On row & column vectors, do you really want to think of them as {1,...,n)->R? They often represent linear maps from R^n to R or R to R^n, which are very different types. Similarly, instead of working with matrices, how about linear maps from R^n to R^m? In this view, column and row vectors, matrices, and often scalars are useful as representations of linear maps. I've played around some with this idea of working with linear maps instead of the common representations, especially in the context of derivatives (including higher-dimensional and higher-order), where it is the view of calculus on manifolds. It's a lovely, unifying approach and combines all of the various chain rules into a single one. I'd love to explore it more thoroughly with one or more collaborators. Cheers, - Conal -----Original Message----- Henning Thielemann wrote:
On Wed, 29 Jun 2005, Jacques Carette wrote:
9. There are row vectors and column vectors, and these are different types. You get type errors if you mix them incorrectly.
What do you mean with "row vectors and column vectors are different types"? Do you mean that in a well designed library they should be distinguished? I disagree to this point of view which is also represented by MatLab.
This was hotly debated during our design sessions too. It does appear to be a rather odd decision, I agree! It was arrived at after trying many alternatives, and finding all of them wanting. Mathematically, vectors are elements of R^n which does not have any inherent 'orientation'. They are just as simply abstracted as functions from {1,...,n) -> R, as you point out. But matrices are then just linear operators on R^n, and if you go there, then you have to first specify a basis for R^n before you can write down a representation for any matrix. Not useful. However, when you step away from the pure mathematics point of view and instead look at actual mathematical usage, things change significantly. Mathematics is full of abuse of notation, and to make usage of matrices and vectors both convenient yet 'safe' required us to re-examine these many de facto conventions that mathematicians routinely use. And try to adapt our design to find the middle road between safety and usefulness. One of the hacks in Matlab, because it only has matrices, is that vectors are also 1xN and Nx1 matrices. In Maple, these are different types. This helps a lot, because then a dot product is a function from R^n x R^n -> R. 1x1 matrices are not reals in Maple. Nor are Nx1 matrices Vectors.
If we instead distinguish row and column vectors because we treat them as matrices, then the quadratic form x^T * A * x denotes a 1x1 matrix, not a real.
But if you consider x to be a vector without orientation, writing down x^T is *completely meaningless*! If x is oriented, then x^T makes sense. Also, if x is oriented, then x^T * (A * x) = (x^T * A) * x. What is the meaning of (x * A) for a 'vector' x ? It gets much worse when the middle A is of the form B.C. To ensure that everything is as associative as can be, but no more, is very difficult. I don't have the time to go into all the details of why this design 'works' (and it does!), giving the 'expected' result to all meaningful linear algebra operations, but let me just say that this was the result of long and intense debate, and was the *only* design that actually allowed us to translate all of linear algebra idioms into convenient notation. Please believe me that this design was not arrived at lightly! Jacques _______________________________________________ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe