The Basis function computes a basis of the set of Vectors described by A, and returns that basis and the nullspace.

The orient parameter is specified if A is a mod m Matrix.  It indicates to the Basis command whether the vectors of the input Matrix are in rows (orient=row) or columns (orient=column).

The basis and nullsp parameters indicate the format in which the respective Vectors are returned by Basis. They can be any of the following values:

If neither basis or nullsp are false, both the basis and nullspace Vectors are returned as a two element sequence. If both are false, an error results.

For example, if Basis is called with basis=false and nullsp=column, the output would be a list of column Vectors for the nullspace.

The Basis command works with a copy of the input data, using the MatBasis command to perform the computation, so if data storage or efficiency are a concern, the MatBasis command should be used.

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

$\mathrm{with}\left(\mathrm{LinearAlgebra}\left[\mathrm{Modular}\right]\right)\:$

$p≔2741$

$p≔2741$

$A≔\left\{\mathrm{seq}\left(\mathrm{Mod}\left(p\,\mathrm{Vector}\left[\mathrm{column}\right]\left(5\,i\mapsto \mathrm{rand}\left(\right)\right)\,\mathrm{float}\left[8\right]\right)\,i=1..3\right)\right\}$

$A≔\left\{\left[\begin{array}{c}164.\\ 1946.\\ 211.\\ 49.\\ 2418.\end{array}\right]\,\left[\begin{array}{c}2543.\\ 1568.\\ 127.\\ 356.\\ 581.\end{array}\right]\,\left[\begin{array}{c}430.\\ 1549.\\ 2376.\\ 1511.\\ 1839.\end{array}\right]\right\}$

$B,N≔\mathrm{Basis}\left(p\,A\,\mathrm{column}\,\mathrm{row}\right)$

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

$\mathrm{seq}\left(\mathrm{seq}\left(\mathrm{Multiply}\left(p\,i\,j\right)\,i=N\right)\,j=B\right)$

$0.,0.,0.,0.,0.,0.$

$A≔\mathrm{Mod}\left(p\,\mathrm{Matrix}\left(3\,5\,\left(i\,j\right)\mapsto \mathrm{rand}\left(\right)\right)\,\mathrm{integer}\left[\right]\right)$

$A≔\left[\begin{array}{ccccc}30& 1480& 754& 1049& 423\\ 1980& 2533& 1439& 67& 2051\\ 2635& 353& 2657& 1617& 2198\end{array}\right]$

$N≔\mathrm{Basis}\left(p\,A\,\mathrm{row}\,\mathrm{false}\,\mathrm{column}\right)$

$N≔\left[\left[\begin{array}{c}1279\\ 1956\\ 2481\\ 1\\ 0\end{array}\right]\,\left[\begin{array}{c}327\\ 2464\\ 701\\ 0\\ 1\end{array}\right]\right]$

$\mathrm{seq}\left(\mathrm{Multiply}\left(p\,A\,i\right)\,i=N\right)$

$\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right],\left[\begin{array}{c}0\\ 0\\ 0\end{array}\right]$

$N≔\mathrm{Basis}\left(p\,A\,\mathrm{row}\,\mathrm{false}\,\mathrm{Matrix}\left[\mathrm{column}\right]\right)$

$N≔\left[\begin{array}{cc}1279& 327\\ 1956& 2464\\ 2481& 701\\ 1& 0\\ 0& 1\end{array}\right]$

$\mathrm{Multiply}\left(p\,A\,N\right)$

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