Sage Reference Manual: Matrices and Spaces of Matrices - Mirrors

Sage Reference Manual: Matrices and Spaces of Matrices, Release 6.1.1
[1 0 0 0 0 1]
[0 1 1 0 0 0]
[0 0 0 1 0 0]
[0 0 0 0 1 1]
sage: A*E.transpose() == zero_matrix(GF(2), 3, 4)
True
Since GF(2) is a field we can route this computation to the generic code and obtain the ‘pivot’ form of the
basis. The algorithm keywords, ‘pluq’, ‘default’ and unspecified, all have the same effect as there is
no optional behavior.
sage: A = matrix(GF(2),[[0, 1, 1, 0, 0, 0],
... [1, 0, 0, 0, 1, 1,],
... [1, 0, 0, 0, 1, 1]])
sage: P = A.right_kernel_matrix(algorithm=’generic’, basis=’pivot’); P
[0 1 1 0 0 0]
[0 0 0 1 0 0]
[1 0 0 0 1 0]
[1 0 0 0 0 1]
sage: A*P.transpose() == zero_matrix(GF(2), 3, 4)
True
sage: DP = A.right_kernel_matrix(algorithm=’default’, basis=’pivot’); DP
[0 1 1 0 0 0]
[0 0 0 1 0 0]
[1 0 0 0 1 0]
[1 0 0 0 0 1]
sage: A*DP.transpose() == zero_matrix(GF(2), 3, 4)
True
sage: A.right_kernel_matrix(algorithm=’pluq’, basis=’echelon’)
[1 0 0 0 0 1]
[0 1 1 0 0 0]
[0 0 0 1 0 0]
[0 0 0 0 1 1]
We test that the mod 2 code is called for matrices over GF(2).
sage: A = matrix(GF(2),[[0, 1, 1, 0, 0, 0],
... [1, 0, 0, 0, 1, 1,],
... [1, 0, 0, 0, 1, 1]])
sage: set_verbose(1)
sage: A.right_kernel(algorithm=’default’)
verbose ...
verbose 1 () computing right kernel matrix over integers mod 2 for 3x6 matrix
verbose 1 () done computing right kernel matrix over integers mod 2 for 3x6 matrix
...
Vector space of degree 6 and dimension 4 over Finite Field of size 2
Basis matrix:
[1 0 0 0 0 1]
[0 1 1 0 0 0]
[0 0 0 1 0 0]
[0 0 0 0 1 1]
sage: set_verbose(0)
Over Arbitrary Fields:
For kernels over fields not listed above, totally general code will compute a set of basis vectors in the pivot
format. These could be returned as a basis in echelon form.
244 Chapter 7. Base class for matrices, part 2