Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
sage: A = MatrixSpace(IntegerRing(), 2)([1,2,3,4])
sage: A.row_module()
Free module of degree 2 and rank 2 over Integer Ring
Echelon basis matrix:
[1 0]
[0 2]
row_space(base_ring=None)
Return the row space of this matrix. (Synonym for self.row_module().)
EXAMPLES:
sage: t = matrix(QQ, 3, range(9)); t
[0 1 2]
[3 4 5]
[6 7 8]
sage: t.row_space()
Vector space of degree 3 and dimension 2 over Rational Field
Basis matrix:
[ 1 0 -1]
[ 0 1 2]
sage: m = Matrix(Integers(5),2,2,[2,2,2,2]);
sage: m.row_space()
Vector space of degree 2 and dimension 1 over Ring of integers modulo 5
Basis matrix:
[1 1]
rref(*args, **kwds)
Return the reduced row echelon form of the matrix, considered as a matrix over a field.
If the matrix is over a ring, then an equivalent matrix is constructed over the fraction field, and then row
reduced.
All arguments are passed on to :meth:echelon_form.
Note: Because the matrix is viewed as a matrix over a field, every leading coefficient of the returned
matrix will be one and will be the only nonzero entry in its column.
EXAMPLES:
sage: A=matrix(3,range(9)); A
[0 1 2]
[3 4 5]
[6 7 8]
sage: A.rref()
[ 1 0 -1]
[ 0 1 2]
[ 0 0 0]
Note that there is a difference between rref() and echelon_form() when the matrix is not over a
field (in this case, the integers instead of the rational numbers):
sage: A.base_ring()
Integer Ring
sage: A.echelon_form()
[ 3 0 -3]
[ 0 1 2]
[ 0 0 0]
250 Chapter 7. Base class for matrices, part 2