The Linsolve function is a placeholder for representing the solution x to the linear system $Ax=b$.

The call Linsolve(A,b) mod n computes the solution vector b if it exists of the linear system $Ax=b$ over a finite ring of characteristic n. This includes finite fields, $\mathrm{GF}\left(p\right)$, the integers mod p, and $\mathrm{GF}\left(p^k\right)$ where elements of $\mathrm{GF}\left(p^k\right)$ are expressed as polynomials in RootOfs.

If an optional third parameter r is specified, and it is a name, it is assigned the rank of the matrix A.

A linear system with an infinite set of solutions will be parametrized in terms of variables.  Maple uses the global names _t[1], _t[2], ...  are used by default.  If an optional fourth parameter t is specified, and it is a name, the names t[1], t[2], etc. will be used instead.

$A≔\mathrm{Matrix}\left(\left[\left[1\,2\,3\right]\,\left[1\,3\,0\right]\,\left[1\,4\,3\right]\right]\right)$

$A≔\left[\begin{array}{ccc}1& 2& 3\\ 1& 3& 0\\ 1& 4& 3\end{array}\right]$

$b≔\mathrm{Vector}\left(\left[1\,2\,3\right]\right)$

$b≔\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right]$

$x≔\mathrm{Linsolve}\left(A\,b\right)\mathbf{mod}5$

$x≔\left[\begin{array}{c}4\\ 1\\ 0\end{array}\right]$

$A·x-b\mathbf{mod}5$

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

$x≔\mathrm{Linsolve}\left(A\,b\,'r'\,t\right)\mathbf{mod}6$

$x≔\left[\begin{array}{c}5+3t_3\\ 1+3t_3\\ t_3\end{array}\right]$

$r$

$2$

$A·x-b\mathbf{mod}6$

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

$\mathrm{alias}\left(a=\mathrm{RootOf}\left(y^4+y+1\right)\mathbf{mod}2\right)\:$

$A≔\mathrm{Matrix}\left(\left[\left[1\,a\,a^2\right]\,\left[1\,a^2\,1\right]\,\left[1\,a^3\,a^2\right]\right]\right)$

$A≔\left[\begin{array}{ccc}1& a& a^2\\ 1& a^2& 1\\ 1& a^3& a^2\end{array}\right]$

$b≔\mathrm{Vector}\left(\left[1\,a\,a^2\right]\right)$

$b≔\left[\begin{array}{c}1\\ a\\ a^2\end{array}\right]$

$x≔\mathrm{Linsolve}\left(A\,b\right)\mathbf{mod}2$

$x≔\left[\begin{array}{c}0\\ a^3+1\\ 0\end{array}\right]$

$z≔A·x-b\mathbf{mod}2$

$z≔\left[\begin{array}{c}1+a\left(a^3+1\right)\\ a^2\left(a^3+1\right)+a\\ a^3\left(a^3+1\right)+a^2\end{array}\right]$

$\mathrm{Expand}\left(\mathrm{convert}\left(z\,\mathrm{list}\right)\right)\mathbf{mod}2$

$\left[0\,0\,0\right]$