3. 基底と次元

Pythonで学ぶ線形代数学

code: lincombi.py

from vpython import vec, arrow, color, points

from numpy.random import normal

x = vec(*normal(0, 1, 3))

arrow(pos=vec(0, 0, 0), axis=x, color=color.red)

y = vec(*normal(0, 1, 3))

arrow(pos=vec(0, 0, 0), axis=y, color=color.red)

W = a * x + b * y for (a, b) in normal(0, 1, (1000, 2))

points(pos=W, radius=2)

ソースコード：lincombi.py

https://gyazo.com/20ccc2b29748905aefdd386211acd332

code: eqn1.py

from sympy import solve

from sympy.abc import a, b, x, y

ans = solve(a + 2 * b - x, 2 * a + 3 * b - y, a, b)

ソースコード：eqn1.py

code: python

>> ans

{b: 2*x - y, a: -3*x + 2*y}

>> ansa

-3*x + 2*y

>> ansb

2*x - y

code: eqn2.py

from sympy import solve

from sympy.abc import a, b, x, y, z

ans = solve(a + 2 * b - x, 2 * a + 3 * b - y, 3 * a + 4 * b - z, a, b)

ソースコード: eqn2.py

code:python

>> ans

[]

code: eqn3.py

from sympy import solve

from sympy.abc import a, b, c, x, y, z

ans = solve([a + 2 * b + 3 * c - x, 2 * a + 3 * b + c - y,

3 * a + 4 * b + 2 * c - z], a, b, c)

ソースコード: eqn3.py

code:python

>> M = [anskey for key in a, b, c]

>> M

-2*x/3 - 8*y/3 + 7*z/3, x/3 + 7*y/3 - 5*z/3, x/3 - 2*y/3 + z/3

>> N = [m.subs(x, 2], y, 3, [z, 5) for m in M]

>> N

7/3, -2/3, 1/3

>> n.evalf(2) for n in N

2.3, -0.67, 0.33

code: eqn4.py

from sympy import solve

from sympy.abc import x, y

ans = solve(x + 2 * y, 2 * x + 3 * y, x, y)

print(ans)

ソースコード: eqn4.py

code: python

{x: 0, y: 0}

code: eqn5.py

from sympy import solve

from sympy.abc import x, y

ans = solve(x + 2 * y, 2 * x + 4 * y, x, y)

print(ans)

ソースコード: eqn5.py

code: python

{x: -2*y}

code: arrow2d.py

import matplotlib.pyplot as plt

o, a, b, c = (0, 0), (1, 2), (2, 3), (2, 4)

arrows = o, a, 'r', 0.1], o, b, 'g', 0.05, [o, c, 'b', 0.05

for p, v, c, w in arrows:

plt.quiver(p0, p1, v0, v1, units='xy',

scale=1, color=c, width=w)

plt.axis('scaled'), plt.xlim(0, 5), plt.ylim(0, 5), plt.show()

ソースコード: ：arrow.2dpy

https://gyazo.com/8729a1c4491de2284748de8eae079498

code: eqn6.py

from sympy import solve

from sympy.abc import x, y, z

ans1 = solve(x + 2 * y + 3 * z, 2 * x + 3 * y + 4 * z, 3 * x + 4 * y + 5 * z,

x, y, z)

print(ans1)

ans2 = solve(x + 2 * y + 3 * z, 2 * x + 3 * y + z, 3 * x + y + 2 * z,

x, y, z)

print(ans2)

ソースコード: eqn6.py

code:python

{y: -2*z, x: z}

{z: 0, y: 0, x: 0}

code: arrow3d.py

from vpython import vec, arrow, mag

o = vec(0, 0, 0)

for p in (1, 2, 3), (2, 3, 4), (3, 4, 5), (3, 1, 2):

v = vec(*p)

arrow(pos=o, axis=v, color=v, shaftwidth=mag(v) * 0.02)

ソースコード: arrow3d.py

https://gyazo.com/a1aa8b01d2967e43c056483318739e9b

code: lena3.py

from vpython import canvas, vec, curve, arrow, color, points

from numpy import array, linspace, sin, cos, pi, random

canvas(background=color.white, foreground=color.black)

for v in vec(1, 0, 0), vec(0, 1, 0), vec(0, 0, 1):

curve(pos=-v, v, color=v)

with open('lena.txt', 'r') as fd:

XY = eval(fd.read())

random.seed(123)

a = vec(*random.normal(0, 1, 3))

arrow(pos=vec(0, 0, 0), axis=a, shaftwidth=0.1)

b = vec(*random.normal(0, 1, 3))

arrow(pos=vec(0, 0, 0), axis=b, shaftwidth=0.1)

P = x * a + y * b for (x, y) in XY

Q = cos(t) * a + sin(t) * b for t in linspace(0, 2 * pi, 101)

points(pos=P, radius=2, color=color.cyan)

curve(pos=Q, color=color.magenta)

ソースコード: lena3.py

https://gyazo.com/8bc425427abeb8f1649de8a95b9be788

code: rank.py

from numpy.linalg import matrix_rank

def f(*x): return matrix_rank(x)

a, b, c = (1, 2), (2, 3), (2, 4)

print(f(a, b), f(b, c), f(a, c), f(a, b, c))

a, b, c, d = (1, 2, 3), (2, 3, 4), (3, 4, 5), (3, 4, 4)

print(f(a, b), f(a, b, c), f(a, b, d), f(a, b, c, d))

ソースコード: rank.py

code: dim.py

from numpy import array

A = array(1, 2, 3)

B = array(1, 2, 3], [4, 5, 6)

C = array(1, 2, [3, 4, 5, 6], 7, 8)

ソースコード: dim.py

code: random2d.py

import matplotlib.pyplot as plt

from numpy.random import normal

P = normal(0, 1, (1000, 2))

plt.scatter(P:, 0, P:, 1, s=4)

plt.axis('equal'), plt.show()

ソースコード: random2d.py

https://gyazo.com/f3fc5f68b36bee01030756a14edfe727

code: random3d.py

from vpython import canvas, color, vec, curve, points

from numpy.random import normal

canvas(background=color.white, foreground=color.black)

for v in vec(5, 0, 0), vec(0, 5, 0), vec(0, 0, 5):

curve(pos=-v, v, color=v)

P = normal(0, 1, (1000, 3))

points(pos=vec(*p) for p in P, radius=4)

ソースコード: random3d.py

https://gyazo.com/0113a3ccaeb10654390095d2f61a8770