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Perhaps the sanest approach is we type every column of X and Y separately...

Typing every column of a matrix is not sufficiently general. Each element of the matrix can have its own type.

Suppose you're solving a linear algebra problem in two unknowns. Say it's an electromechanical system, and we want to find a voltage and a pressure that gives us some desired flow rates. So let x [=] (V, Pa), for example, and let y [=] (mA, kg/s).

Then the matrix-vector product Ax = y requires the units:

    [ mA/V   , mA/Pa   ] * ( V  ) = ( mA   )
    [ kg/V/s , kg/Pa/s ]   ( Pa )   ( kg/s )

Of course those units in A have nicer reductions, but it doesn't teach us anything to figure them out. The point is that the matrix A represents the system design, so the units don't have to have familiar names, let alone be consistent within a column. If the system involves Johnny reading a value off a meter, and then phoning somebody up to tell them to pedal a stationary bike a little faster, that process has some kind of units associated with it, and the matrix A must include those units in order to represent the system.

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