Back to Questions
1.1k
views4
comments
RowVector vs Vector
f#

Is there a particular reason we need to have a RowVector as opposed to just a Vector. Couldnt the RowVector simply be a module of functions as opposed to being a distinct type. I am currently writting some code which manipulates matrices and I find it very annoying that the column methods and row method emit two different types of vectors when they are backed by essentially the same record structure.

I would much prefer them to both sets of operations to emit vectors which can be processed by functinos in the Vector.Generic module or if there is some Row or Column specific stuff needed put those methods in seperate modules Matrix.RowVector / Matrix.ColumnVector

.

I can't speak for whoever designed those types, but from a mathematical point of view, row vectors and column vectors are two different things- for example, when you change the basis of your vector space, your row vectors and column vectors will transform differently (column vectors transform contravariantly, and row vectors transform covariantly, if I remember right), so I can see why maybe the designers made different types.

By on 3/22/2009 7:37 PM ()

a mathematical vector is neither a row nor a column "vector" it's a member of an vector space. If the field of the vector space is K than you can choose a basis for this vector space and get an representation of the vector as a n-tupel over K where n is the dimension of the space - but still this is neither a row nor a column.

So where does the row/col get interesting?

Well if you have a linear mapping between vector spaces you can get a representation of this homeomorphism just like you get one of the vectors by choosing fixed bases for your vectorspaces - this representation is what we know as a matrix - and now in order to use the matrix-multiplication for matrix*vektor or vektor*matrix you map the vector representation to a 1xn or nx1 matrix and these ar called row- or colum-"vectors".

So why choose row-vector?

Well because you only work with those represantations for a very fixed basis (the canonical one - basically the axis in the cartesien coordinate system) and want to use these with a matrix in the form M*v ;)

By on 3/23/2009 1:31 AM ()

I don't mean to be disagreeable, and it's been enough years since I've done math that I could get myself in trouble here, but there are most certainly two types of vectors- contravariant and covariant. If you have a differentiable manifold, and you want to talk about a vector like the velocity vector of a particle moving along the manifold, you are talking about a contravariant vector. That's because you're talking about an element of the tangent space of the manifold. Covariant vectors live in the cotangent space; those are linear operators on the tangent space. So covariant and contravariant vectors live in dual vector spaces, which means you can loosely talk about them the same way because the spaces have the same dimension, and you can represent the elements in each of them by ordered tuples of the same size. But the tuples mean different things- the tangent space and cotangent space have different bases.

The reason contravariant vectors are associated with columns and covariant vectors are associated with rows is really just an artifact of the way matrix multiplication is defined, along with the notational gimmick which allows you to say "I want to apply the linear transformation T to the vector v, and to do that I can just represent T as a matrix and write T*v = v' ."

By on 3/23/2009 6:02 AM ()

I don't mean to be disagreeable, and it's been enough years since I've done math that I could get myself in trouble here, but there are most certainly two types of vectors- contravariant and covariant. If you have a differentiable manifold, and you want to talk about a vector like the velocity vector of a particle moving along the manifold, you are talking about a contravariant vector. That's because you're talking about an element of the tangent space of the manifold. Covariant vectors live in the cotangent space; those are linear operators on the tangent space. So covariant and contravariant vectors live in dual vector spaces, which means you can loosely talk about them the same way because the spaces have the same dimension, and you can represent the elements in each of them by ordered tuples of the same size. But the tuples mean different things- the tangent space and cotangent space have different bases.

The reason contravariant vectors are associated with columns and covariant vectors are associated with rows is really just an artifact of the way matrix multiplication is defined, along with the notational gimmick which allows you to say "I want to apply the linear transformation T to the vector v, and to do that I can just represent T as a matrix and write T*v = v' ."

First: sorry I hit the wrong "reply"-Button the answer was not exactly to your message.

But anyway: you are right - BUT as you said yourself you are talking about vectors in different vectorspaces in a very special situation. So here you introduce "covariant" and "contravariant" to make talking about those things easier.

In the general case of a vectorspace those labels make IMHO no sense.

By on 3/23/2009 11:12 PM ()

    Topic tags

    IntelliFactory Offices Copyright (c) 2011-2012 IntelliFactory. All rights reserved.
    Home | Products | Consulting | Trainings | Blogs | Jobs | Contact Us | Terms of Use | Privacy Policy | Cookie Policy     		Built with WebSharper