Sage.ipynb

Kernel: SageMath 7.6

R3 = VectorSpace(QQ, 3)
(b1, b2, b3) = R3.basis()
b1, b2, b3

vector1 = R3([-1, 2, 7])
vector2 = R3([4, -9, 2])
var('a b')
a * vector1 + b * vector2

(-a + 4*b, 2*a - 9*b, 7*a + 2*b)

sqrt(vector1 * vector1)
vector1.norm()
vector1.norm(2)
vector1.norm(Infinity)

7

2 * vector1
vector1 * vector2
vector1.dot_product(vector2)
vector1.cross_product(vector2)

(67, 30, 1)

u = vector(QQ, [1, 2/7, 10/3])
u[1]

2/7

u[:2]

(1, 2/7)

u[2] = numerical_approx(pi, digits=5)
u

(1, 2/7, 355/113)

M4 = MatrixSpace(QQ, 4)
show(M4.identity_matrix())

M34 = MatrixSpace(QQ, 3, 4)
show(M34.basis())

A = M4.matrix([[0, -1, -1, 1], [1, 1, 1, 1], [2, 4, 1, -2],
       [3, 1, -2, 2]])
b = vector(QQ, [0, 6, -1, 3])
show(A)
show(b)
A.solve_right(b)
A\b

(2, -1, 3, 2)

A[1]

(1, 1, 1, 1)

A[:,0]

[0]
[1]
[2]
[3]

A[:,1] = vector([1,1,1,0])
A

[ 0  1 -1  1]
[ 1  1  1  1]
[ 2  1  1 -2]
[ 3  0 -2  2]

S5 = SymmetricGroup(5)
S5.cardinality()
S5.list() #ciklička notacija!

[(),
 (1,2),
 (1,2,3,4,5),
 (1,3,4,5),
 (2,3,4,5),
 (1,3,5,2,4),
 (1,2,4)(3,5),
 (1,3,4,5,2),
 (1,3,5)(2,4),
 (1,4,2,5,3),
 (1,4)(2,3,5),
 (1,3)(2,5,4),
 (2,4)(3,5),
 (1,4,3)(2,5),
 (1,5,4,3,2),
 (1,4)(2,5,3),
 (1,5,3)(2,4),
 (1,4)(3,5),
 (1,4,2,3,5),
 (1,5,3,2,4),
 (1,3,2,5,4),
 (1,5,4,2),
 (1,2,5,4,3),
 (1,4,3,2,5),
 (2,5,4,3),
 (1,5,4,2,3),
 (1,4,3,2),
 (1,4,2)(3,5),
 (1,5,4,3),
 (1,5,3,2),
 (1,5,2,4,3),
 (2,5,3),
 (4,5),
 (2,3),
 (1,5),
 (2,4,3),
 (1,3,2),
 (3,4),
 (2,5,4),
 (1,5,4)(2,3),
 (1,5,2),
 (1,5,3),
 (1,5)(2,4,3),
 (1,4,2),
 (1,4,3),
 (1,5,4),
 (1,2,4,3),
 (1,2)(4,5),
 (1,4),
 (1,2,3,5),
 (1,2,5,3),
 (1,2)(3,4),
 (1,5)(2,3),
 (3,4,5),
 (1,2,4,5),
 (1,3),
 (2,3)(4,5),
 (1,2,5),
 (1,5)(3,4),
 (2,4),
 (2,3,4),
 (1,2,3),
 (1,2,5,4),
 (2,5),
 (1,4,5),
 (1,2,3,4),
 (1,2,3)(4,5),
 (1,5)(2,3,4),
 (1,4,5)(2,3),
 (1,5,2,3,4),
 (1,2,3,5,4),
 (1,3,4),
 (1,4,5,2),
 (2,4,5),
 (1,2,5)(3,4),
 (1,2)(3,4,5),
 (1,3,2,4,5),
 (1,3,2,5),
 (1,5,2)(3,4),
 (1,2,4,3,5),
 (1,3,5),
 (1,3,4,2),
 (1,3,4)(2,5),
 (1,3)(2,4,5),
 (1,5,2,3),
 (1,2,4),
 (1,3,2)(4,5),
 (2,3,5),
 (1,3)(4,5),
 (1,4)(2,5),
 (1,3,5,2),
 (2,5)(3,4),
 (3,5),
 (2,4,5,3),
 (1,3)(2,5),
 (1,2,5,3,4),
 (1,5,3,4),
 (1,3,5,4),
 (2,3,5,4),
 (1,4,3,5),
 (1,3)(2,4),
 (1,2,4,5,3),
 (2,4,3,5),
 (1,4,5,2,3),
 (1,5,2,4),
 (1,3,4,2,5),
 (1,3,2,4),
 (1,2)(3,5),
 (1,4,5,3,2),
 (2,5,3,4),
 (1,4,2,5),
 (1,4,3,5,2),
 (1,4,5,3),
 (1,4)(2,3),
 (1,3,5,4,2),
 (1,5)(2,4),
 (1,5,3,4,2),
 (1,4,2,3),
 (3,5,4),
 (1,2)(3,5,4)]

S3 = SymmetricGroup(3)
S3.cayley_table()

*  a b c d e f
 +------------
a| a b c d e f
b| b a f e d c
c| c e d a f b
d| d f a c b e
e| e c b f a d
f| f d e b c a

g = graphs.CompleteGraph(5)
show(g)