In [1]:
%load_ext sage
In [2]:
2-5
Out[2]:
-3
In [3]:
C = matrix( [ [1,0,0], [1,0,0], [1,0,0] ])
In [4]:
print C

[1 0 0]
[1 0 0]
[1 0 0]
In [5]:
C[1,0]=2
In [6]:
C.nullity()
Out[6]:
2
In [7]:
C.kernel().basis()
Out[7]:
[
(1, 0, -1),
(0, 1, -2)
]
In [8]:
a = matrix(
[
[1, 0, 0, 0, -1, 1, 1, 0, 0, 0, 0],
[1, 0, 0, 0, 0, 0, 0, 1, 1, -1, 0],
[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1],
[0, 1, 0, 0, -1, 1, 0, 0, 0, 0, 0],
[0, 1, -1, 0, 0, 0, 0, 0, 1, 0, 0],
[0, 0, -1, 1, 0, 0, 1, 0, 0, 0, 0],
[0, 0, 0, 1, 0, 0, 0, 1, 0, -1, 0],
[0, 1, 0, 0, 0, 0, -1, 0, 0, 0, 1],
[0, 0, 0, 0, 1, 0, -1, 1, 0, 0, 0],
[0, 0, 0, 0, 0, 1, 0, 1, 0, 0, -1],
[0, 0, -1, 0, 1, 0, 0, 0, 0, 1, 0],
[0, 0, 0, 0, 0, 1, 0, 0, -1, 1, 0],
[0, 0, 0, 1, 0, 0, 0, 0, -1, 0, 1]
]
)
print a

[ 1  0  0  0 -1  1  1  0  0  0  0]
[ 1  0  0  0  0  0  0  1  1 -1  0]
[ 1  0  0  0  0  0  0  0  0  0  1]
[ 0  1  0  0 -1  1  0  0  0  0  0]
[ 0  1 -1  0  0  0  0  0  1  0  0]
[ 0  0 -1  1  0  0  1  0  0  0  0]
[ 0  0  0  1  0  0  0  1  0 -1  0]
[ 0  1  0  0  0  0 -1  0  0  0  1]
[ 0  0  0  0  1  0 -1  1  0  0  0]
[ 0  0  0  0  0  1  0  1  0  0 -1]
[ 0  0 -1  0  1  0  0  0  0  1  0]
[ 0  0  0  0  0  1  0  0 -1  1  0]
[ 0  0  0  1  0  0  0  0 -1  0  1]
In [9]:
a.transpose().kernel().basis()
Out[9]:
[
(1, 0, 0, 1, 0, 0, -1, -1, 0, 0, -1),
(0, 1, 0, -1, 0, -1, 1, 1, -1, 0, 0),
(0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0),
(0, 0, 0, 0, 1, 1, 0, -1, 0, -1, 0)
]
In [10]:
a.rank()
Out[10]:
7
In [11]:
print a[12,0]

0
In [12]:
C.transpose().kernel().basis()
Out[12]:
[
(0, 1, 0),
(0, 0, 1)
]
In [15]:
a.transpose().kernel().basis()
Out[15]:
[
(1, 0, 0, 1, 0, 0, -1, -1, 0, 0, -1),
(0, 1, 0, -1, 0, -1, 1, 1, -1, 0, 0),
(0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 0),
(0, 0, 0, 0, 1, 1, 0, -1, 0, -1, 0)
]
In [17]:
print type(b)

<class 'sage.structure.sequence.Sequence_generic'>
In [18]:
d=matrix(b)
In [19]:
print d

[ 1  0  0  1  0  0 -1 -1  0  0 -1]
[ 0  1  0 -1  0 -1  1  1 -1  0  0]
[ 0  0  1  1  0  0  0  0  1  1  0]
[ 0  0  0  0  1  1  0 -1  0 -1  0]
In [ ]: