24.10 Matrices as Basis of a Row Space

returns a basis for the row space generated by the rows of mat in the form of an immutable matrix.

Does the same as BaseMat, with the difference that it may destroy the matrix mat. The matrix mat must be mutable.

gap> BaseMat(mat);
[ [ 1, 0, -1 ], [ 0, 1, 2 ] ]
gap> mm:= [[1,2,3],[4,5,6],[5,7,9]];; 
gap> BaseMatDestructive( mm );
[ [ 1, 0, -1 ], [ 0, 1, 2 ] ]
gap> mm;
[ [ 1, 0, -1 ], [ 0, 1, 2 ], [ 0, 0, 0 ] ]

Let V be the row space generated by the rows of mat (over any field that contains all entries of mat). BaseOrthogonalSpaceMat( mat ) computes a base of the orthogonal space of V.

The rows of mat need not be linearly independent.

performs Zassenhaus' algorithm to compute bases for the sum and the intersection of spaces generated by the rows of the matrices M1, M2.

returns a list of length 2, at first position a base of the sum, at second position a base of the intersection. Both bases are in semi-echelon form (see Echelonized matrices).

gap> SumIntersectionMat(mat,[[2,7,6],[5,9,4]]);
[ [ [ 1, 2, 3 ], [ 0, 1, 2 ], [ 0, 0, 1 ] ], [ [ 1, -3/4, -5/2 ] ] ]

find vectors extending mat to a basis spanning the span of bas. Both bas and mat must be matrices of full (row) rank. It returns a record with the following components:

subspace

is a basis of the space spanned by mat in upper triangular form with leading ones at all echelon steps and zeroes above these ones.

factorspace

is a list of extending vectors in upper triangular form.

factorzero

is a zero vector.

heads

is a list of integers which can be used to decompose vectors in the basis vectors. The ith entry indicating the vector that gives an echelon step at position i. A negative number indicates an echelon step in the subspace, a positive number an echelon step in the complement, the absolute value gives the position of the vector in the lists subspace and factorspace.

gap> BaseSteinitzVectors(IdentityMat(3,1),[[11,13,15]]);
rec(
  factorspace := [ [ 0, 1, 15/13 ], [ 0, 0, 1 ] ],
  factorzero := [ 0, 0, 0 ],
  subspace := [ [ 1, 13/11, 15/11 ] ],
  heads := [ -1, 1, 2 ] )

See also chapter Integral Matrices and Lattices

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