The MatBasis function computes a basis of the set of vectors in the first nrow rows of A. Optionally, a basis for the nullspace can also be computed. On successful completion, the number of rows in A containing vectors (labeled r) is returned (the dimension of the basis).

Computation of the basis does not require that m be a prime, but computation of the nullspace does. In some cases, it is possible to compute the nullspace with m composite. If this is not possible, the function returns an error indicating that the algorithm failed because m is composite.

To request a nullspace, set nullflag=true. If the nullspace is requested, the input Matrix must have at least as many rows as columns, and the basis of the nullspace is specified in the trailing $n-r+1..n$ rows of the Matrix, where $n$ is the number of columns, and $r$ is the return value (the dimension of the basis for the input vectors).

This command is part of the LinearAlgebra[Modular] package, so it can be used in the form MatBasis(..) only after executing the command with(LinearAlgebra[Modular]).  However, it can always be used in the form LinearAlgebra[Modular][MatBasis](..).

with(LinearAlgebra[Modular]):

p := 2741;

$p≔2741$

A := Mod(p,Matrix(5,5,(i,j)->rand()),integer[]):

Fill(p,A,4..5):

A;

$\left[\begin{array}{ccccc}2543& 1568& 127& 356& 581\\ 430& 1549& 2376& 1511& 1839\\ 164& 1946& 211& 49& 2418\\ 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0\end{array}\right]$

r := MatBasis(p,A,3,true):

A,r;

$\left[\begin{array}{ccccc}1& 0& 0& 1972& 1878\\ 0& 1& 0& 1578& 1735\\ 0& 0& 1& 1724& 2166\\ 769& 1163& 1017& 1& 0\\ 863& 1006& 575& 0& 1\end{array}\right],3$

Multiply(p,A,1..r,A,r+1..5,'transpose');

$\left[\begin{array}{cc}0& 0\\ 0& 0\\ 0& 0\end{array}\right]$

A := Mod(p,Matrix(5,5,(i,j)->rand()),float[8]):

for i to 5 do
   A[i,4] := modp(trunc(p-A[i,1]),p);
   A[i,5] := modp(2*trunc(p-A[i,3]),p);
end do:
A;

$\left[\begin{array}{ccccc}2635.& 353.& 2657.& 106.& 168.\\ 2587.& 1857.& 827.& 154.& 1087.\\ 1720.& 1181.& 493.& 1021.& 1755.\\ 2209.& 884.& 1207.& 532.& 327.\\ 26.& 2325.& 518.& 2715.& 1705.\end{array}\right]$

r := MatBasis(p,A,5,true):

A,r;

$\left[\begin{array}{ccccc}1.& 0.& 0.& 2740.& 0.\\ 0.& 1.& 0.& 0.& 0.\\ 0.& 0.& 1.& 0.& 2739.\\ 1.& 0.& 0.& 1.& 0.\\ 0.& 0.& 2.& 0.& 1.\end{array}\right],3$

Multiply(p,A,1..r,A,r+1..5,'transpose');

$\left[\begin{array}{cc}0.& 0.\\ 0.& 0.\\ 0.& 0.\end{array}\right]$