Lineáris leképezések

42. feladat (χ)

A = matrix(QQ, [[1,1],[-1,1],[0,0]])
pretty_print("Im χ genrsz:",A,"→" ,A.rref())
pretty_print(" ")
pretty_print("Ker χ egyrsz:",A.transpose(),"→" ,A.transpose().rref())

42. feladat (ω)

A = matrix(QQ, [[1,0,1,-1],[-1,1,0,0],[0,-1,-1,1]])
pretty_print("Im ω genrsz:",A,"→" ,A.rref())
pretty_print(" ")
pretty_print("Ker ω egyrsz:",A.transpose(),"→" ,A.transpose().rref())

44. feladat (ζ)

A = matrix(GF(3), [[1,1,2,1],[2,1,0,2]])
pretty_print("Im ζ genrsz:",A,"→" ,A.rref())
pretty_print(" ")
pretty_print("Ker ζ egyrsz:",A.transpose(),"→" ,A.transpose().rref())

44. feladat (η)

A = matrix(GF(3), [[1,1,0],[1,2,2],[2,1,1],[1,0,1]])
pretty_print("Im η genrsz:",A,"→" ,A.rref())
pretty_print(" ")
pretty_print("Ker η egyrsz:",A.transpose(),"→" ,A.transpose().rref())

44. feladat (ϑ)

A = matrix(GF(5), [[3,1,2],[0,4,2],[0,1,3]])
pretty_print("Im 𝜗 genrsz:",A,"→" ,A.rref())
pretty_print(" ")
pretty_print("Ker 𝜗 egyrsz:",A.transpose(),"→" ,A.transpose().rref())

45. feladat (σ)

A = matrix([[1,2-i],[2,1],[3*i,1+3*i]])
pretty_print("Im σ genrsz:",A,"→" ,A.rref())
pretty_print(" ")
pretty_print("Ker σ egyrsz:",A.transpose(),"→" ,A.transpose().rref())

45. feladat (τ)

A = matrix([[1,1,0],[1,2,2],[2,1,1],[1,0,1]])
pretty_print("Im τ genrsz:",A,"→" ,A.rref())
pretty_print(" ")
pretty_print("Ker τ egyrsz:",A.transpose(),"→" ,A.transpose().rref())

45. feladat (υ)

A = matrix(QQ, [[1,1,-5],[0,4,-8],[0,1,-2]])
pretty_print("Im υ genrsz:",A,"→" ,A.rref())
pretty_print(" ")
pretty_print("Ker υ egyrsz:",A.transpose(),"→" ,A.transpose().rref())

54. feladat

Egy ℰ bázisban megadott vektor koordinátáinak kiszámítása az ℱ bázisban:

egyrsz = matrix(QQ, [[1,-2,-1],[2,1,8]])
egyrsz.subdivide(None,[2])
pretty_print(egyrsz,"→" ,egyrsz.rref())

Rajz:

f1 = arrow2d((0,0), (1,2), 4, color='blue')
f2 = arrow2d((0,0), (-2,1), 4, color='green')
v = arrow2d((0,0), (-1,8), 4, color='red')
w = arrow2d((0,0), (7,4), 4, color='orange')
f12 = arrow2d((0,0), (2,4), 4, color='blue')
f13 = arrow2d((0,0), (3,6), 4, color='blue')
f21 = arrow2d((3,6), (1,7), 4, color='green')
f22 = arrow2d((3,6), (-1,8), 4, color='green')
f2m1 = arrow2d((3,6), (5,5), 4, color='green')
f2m2 = arrow2d((3,6), (7,4), 4, color='green')
var('x,y')
t=implicit_plot(y==2*x, (x,-1,9), (y,-1,9), color='lightblue')
t+f1+f2+v+w
#t+f13+f12+f1+f22+f21+f2+v+w
#t+f13+f12+f1+f22+f21+f2m1+f2m2+f2+v+w

A leképezés mátrixa a bázisáttérés formulájával:

B = matrix(QQ, [[1,0],[0,-1]])
FE = matrix(QQ, [[1,2],[-2,1]])
pretty_print("[φ]_F:",B)
pretty_print(" ")
pretty_print("[F→E]:",FE)
pretty_print(" ")
pretty_print("[E→F]:",FE^(-1))
pretty_print(" ")
pretty_print("[φ]_E:",FE^(-1)*B*FE)

55. feladat

Egy ℰ bázisban megadott vektor koordinátáinak kiszámítása az ℱ bázisban:

egyrsz = matrix(QQ, [[1,-2,-1],[2,1,8]])
egyrsz.subdivide(None,[2])
pretty_print(egyrsz,"→" ,egyrsz.rref())

