God bless Eigen, I am using it to implement the state matrices of a Kalman Filter and it's a joy to use its APIs.
* If a vector is an n-by-1 array (a "matrix"), why not an n-by-1-by-1-by-1-by-1?
* What is a row vector in something like a function space? Is sin(t) a row or column vector?
* You _can_ think of a matrix as acting "to the right (on a column vector)" and "to the left (on a row vector)", but this disrupts the notion of a matrix as a linear function acting on a thing (since it can act on two things). It's much more consistent to define a matrix acting on a vector, m(x), independent of "direction" and then define the transformations that let you go in the opposite direction (take the transpose of m which is a whole new matrix).
* Are physical quantities like velocity a row or column vector? Convention dictates they're column vectors, so in what scenario would it become a row vector? If I have a velocity, v1, that's a row vector and another, v2, that's a column vector, is it appropriate to calculate the projection of v1 onto v2? Should the projection operation be "aware" of the orientations? If all vectors are the same ("column vectors"), this isn't a problem -- you can always define the projection operation without special cases.
* In matlab specifically, if you have an n-by-1 array `a` and a 1-by-n array `b` with the same elements, then `a(i) == b(i)` but `a` and `b` are not the same thing.
Or equivalently https://jiahao.github.io/talk/2017-06-juliacon/
There are basically two self-consistent choices for the design of linear algebra libraries.
But tge nice part is that these vectors can still interact well with parts of the code that expect a matrix.