[Haskell-cafe] matrix computations based on the GSL
Benjamin Franksen
benjamin.franksen at bessy.de
Thu Jun 30 17:23:56 EDT 2005
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
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