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

## Synopsis

• Usage:
M = affineGeometry(n, p)
• Inputs:
• n, an integer, the dimension of the ambient vector space
• p, an integer, a prime
• Outputs:
• , the affine geometry of rank n+1 over F_p

## 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 : representationOf 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