radionica_vektori_matrice -- Sage

automatic_names(True)

a=[1,2,3] b=[6,-4,2]

a+b

2*a

a-b

a*b

a[1]=-10

c=(1,2,3) d=(6,-4,2)

c+d

c-d

c[1]=-9

Vektori

help(vector)

a=vector((1,2,3)) b=vector([6,-4,2])

a.parent()

a1=vector(QQ,(1,2,3))

a1.parent()

a+b

2*a

a-b

2/3*a-5*b

a.norm()

norm(a)

a.dot_product(b)

a.cross_product(b)

def SP(v1,v2): v1=vector(v1) v2=vector(v2) return v1.dot_product(v2)

SP((1,2,3),[0,-2,1])

SP(a,b)

$\vec{a}\times(\vec{b}+\vec{c})=\vec{a}\times\vec{b}+\vec{a}\times\vec{c}$

a=vector((a_x,a_y,a_z)) b=vector((b_x,b_y,b_z)) c=vector((c_x,c_y,c_z))

lijeva_strana=a.cross_product(b+c) desna_strana=a.cross_product(b)+a.cross_product(c)

lijeva_strana

desna_strana

lijeva_strana-desna_strana

rez=lijeva_strana-desna_strana

rez

rez.apply_map(lambda t:expand(t))

Odredite $m\in\mathbb{R}$ tako da vektor $\vec{a}$ zatvara jednake kutove s vekltorima $\vec{b}$ i $\vec{c}$

a=vector((2*m,1,1-m)) b=vector((-1,3,0)) c=vector((5,-1,8))

jednadzba=a.dot_product(b)/(norm(a)*norm(b))==a.dot_product(c)/(norm(a)*norm(c)) jednadzba

solve(jednadzba,m)

vektor nad simboličkim prstenom

v=vector((2^(1/3),-4,1))

v.parent()

vektor nad $\mathbb{R}$

w=vector(RR,(2^(1/3),-4,1))

w1=vector(RealField(200),(2^(1/3),-4,1))

vektor definiran preko formule

vector(i^2 for i in xrange(5))

vektori nad konačnim poljem

v1=vector(GF(3),(1,1,0)) v2=vector(GF(3),(2,1,2))

v1+v2

3*v1

5*v1

vector(GF(3),(i^2 for i in xrange(5)))

Matrice

help(matrix)

A=matrix([[2,3,4],[-1,-5,1]]) B=matrix([[8,5,2],[0,-2,7]])

A1=matrix(2,3,[2,3,4,-1,-5,1])

A[0,0]

A[1,1]

A+B

A-B

A*B

C=matrix([[3,1],[6,2],[-5,4]])

A*C

-3/2*A

A.T

A.apply_map(sin)

A.apply_map(lambda t:t-cos(t))

A.parent()

A2=matrix(QQ,[[2,3,4],[-1,-5,0]])

A3=matrix(QQ,2,3,[2,3,4,-1,-5,0])

A2.parent()

matrice nad prstenom

A=matrix([[2,3,15],[0,16,22]]) B=matrix([[19,25,18],[19,23,7]])

A,B,A+B,A-B

A=matrix(Integers(26),[[2,3,15],[0,16,22]]) B=matrix(Integers(26),[[19,25,18],[19,23,7]])

A,B,A+B,A-B

matrice definirane formulom

matrix(3,4,lambda i,j:i^2-j^2)

matrix(3,4,lambda i,j:(i+1)^2-(j+1)^2)

matrix(GF(3),3,4,lambda i,j:(i+1)^2-(j+1)^2)

matrix(3,4,lambda i,j: cos((i+1)*pi/2) if i>j else log(i+j+2,2))

Ispisivanje velikih matrica

M=matrix(15,30,lambda i,j:(i-j)/(i+j+1))

print M.str()

f=open('M.txt','w') f.write(M.str()) f.close()

M2=matrix(300,200,lambda i,j:2*i-j)

f=open('M2.txt','w') f.write(M2.str()) f.close()

Kvadratne matrice

A=matrix([[5,3],[19,1]]) B=matrix(Integers(26),[[5,3],[19,1]])

