%load_ext sage

2-5

-3

C = matrix( [ [1,0,0], [1,0,0], [1,0,0] ])

print C

[1 0 0]
[1 0 0]
[1 0 0]

C[1,0]=2

C.nullity()

2

C.kernel().basis()

[
(1, 0, -1),
(0, 1, -2)
]

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]

a.transpose().kernel().basis()

[
(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)
]

a.rank()

7

print a[12,0]

0

C.transpose().kernel().basis()

[
(0, 1, 0),
(0, 0, 1)
]

a.transpose().kernel().basis()

[
(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)
]

print type(b)

<class 'sage.structure.sequence.Sequence_generic'>

d=matrix(b)

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]