추가자료

예 2.6.11 - (1)

A = matrix(3, [1,2,3,-1,0,4,3,1,2])

inv_A = A^(-1)

print('inverse matrix of A')

print(inv_A)

예 2.6.11 - (2)

A = matrix( 3, [1,2,3,-1,0,4,3,1,2])

print(A.is_invertible())

예 2.6.11 - (3)

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

print(A.is_invertible())

예 2.6.12

A = matrix([[2, 3], [-1, 2]])

A_inv = A^(-1)

print("A^(-1):")

print(A_inv)

v = matrix([[8], [3]])

print("solution: ")

print(A_inv*v)

예 2.7.16 - (1)

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

P, L, U = A.LU()

print('P =')

print(P)

print('')

print('L =')

print(L)

print('')

print('U =')

print(U)

예 2.7.16 - (2)

B=matrix(RDF,[[2,-5,5], [1,-2, 3],[-1,3,0]])

P, L, U = B.LU()

print('P =')

print(P)

print('')

print('L =')

print(L)

print('')

print('U =')

print(U)

예 3.1.7

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

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

print('determinant of A')

print(A.det())

print('')

print('determinant of B')

print(B.det())

예 3.2.3

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

A1 = A.row(0)

A2 = A.row(1)

A3 = A.row(2)

A = matrix([A1, A2, A3])

print("A =")

print(A)

print('')

S12=matrix([A2,A1,A3])

print("S12 =")

print(S12)

print('')

S13=matrix([A3,A2,A1]);S13

print("S13 =")

print(S13)

print('')

print("det(A)=", A.det())

print("det(S12)=", S12.det())

print("det(S13)=", S13.det())

예 3.2.9

A = matrix([[-4, 5, 3], [12, -15, -9], [2, 5, 5]])

print("det(A)=", A.det())

예 3.2.13

A = matrix([[2, -3, 10], [1, 2, -2], [0, 1, -3]])

print("det(A)=", A.det())

예 3.2.18

A = matrix([[1, 2, -2, 24], [-1, 1, 0, 2], [0, 2, 0, 3], [3, 4, 0, -2]])

print("det(A)=", A.det())

예 3.2.19

A = matrix([[-3, 2, -3], [5, -4, 0], [2, -1, 6]])

print("det(A)=", A.det())

예 3.3.3

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

print(A.adjugate())

print('')

print(A.det())

print('')

print(A*A.adjugate())

예 3.4.2

A = matrix([[1, 2, 1], [2, 3, 1], [1, -1, 1]])

print("Area =", A.det()/2)

예 3.4.6

x, y , z = var("x y z")

A = matrix([[x, y, z, 1], [0, 1, 0, 1], [-1, 3, 2, 1], [0, 0, 1, 1]])

print( (A.det()).full_simplify(), " = " , 0)

예 4.2.12

v1 = vector([1,0])

v2 = vector([1/2, 1])

v = vector([0,1])

A = matrix([v1, v2]).transpose()

A_rref = A.augment(v).rref()

print('A_rref =')

print(A_rref)

print('')

print('solution :')

print(A_rref[:,-1])

예 4.2.14

v1=matrix(3, 1, [1, -1, 3])

v2=matrix(3,1,[2, 0, 1])

v3=matrix(3,1,[0, -2,6])

v=matrix(3,1,[12, -5, 20])

print('reduced row echelon form =')

sol = v1.augment(v2).augment(v3).augment(v).rref()

print(sol)

print('')

print('solution :')

print(sol[:,-1])

예 4.3.3

A = matrix([[2, 3, 8, -5], [-1, 2, 3, 4], [5, 4, 13, -14]])

print(A.rref())

print('')

print(A.transpose().rref())

예 4.3.4

A = matrix([[1, 1, 2], [0, 1, 2], [-1, 1, 2], [1, 3, 1]])

print(A.rref())

예 4.3.13

A=matrix(QQ, 5, 7, [[1,3,2,2,2, 7,4], [1, 2, 5, -1,3, 0,0],

[5,2,-1,4,4,4,4], [0,-2,5,5,5,5,5], [4, 2, 0,0,3,3,3]])

print('A =')

print(A)

print('')

print(A.right_kernel())

print("null(A)=", A.right_kernel().dimension())

print("rank(A)=", A.rank())

A.right_kernel().dimension()+A.rank()==7

예 5.1.9

def ip(f, g): return integral(f*g, x, 0, 1)

print( "d(1, x)=", sqrt(ip(1-x, 1-x)))

print( "d(1, x^2)=", sqrt(ip(1-x^2, 1-x^2)))

print( "d(1, x^3)=", sqrt(ip(1-x^3, 1-x^3)))

