Henning Thielemann wrote:
On Fri, 8 Jul 2005, Keean Schupke wrote:
So the linear operator is translation (ie: + v)... effectively 'plus' could be viewed as a function which takes a vector and returns a matrix (operator)
(+) :: Vector -> Matrix
Since a matrix _is_ not a linear map but only its representation, this would not make sense. As I said (v+) is not a linear map thus there is no matrix which represents it. A linear map f must fulfill f 0 == 0
But since v+0 == v the function (v+) is only a linear map if 'v' is zero.
I can't see how to fit in your vector extension by the 1-component.
Eh? Translation is a linear operation no? Adding vectors translates the first by the second (or the second by the first - the two are isomorphic)... A translation can be represented by the matrix: 1 0 0 0 0 1 0 0 0 0 1 0 dx dy dz 1 So the result of "v+" is this matrix. In other words this matrix is the 'vector addition operator'... providing you pad the vectors with an additional '1' at the end. So if: translate :: Vector -> Matrix [x,y,z,1] = [[1,0,0,0],[0,1,0,0],[0,0,1,0],[x,y,z,1]] we can create a matrix representing translation from: translate [3,4,5,1] and can apply this translation to another vector: mapply (translate [3,4,5,1]) [2,3,4,1] = [5,7,9,1] All I was saying is that following this, partial application of vector addition: [3,4,5] + [2,3,4] = [5,7,9] but partially applying: ([3,4,5] +) would be a the matrix defined above as (translate [3,4,5,1]) ... Of course this has the drawback that you need an extra dimension in you vectors and matrices to cope with translation. Anyway I have more or less convinced myself that separating vectors and matrices is the right thing to do... I was just trying to define vector addition in terms of a matrix operation for neatness. Keean.