LinearDependence(S, R)

lindep.spad line 1 [edit on github]

• S : IntegralDomain
• R : LinearlyExplicitOver(S)

Test for linear dependence.

linearDependence : Vector(R) -> Union(Vector(S), "failed")

linearDependence([v1, ..., vn]) returns [c1, ..., cn] if c1*v1 + ... + cn*vn = 0 and not all the ci's are 0, "failed" if the vi's are linearly independent over S.

linearlyDependent? : Vector(R) -> Boolean

linearlyDependent?([v1, ..., vn]) returns true if the vi's are linearly dependent over S, false otherwise.

particularSolution : (Matrix(R), Vector(R)) -> Union(Vector(S), "failed") if S has Field

particularSolution([v1, ..., vn], u) returns [c1, ..., cn] such that c1*v1 + ... + cn*vn = u, "failed" if no such ci's exist in S.

particularSolution : (Vector(R), R) -> Union(Vector(S), "failed") if S has Field

particularSolution([v1, ..., vn], u) returns [c1, ..., cn] such that c1*v1 + ... + cn*vn = u, "failed" if no such ci's exist in S.

particularSolution : (Matrix(R), Vector(R)) -> Union(Vector(Fraction(S)), "failed") if S hasn't Field

particularSolution([v1, ..., vn], u) returns [c1, ..., cn] such that c1*v1 + ... + cn*vn = u, "failed" if no such ci's exist in the quotient field of S.

particularSolution : (Vector(R), R) -> Union(Vector(Fraction(S)), "failed") if S hasn't Field

particularSolution([v1, ..., vn], u) returns [c1, ..., cn] such that c1*v1 + ... + cn*vn = u, "failed" if no such ci's exist in the quotient field of S.

solveLinear : (Matrix(R), Vector(R)) -> Record(particular : Union(Vector(S), "failed"), basis : List(Vector(S))) if S has Field

solveLinear([v1, ..., vn], u) returns solution of the system c1*v1 + ... + cn*vn = u and and a basis of the associated homogeneous system c1*v1 + ... + cn*vn = 0

solveLinear : (Vector(R), R) -> Record(particular : Union(Vector(S), "failed"), basis : List(Vector(S))) if S has Field

solveLinear([v1, ..., vn], u) returns solution of the system c1*v1 + ... + cn*vn = u and and a basis of the associated homogeneous system c1*v1 + ... + cn*vn = 0

solveLinear : (Matrix(R), Vector(R)) -> Record(particular : Union(Vector(Fraction(S)), "failed"), basis : List(Vector(Fraction(S)))) if S hasn't Field

solveLinear([v1, ..., vn], u) returns solution of the system c1*v1 + ... + cn*vn = u and and a basis of the associated homogeneous system c1*v1 + ... + cn*vn = 0

solveLinear : (Vector(R), R) -> Record(particular : Union(Vector(Fraction(S)), "failed"), basis : List(Vector(Fraction(S)))) if S hasn't Field

solveLinear([v1, ..., vn], u) returns solution of the system c1*v1 + ... + cn*vn = u and and a basis of the associated homogeneous system c1*v1 + ... + cn*vn = 0