예 5.2.3

(1)

u = vector([5, -5, 5])

v = vector([1, -1, 1])

print(((u*v)/(v*v))*v)

(2)

u = vector([5, 5, 0])

v = vector([1, -1, 1])

print(((u*v)/(v*v))*v)

(3)

u = vector([5, 5, 5])

v = vector([1, -1, 1])

print(((u*v)/(v*v))*v)

예 5.2.6

u=vector([17, -12, 10])

v=vector([1, -1, 1])

print(((u*v)/(v*v))*v)

예 5.2.7

def ip(f, g): return integral(f*g, x, -1, 1)

(ip(x^2+2*x, x)/ip(x, x))*x

예 5.3.6

v1 = vector([2, 3, 2, -2])

v2 = vector([1, 0, 0, 1])

v3 = vector([-1, 0, 2, 1])

v4 = vector([-1, 2, -1, 1])

(v1*v2==0)and(v1*v3==0)and(v1*v4==0)and(v2*v3 ==0)and(v2*v4==0)and(v3*v4==0)

예 5.3.8

def coordinate_vector(v1,v2, v3, w): return v1*w, v2*w, v3*w

v1=vector([3/5, 4/5, 0])

v2=vector([-4/5, 3/5, 0])

v3=vector([0,0,1])

w=vector([5,-5,2])

coordinate_vector(v1,v2, v3, w)

예 5.4.2

(1)

v1=vector([1, 0])

v2=vector([2, 1])

w1=v1; print("w1=",w1)

w2= v2-((v2*w1)/(w1*w1))*w1; print("w2=", w2)

u1=w1/norm(w1); print("u1=", u1)

u2=w2/norm(w2); print("u2=", u2)

(2)

v1=vector([1, 1])

v2=vector([2, 1])

w1=v1; print("w1=",w1)

w2= v2-((v2*w1)/(w1*w1))*w1; print("w2=", w2)

u1=w1/norm(w1); print("u1=", u1)

u2=w2/norm(w2); print("u2=", u2)

예 5.4.5

(1)

v1=vector([2,4,0])

v2=vector([1,3,3])

w1=v1; print("w1=",w1)

w2= v2-((v2*w1)/(w1*w1))*w1; print("w2=", w2)

u1=w1/norm(w1); print("u1=", u1)

u2=w2/norm(w2); print("u2=", u2)

(2)

v2=vector([1,3,3])

A=matrix(2,3,[v1,v2])

[G, mu]=A.gram_schmidt() #mu*G=A

print("G:")

print(G)

print('mu:')

print(mu)

w1=G.row(0) # G의 1행

w2=G.row(1) # G의 2행

print('')

u1=G.row(0)/G.row(0).norm();print("u1=", u1)

u2=G.row(1)/G.row(1).norm();print("u2=", u2)

예 5.4.6

v1=vector([1,1,1,-1])

v2=vector([2,-1,-1,1])

v3=vector([0,3,3,-3])

v4=vector([-1,2,2,1])

A=matrix(4,4,[v1,v2, v3, v4])

[G, mu]=A.gram_schmidt(); print(G)

u1=G.row(0)/G.row(0).norm()

u2=G.row(1)/G.row(1).norm()

u3=G.row(2)/G.row(2).norm()

print(u1, u2, u3)

예 5.4.7

def ip(f, g): return integral(f*g, x, -1, 1)

v1=1

v2=x

v3=x*x

v4=x*x*x

w1=v1; print("w1=",w1)

w2= v2-(ip(v2,w1)/ip(w1,w1))*w1;

print("w2=", w2)

w3 = v3 - (ip(v3, w1)/ip(w1,w1))*w1 -(ip(v3,w2)/ip(w2,w2))*w2

print('w3=',w3)

w4 = v4 - (ip(v4, w1)/ip(w1,w1))*w1 -(ip(v4,w2)/ip(w2,w2))*w2 - (ip(v4,w3)/ip(w3,w3))*w3

print('w4=',w4)

예 5.5.7

v = vector([13, 2, 3, 7])

e1 = vector([1, 0, 0, 0])

e2 = vector([0, 1, 0, 0])

e3 = vector([0, 0, 1, 0])

proj = (v*e1)*e1 + (v*e2)*e2 + (v*e3)*e3

print(proj)

예 5.5.9

(1)

v1=vector([1, 0, 0])

v2=vector([0, 1, 0])

A=matrix(2,3,[v1,v2])

[G, mu]=A.gram_schmidt(); print(G)

u1=G.row(0)/G.row(0).norm()

u2=G.row(1)/G.row(1).norm()

v = vector([1, -2, 3])

proj = (v*u1)*u1 + (v*u2)*u2

print(proj)

(2)

v1=vector([0, 3, 1])

v2=vector([2, 0, 0])

