Solutions of Systems of Linear Equations

IsConsistent(A, w): ModMatRngElt, ModTupRng -> BoolElt, ModTupRngElt, ModTupRng

Given a matrix \(A\) belonging to \(M_n(R)\) and a vector \(w\) belonging to the tuple module \(R^{(n)}\), return true iff the system of linear equations \(v*A = w\) is consistent. If the system is consistent, then the function will also return:

(a)

A particular solution \(v\);

(b)

The kernel \(K\) of \(A\) so that \((v + k) * A = w\) for \(k\in K\).

IsConsistent(A, W): ModMatRngElt, [ ModTupRng ] -> BoolElt, [ ModTupRngElt ], ModTupRng

Given a matrix \(A\) belonging to \(M_n(R)\) and a sequence \(W\) of vectors belonging to the tuple module \(R^{(m)}\), return true iff the system of linear equations \(V[i]*A = W[i]\) for each \(i\) is consistent. If the systems are all consistent, then the function will also return:

(a)

A solution sequence \(V\);

(b)

The kernel \(K\) of \(A\) so that \((V[i] + k) * A = W[i]\) for \(k\in K\).

Solution(A, w): ModMatRngElt, ModTupRng -> ModTupRngElt, ModTupRng

Given a matrix \(A\) belonging to \(M_n(R)\) and a vector \(w\) belonging to the tuple module \(R^{(n)}\), solve the system of linear equations \(v*A = w\). The function returns two values:

(a)

A particular solution \(v\);

(b)

The kernel \(K\) of \(A\) so that \((v + k) * A = w\) for \(k\in K\).

Solution(A, W): ModMatRngElt, [ ModTupRng ] -> [ ModTupRngElt ], ModTupRng

Given a matrix \(A\) belonging to \(M_n(R)\) and a sequence \(W\) of vectors belonging to the tuple module \(R^{(n)}\), solve the system of linear equations \(V[i]*A = W[i]\) for each \(i\). The function returns two values:

(a)

A solution sequence \(V\);

(b)

The kernel \(K\) of \(A\) so that \((V[i] + k) * A = W[i]\) for \(k\in K\).