python解决线性代数

假设你有一个鸡、犀牛和山羊的动物园。如果有12个头，38只脚和10只角，每种动物有多少只？解决这个问题的一个方法是将每个属性的方程式转化为矩阵。

低阶问题初探：

回顾题目：现有一笼子，里面有鸡和兔子若干只，数一数，共有头14个，腿38条，球鸡和兔子各有多少只？

假设有x只鸡，y只兔子

x + y = 14
2x + 4y = 38

参考一个代码

import matplotlib.pyplot as plt
import numpy as np
from mpl_toolkits.mplot3d import Axes3D
import scipy as sp
import scipy.linalg
import sympy as sy
sy.init_printing() 
np.set_printoptions(precision=3)
np.set_printoptions(suppress=True)

x = np.linspace(-5, 5, 100)
y1 = -x + 14
y2 = -0.5*x + 19*0.5

fig, ax = plt.subplots(figsize = (12, 7))
ax.scatter(9, 5, s = 200, zorder=5, color = 'r', alpha = .8) 
ax.plot(x, y1, lw =3, label = '$x+y=14$')
ax.plot(x, y2, lw =3, label = '$2x+4y=38$')
ax.plot([1, 1], [0, 5], ls = '--', color = 'b', alpha = .5)
ax.plot([-5, 1], [5, 5], ls = '--', color = 'b', alpha = .5)
ax.set_xlim([-5, 5])
ax.set_ylim([0, 12])

ax.legend()
s = '$(1,5)$'
ax.text(9, 5.5, s, fontsize = 20)
ax.set_title('Solution of $x+y=14$, $2x+4y=38$', size = 22)
ax.grid()

image.png

x, y = [-1, 0, 1], [-1, 0, 1]
X, Y = np.meshgrid(x, y)
fig, ax = plt.subplots(figsize = (12, 7))
ax.scatter(X, Y, s = 200, color = 'red')
ax.axis([-2, 3.01, -2.01, 2])
ax.spines['left'].set_position('zero') # alternative position is 'center'
ax.spines['right'].set_color('none')
ax.spines['bottom'].set_position('zero')
ax.spines['top'].set_color('none')
ax.grid()
plt.show()

image.png

3个未知数和3个方程式

x + y + z = 12

2x + 4y + 4*z = 38

0x + 1y + 2*z = 10

import numpy as np
import matplotlib as plt
import matplotlib.pyplot as plt


'''
x + y + z = 12
2*x + 4*y + 4*z = 38
0*x + 1*y + 2*z = 10
'''

x1 = np.linspace(0, 35, 20)
x2 = np.linspace(0, 20, 20)
X1, X2 = np.meshgrid(x1, x2)

fig = plt.figure(figsize = (9, 9))
ax = fig.add_subplot(111, projection = '3d')

X3 = 12 - X2 - X1
ax.plot_surface(X1, X2, X3, cmap ='winter', alpha = 1) 

X3 = 19*0.5 - X2 - 0.5*X1
ax.plot_surface(X1, X2, X3, cmap ='summer', alpha = 1)

X3 = 5 - 0.5*X2
ax.plot_surface(X1, X2, X3, cmap ='spring', alpha = 1)

#numpy 求解交点
A = np.array([[1,1,1],[2,4,4],[0,1,2]])
B = np.array([12,38,10])
dot = np.linalg.solve(A,B)

x,y,z = [int(i) for i in dot]
print(x,y,z)
#ax.scatter(29, 16, 3, s = 200, color = 'black')
ax.scatter(x, 0, 0, s = 200, color = 'black')
ax.scatter(0, y, 0, s = 200, color = 'black')
ax.scatter(0, 0, z, s = 200, color = 'black')
plt.show()

image.png

求解结果：5，4 ，3

5只鸡
4只犀牛
3只羊