A=matrix(2,3,[v1,v2])

[G, mu]=A.gram_schmidt(); print(G)

u1=G.row(0)/G.row(0).norm()

u2=G.row(1)/G.row(1).norm()

v = vector([1, 2, 3])

proj = (v*u1)*u1 + (v*u2)*u2

print(proj)

예 5.5.11

v1=vector([1, 1, 1])

v2=vector([1, 2, 3])

v = vector([0, 1, 4])

A=matrix(2,3,[v1,v2])

[G, mu]=A.gram_schmidt(); print(G)

u1=G.row(0)/G.row(0).norm()

u2=G.row(1)/G.row(1).norm()

proj = (v*u1)*u1 + (v*u2)*u2

print(proj)

B = matrix(3, 3, [v1, v2, proj]).transpose()

print(B)

print('')

print(B.rref())

예 5.6.3

n=2

A=matrix([[1,1],[1,2],[1,3]])

b=vector([0, 1, 4])

x=(A.transpose()*A).augment(A.transpose()*b).rref().column(n)

print(x)

예 5.6.4

n=2

A=matrix([[1,1],[0,-1],[2,0],[0,1]])

b=vector([2,1,0,1])

x=(A.transpose()*A).augment(A.transpose()*b).rref().column(n)

print(x)

print(A*x)

예 5.6.7

A=matrix(QQ, 6,3,[1,0,0,1,1,1,1,2,4,1,3,9,1,4,16,1,5,25])

b=matrix(QQ, 6,1,[51,73,94,114,132,152])

(A.transpose()*A).augment(A.transpose()*b).rref()

예 6.2.4

(1)

x, y = var('x y')

h(x, y) = [x-y, x+y]

T = linear_transformation(QQ^2, QQ^2, h) # define LT

C = T.matrix(side='right') # standard matrix

print(C)

print("\n")

(2)

k(x, y) = [2*x -y, x-2*y, x+y]

L = linear_transformation(QQ^2, QQ^3, k)

D = L.matrix(side= "right")

print(D)

예 6.4.3

x, y = var('x y')

h(x, y) = [2*x-y, x-2*y, x+y]

T = linear_transformation(QQ^2, QQ^3, h)

C = T.matrix(side='right')

print(C)

print("reduce echelon form:")

print(C.rref())

print("Nullspace:", T.kernel())

예 6.4.7

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

print("Image of a linear transformation:")

print(A.image())

print("Kernel of a linear transformation:")

print(A.right_kernel())

예 6.6.3

U = QQ^4

V = QQ^5

A=matrix(QQ, [[1,0,0,0], [0,1,0,0],[0,0,1,0],[0,0,0,1], [1,1/2,1/3,1/4]])

T = linear_transformation(U, V, A, side='right')

print (T.image())

print("\n")

print (T.kernel())

print("\n")

x0 = vector(QQ, [2, 0, -3, 10])

print (T(x0))

예 6.7.2

x,y,z=var('x y z')

A(x,y,z)=(2*x-z, y-2*z,x-2*z)

B(x,y,z)=(x+y, y+z)

T1=linear_transformation(QQ^3, QQ^3,A)

T2=linear_transformation(QQ^3, QQ^2,B)

C = T1.matrix(side='right')

D = T2.matrix(side='right')

print ("[T1]=")

print (C)

print ("[T2]=")

print (D)

print ("[T2*T1]=")

print (D*C)

예 6.8.3

v1=matrix(QQ, 2,1, [1,0])

v2=matrix(QQ, 2, 1, [1/2, 1])

w1=matrix(QQ, 2, 1, [1,0])

w2=matrix(QQ, 2, 1, [0, 1])

A=v1.augment(v2).augment(w1).augment(w2)

A.rref()

예 7.1.7

A = matrix( [[5, 3], [2,0]] )

f = A.charpoly()

print(f)

예 7.1.8

A = matrix( [[1, 2], [3, 4]] )

f = A.charpoly()

print(f)

예 7.1.9

A = matrix( [[1, 0, 0], [3, 1, 0], [1, 1, 2]] )

f = A.charpoly()

print(f)

예 7.1.12

A=matrix([[-2, 0, -6], [0, 1, 0], [-4, 1, 0]])

f = A.charpoly()

print(f)

solve(x^3 + x^2 - 26*x + 24 , x)

print(A.eigenvalues())

print(A.eigenvectors_right())

예 7.1.13

A=matrix([[2, 1, -1], [0,0,9], [0,-1, 6]])

print(A.eigenvalues())

print(A.eigenvectors_right())

B=matrix([[2, 1, -1], [0,4,2], [0,-2,0]])

print(B.eigenvalues())

print(B.eigenvectors_right())