A.det()

B.det()

A^-1

B^-1

det(A)

A2=matrix([[14,3],[25,1]]) B2=matrix(Integers(26),[[14,3],[25,1]])

A2.det(),B2.det()

A2^-1

B2^-1

C=matrix(Integers(26),[[3,2],[1,5]])

C.det()

C^-1

reset('a')

det(matrix(5,5,[a,a,a,a,a,-2,a,a,a,a,2,0,a,a,a,-4,0,0,a,a,4,0,0,0,a]))

svojstvene vrijednosti

A=matrix([[5,-6,-6],[-1,4,2],[3,-6,-4]]);A

A.charpoly()

A.charpoly(t)

A.fcp()

A.minpoly()

A.minpoly().factor()

A.eigenvalues()

A.eigenvectors_right()

A.eigenspaces_right()

A.eigenspaces_right(algebraic_multiplicity=True)

D,P=A.eigenmatrix_right()

D,P

P^-1*A*P

P^-1*A*P==D

A.eigenspaces_left()

matrix([[1,-2,-2]])*A

M=matrix(QQ,[[0,-1,0],[1,0,0],[0,0,2]])

M.eigenvalues()

M.parent()

M.eigenvalues(extend=False)

jezgra i slika

A=matrix([[2,3,5,-3,-2],[3,4,3,-1,-3],[5,6,-1,3,-5]]) B=matrix(QQ,[[2,3,5,-3,-2],[3,4,3,-1,-3],[5,6,-1,3,-5]])

A.right_kernel()

B.right_kernel()

A.echelon_form()

B.echelon_form()

A.rank()

A.right_nullity()

B1=matrix([[1,2,-1,3,4],[2,4,-2,6,8],[1,3,2,2,6],[1,4,5,1,8],[2,7,3,3,9]]) B2=matrix(QQ,[[1,2,-1,3,4],[2,4,-2,6,8],[1,3,2,2,6],[1,4,5,1,8],[2,7,3,3,9]])

B1.echelon_form()

B2.echelon_form()

Vektorski prostori

V=VectorSpace(QQ,3);V

V1=span([[1,0,0],[0,1,0],[0,0,1]],QQ);V1

V.basis()

V.basis_matrix()

V.dimension()

vector((1,3,2)) in V, vector((1,2/3,1.41)) in V, vector((1,2/3,2^(1/3))) in V, vector((1,2/3,1.41,5)) in V

promjena baze u vektorskom prostoru

V=V.span_of_basis([[1,0,0],[1,2,0],[3,4,5]]);V

V.basis()

V.coordinates((1,1,4))

još vektorskih prostora

U=VectorSpace(RealField(200),4);U

U.basis_matrix()

C=VectorSpace(CC,4);C

C.basis_matrix()

C1=VectorSpace(ComplexField(200),3);C1

C1.basis_matrix()

W=VectorSpace(SR,5)

W.basis_matrix()

vector((1,3,2/3,5.67,1)) in W, vector((2,5^(2/3),1,0.33,2.2)) in W, vector((2,sin(3),7,2.1,log(2,3))) in W,vector((3,3+I,2,1,0)) in W

T=VectorSpace(QQbar,3);T

vektorski prostori nad konačnim poljem

GF(3).list()

K=VectorSpace(GF(3),4);K

K.list()

K.cardinality()

generiranje potprostora

U=span([[1,2,-1,3],[2,4,1,-2],[3,6,3,-7]],QQbar) W=span([[1,2,-4,11],[2,4,-5,14]],QQbar)

U==W

U1=(QQbar^4).subspace([[1,2,-1,3],[2,4,1,-2],[3,6,3,-7]]) W1=(QQbar^4).subspace([[1,2,-4,11],[2,4,-5,14]])

W2=(QQbar^4).subspace_with_basis([[1,2,-4,11],[2,4,-5,14]]);W2

W2.basis()

U2=(QQbar^4).subspace_with_basis([[1,2,-1,3],[2,4,1,-2],[3,6,3,-7]])

radionica_vektori_matrice