Rajz:

f1 = arrow2d((0,0), (1,2), 4, color='blue')
f2 = arrow2d((0,0), (-2,1), 4, color='green')
v = arrow2d((0,0), (-1,8), 4, color='red')
w = arrow2d((0,0), (3,6), 4, color='orange')
f12 = arrow2d((0,0), (2,4), 4, color='blue')
f13 = arrow2d((0,0), (3,6), 4, color='blue')
f21 = arrow2d((3,6), (1,7), 4, color='green')
f22 = arrow2d((3,6), (-1,8), 4, color='green')
var('x,y')
t=implicit_plot(y==2*x, (x,-1,9), (y,-1,9), color='lightblue')
t+v+w+f1+f2
#t+v+w+f13+f12+f1+f22+f21+f2

A leképezés mátrixa a bázisáttérés formulájával:

B = matrix(QQ, [[1,0],[0,0]])
FE = matrix(QQ, [[1,2],[-2,1]])
pretty_print("[φ]_F:",B)
pretty_print(" ")
pretty_print("[F→E]:",FE)
pretty_print(" ")
pretty_print("[E→F]:",FE^(-1))
pretty_print(" ")
pretty_print("[φ]_E:",FE^(-1)*B*FE)

56. feladat

B = matrix(QQ, [[-1,0],[0,1]])
FE = matrix(QQ, [[3,2],[2,-3]])
pretty_print("[φ]_F:",B)
pretty_print(" ")
pretty_print("[F→E]:",FE)
pretty_print(" ")
pretty_print("[E→F]:",FE^(-1))
pretty_print(" ")
pretty_print("[φ]_E:",FE^(-1)*B*FE)

57. feladat

B = matrix(QQ, [[1,0],[0,0]])
FE = matrix(QQ, [[4,3],[3,-4]])
pretty_print("[φ]_F:",B)
pretty_print(" ")
pretty_print("[F→E]:",FE)
pretty_print(" ")
pretty_print("[E→F]:",FE^(-1))
pretty_print(" ")
pretty_print("[φ]_E:",FE^(-1)*B*FE)

58. feladat

B = matrix(QQ, [[14,-8],[16,-2]])
FE = matrix(QQ, [[1,2],[3,1]])
pretty_print("[φ]_F:",B)
pretty_print(" ")
pretty_print("[F→E]:",FE)
pretty_print(" ")
pretty_print("[E→F]:",FE^(-1))
pretty_print(" ")
pretty_print("[φ]_E:",FE^(-1)*B*FE)

59. feladat

A = matrix(QQ, [[1,0,0],[-1,-1,-1],[1,2,2]])
FE = matrix(QQ, [[2,1,1],[1,2,1],[1,1,1]])
pretty_print("[φ]_E:",A)
pretty_print(" ")
pretty_print("[F→E]:",FE)
pretty_print(" ")
pretty_print("[E→F]:",FE^(-1))
pretty_print(" ")
pretty_print("[φ]_F:",FE*A*FE^(-1))

60. feladat

B = matrix(QQ, [[-1,0,0],[0,1,0],[0,0,1]])
FE = matrix(QQ, [[1,2,3],[-2,1,0],[-3,0,1]])
pretty_print("[φ]_F:",B)
pretty_print(" ")
pretty_print("[F→E]:",FE)
pretty_print(" ")
pretty_print("[E→F]:",FE^(-1))
pretty_print(" ")
pretty_print("[φ]_E:",FE^(-1)*B*FE)

61. feladat

A = matrix(QQ, [[5,6,2],[-3,-3,-2],[-2,-2,-1]])
FE = matrix(QQ, [[1,2,-1],[1,1,1],[0,1,-1]])
pretty_print("[φ]_E:",A)
pretty_print(" ")
pretty_print("[F→E]:",FE)
pretty_print(" ")
pretty_print("[E→F]:",FE^(-1))
pretty_print(" ")
pretty_print("[φ]_F:",FE*A*FE^(-1))

62. feladat

B = matrix(QQ, [[1,0,0],[0,-1,0],[0,0,-1]])
FE = matrix(QQ, [[1,1,-1],[-2,3,1],[1,-1,0]])
pretty_print("[φ]_F:",B)
pretty_print(" ")
pretty_print("[F→E]:",FE)
pretty_print(" ")
pretty_print("[E→F]:",FE^(-1))
pretty_print(" ")
pretty_print("[φ]_E:",FE^(-1)*B*FE)