예 7.1.17

A=matrix(3,[[0,-4,0],[1,4,0],[0,0,2]])

print(factor(A.charpoly()))

print(A.eigenvectors_right())

B=matrix([[2, -2, 5], [0, 2, 0], [0,0,2]])

print(factor(B.charpoly()))

print(B.eigenvectors_right())

C=matrix([[2, 5, -7], [0, 2, 1], [0,0,2]])

print(C.charpoly())

print(C.eigenvectors_right())

예 7.2.4

A=matrix([[1,3,0], [0,1,0], [0,0,2]])

print(A.eigenspaces_right())

A=matrix([[1,0,3], [0,1,0], [0,0,2]])

print(A.eigenspaces_right())

A=matrix([[1,0,0], [0,1,3], [0,0,2]])

print(A.eigenspaces_right())

A=matrix([[1,3,4], [0,1,5], [0,0,2]])

print(A.eigenspaces_right())

A=matrix([[1,0,4], [0,1,5], [0,0,2]])

print(A.eigenspaces_right())

예 7.3.1

A=matrix( [[-2,0,-6],[0,1,0],[-4,1,0]])

P=matrix([[1,-2,3],[0,-7,0],[-1,1,2]])

D=P.inverse()*A*P

print(D)

예 7.3.2

A=matrix( [[5,3], [2,0]])

P=matrix([[3,1],[1,-2]])

D=P.inverse()*A*P

print(D)

예 7.3.6

A=matrix( [[5,3,0,0,0], [2,0,0,0,0], [0,0,-2,0,-6], [0,0,0,1,0], [0,0,-4,1,0]])

A.eigenvectors_right()

P= matrix([[1,0,0,1,0],[1/3,0,0,-2,0],[0,1,1,0,1],[0,0,7/2,0,0],[0,-1,-1/2,0,2/3]])

D=P.inverse()*A*P

print(D)

예 7.4.6

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

A.eigenvectors_right()

v1=vector([1,0,1])

v2=vector([1,-2,0])

v3=vector([2,1,-2])

w1=v1

print("w1=",w1)

w2= v2-((v2*w1)/(w1*w1))*w1

print("w2=", w2)

u1=w1/norm(w1); print("u1=", u1)

u2=w2/norm(w2); print("u2=", u2)

u3=v3/norm(v3); print("u3=", u3)

예 7.4.7

A=matrix([[2,-3,0,0],[-3,2,0,0],[0,0,4,2],[0,0,2,7]])

A.eigenvectors_right()

v1=vector([0,0,1,2])

v2=vector([1,-1,0,0])

v3 = vector([0,0,1,-1/2])

v4 =vector([1,1,0,0])

A=matrix(4,4,[v1,v2,v3,v4])

[G, mu]=A.gram_schmidt(); print(G)

u1=G.row(0)/G.row(0).norm()

u2=G.row(1)/G.row(1).norm()

u3=G.row(2)/G.row(2).norm()

u4=G.row(3)/G.row(3).norm()

print(u1)

print(u2)

print(u3)

print(u4)

예 7.5.8

P=matrix(QQ, 3, [0.2,0.2,0.3,0.4,0.5,0.6,0.4,0.3,0.1])

P.eigenvectors_right()

예 7.5.13

(1)

B=matrix ( 5, [0, 1/2, 1/3, 0, 1/5, 1, 0, 1/3, 1/3, 1/5, 0, 1/2, 0, 1/3, 1/5, 0, 0, 1/3, 0, 1/5, 0, 0, 0, 1/3, 1/5])

N=matrix ( 5, [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1])

(85/100)*B + (15/500)*N

(2)

1단계

B=matrix ( 5, [0, 1/2, 1/3, 0, 1/5, 1, 0, 1/3, 1/3, 1/5, 0, 1/2, 0, 1/3, 1/5, 0, 0, 1/3, 0, 1/5, 0, 0, 0, 1/3, 1/5])

N=matrix ( 5, [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1])

G=(85/100)*B + (15/500)*N

G.eigenvectors_right()

2단계

v=vector([1, 5120/3709, 248601/285593, 1533/3709, 81049/285593])

(1/(1 + (5120/3709) + (248601/285593) + (1533/3709) + (81049/285593)))*v

예 8.1.1

(3+4*I).abs()

(3-4*I).abs()

예 8.1.2

z = 2+ 3*I

print(1/z)

w = z.conjugate()/(z*z.conjugate())

print(w)

예 8.2.9

v = vector([2-3*I, 2]); w = vector([5+2*I, 1 -3*I])

print(v*w.conjugate())

v = vector([5+2*I, 1-3*I]); w = vector([5+2*I, 1 -3*I])

print(v*w.conjugate())

예 8.3.2