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affineGeometry -- affine geometry of rank n+1 over F_p

Synopsis

Description

The affine geometry of rank n+1 over F_p is the matroid whose ground set consists of all vectors in a vector space over F_p of dimension n, where independence is given by affine independence, i.e. vectors are dependent iff there is a linear combination equaling zero in which the coefficients sum to zero (equivalently, the vectors are placed in the hyperplane x_0 = 1 in a vector space of dimension n+1, with ordinary linear independence in the larger space).

i1 : M = affineGeometry(3, 2)

o1 = a "matroid" of rank 4 on 8 elements

o1 : Matroid
 
i2 : M === specificMatroid "AG32"

o2 = true
 
i3 : circuits M

o3 = {set {0, 1, 2, 3}, set {0, 1, 4, 5}, set {2, 3, 4, 5}, set {0, 2, 4, 6},
     ------------------------------------------------------------------------
     set {1, 3, 4, 6}, set {1, 2, 5, 6}, set {0, 3, 5, 6}, set {1, 2, 4, 7},
     ------------------------------------------------------------------------
     set {0, 3, 4, 7}, set {0, 2, 5, 7}, set {1, 3, 5, 7}, set {0, 1, 6, 7},
     ------------------------------------------------------------------------
     set {2, 3, 6, 7}, set {4, 5, 6, 7}}

o3 : List
 
i4 : getRepresentation M

o4 = | 1 1 1 1 1 1 1 1 |
     | 0 0 0 0 1 1 1 1 |
     | 0 0 1 1 0 0 1 1 |
     | 0 1 0 1 0 1 0 1 |

             ZZ 4      ZZ 8
o4 : Matrix (--)  <-- (--)
              2         2

See also

Ways to use affineGeometry:

For the programmer

The object affineGeometry is a method function.