent in the sense of adding kilograms to kilojoules; they're just inhomogeneous

you have a potentially large matrix in which potentially every cell has different units. so type systems that require the matrix cells to all be of the same type aren't helpful unless that type is something machine-oriented like `real*8`

moreover this is intrinsic to the problem, or at least the decision to use linear algebra on the problem

an additional difficulty for things like *gemm or lu factorization is that you want them to be applicable to matrices of any size with any type of units of measurement, as long as they're consistent; so they're parametrically polymorphic over units of measurement