On Wednesday 29 June 2005 22:54, Henning Thielemann wrote: > On Wed, 29 Jun 2005, Dan Piponi wrote: > > >On Wed, 29 Jun 2005, Jacques Carette wrote: > > > > > >Distinction of row and column vectors is a misconcept > > > > Row and column vectors are sometimes worth distinguishing because > > they can represent entirely different types of object. For example, > > if a column vector represents an element of a vector space V over a > > field F, then row vectors can be used to represent elements of the > > dual space, V* = {f:V->F, f linear}. Quite different objects and in > > some applications it makes sense to distinguish them. > > If f is a function > f :: a -> a > then, of course, the dual space is again a vector space. But it > contains functionals and they have very different type, namely > f' :: (a -> a) -> a > If dual spaces would represent the concept of transposition then the > dual space of the dual space should be original space. It is not, it > can even not be identified in many cases, only if the space is > reflexive. f'' :: ((a -> a) -> a) -> a > has certainly a type very different from (a -> a)!
IIRC, finite-dimensional spaces are always reflexive, as witnessed by the identification of elements of the dual space f with the vector y in f x = <x,y> (real scalars, here), where the identification is, of course, with respect to a given basis. I bet you know all this very well! Ben _______________________________________________ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe