mapping matrices in column-major order

It's very common to see meantone represented like this

|<1 1 0| <0 1 4|>

This is two bras in a ket. The idea is that the braket product of a
monzo and this is as follows:

|<1 1 0| <0 1 4|> |-1 1 0> = |<1 1 0|-1 1 0> <0 1 4|-1 1 0>> = |0 1>

Has anyone ever considered presenting the mapping matrix in column
major order, though, like this?

<|1 0> |1 1> |0 4>|

Then, you can multiply it out as follows:

<|1 0> |1 1> |0 4>|-1 1 0> = (-1 * |1 0>) + (1 * |1 1>) + (0 * |0 4>) = |0 1>.

In other words, this envisions mapping matrices as extensions of vals,
but in which the coefficients are kets and not scalars. When you
multiply it out, you just multiply the 2-ket by the 2-coefficient of
the monzo, and the 3-ket by the 3-coefficient, etc. The 2-ket is the
2D mapping for 2/1, the 3-ket is the 2D mapping for 3/1, etc.

It's just the other way to look at matrix multiplication - the way in
which you take a weighted sum of column vectors, with the weights
being given by the coefficients in the row vectors on the right.

I find this extremely intuitive and elegant, if not a bit cumbersome
in ASCII. We're just saying that mapping matrices are vals in which
each coefficient is a tempered ket, and the bracket product gives you
a linear combination of those kets. Simple.

Thoughts?

-Mike