2.3 实现属于我们自己的向量

Vector.py

1. class Vector:
2. def __init__(self, lst):
3. self._values = lst
4. #return len
5. def __len__(self):
6. return len(self._values)
7. #return index th item
8. def __getitem__(self, index):
9. return self._values[index]
10. #direct use call this method
11. def __repr__(self):
12. return "Vector({})".format(self._values)
13. #print call this method
14. def __str__(self):
15. return "({})".format(", ".join(str(e) for e in self._values))

main_vector.py

1. import sys
2. import numpy
3. import scipy
4. from playLA.Vector import Vector
5. if __name__ == "__main__":
6. vec = Vector([ 5, 2])
7. print(vec)
8. print(len(vec))
9. print( "vec[0] = {}, vec[1] = {}".format(vec[0], vec[1]))

2.5 实现向量的基本运算

Vector.py

1. class Vector:
2. def __init__(self, lst):
3. self._values = lst
4. #return len
5. def __len__(self):
6. return len(self._values)
7. #return index th item
8. def __getitem__(self, index):
9. return self._values[index]
10. #direct use call this method
11. def __repr__(self):
12. return "Vector({})".format(self._values)
13. #print call this method
14. def __str__(self):
15. return "({})".format(", ".join(str(e) for e in self._values))
16. #vector add method
17. def __add__(self, another):
18. assert len(self) == len(another),"lenth not same"
19. # return Vector([a + b for a, b in zip(self._values, another._values)])
20. return Vector([a + b for a, b in zip(self, another)])
21. #迭代器 设计_values其实是私有成员变量，不想别人访问，所以使用迭代器
22. #单双下划线开头体现在继承上，如果类内内部使用的变量使用单下划线
23. def __iter__(self):
24. return self._values.__iter__()
25. #sub
26. def __sub__(self, another):
27. # return Vector([a + b for a, b in zip(self._values, another._values)])
28. return Vector([a - b for a, b in zip(self, another)])
29. #self * k
30. def __mul__(self, k):
31. return Vector([k * e for e in self])
32. # k * self
33. def __rmul__(self, k):
34. return Vector([k * e for e in self])
35. #取正
36. def __pos__(self):
37. return 1 * self
38. #取反
39. def __neg__(self):
40. return -1 * self

main_vector.py

1. import sys
2. import numpy
3. import scipy
4. from playLA.Vector import Vector
5. if __name__ == "__main__":
6. vec = Vector([ 5, 2])
7. print(vec)
8. print(len(vec))
9. print( "vec[0] = {}, vec[1] = {}".format(vec[0], vec[1]))
10. vec2 = Vector([ 3, 1])
11. print( "{} + {} = {}".format(vec, vec2, vec + vec2))
12. print( "{} - {} = {}".format(vec, vec2, vec - vec2))
13. print( "{} * {} = {}".format(vec, 3, vec * 3))
14. print( "{} * {} = {}".format(3, vec, vec * 3))
15. print( "-{} = {}".format(vec, -vec))
16. print( "+{} = {}".format(vec, +vec))

2.8 实现0向量

Vector.py

1. @classmethod
2. def zero(cls, dim):
3. return cls([0] * dim)

main_vector.py

1. zero2 = Vector.zero( 2)
2. print(zero2)
3. print( "{} + {} = {}".format(vec, zero2, vec + zero2))

3.2实现向量规范

Vector.py

1. # self / k
2. def __truediv__(self, k):
3. return Vector((1 / k) * self)
4. #模
5. def norm(self):
6. return math.sqrt(sum(e**2 for e in self))
7. #归一化
8. def normalize(self):
9. if self.norm() < EPSILON:
10. raise ZeroDivisionError("Normalize error! norm is zero.")
11. return Vector(self._values)/self.norm()

main_vector.py

1. print( "normalize vec is ({})".format(vec.normalize()))
2. print(vec.normalize().norm())
3. try :
4. zero2.normalize()
5. except ZeroDivisionError:
6. print( "cant normalize zero vector {}".format(zero2))

3.3 向量的点乘

3.5实现向量的点乘操作

Vector.py

1. def dot(self, another):
2. assert len(self) == len(another), "Error in dot product. Length of vectors must be same."
3. return sum(a * b for a, b in zip(self, another))

main_vector.py

print(vec.dot(vec2))

3.6向量点乘的应用

3.7numpy中向量的基本使用

main_numpy_vector.py

1. import numpy as np
2. if __name__ == "__main__":
3. print(np.__version__)
4. lst = [ 1, 2, 3]
5. lst[ 0] = "LA"
6. print(lst)
7. #numpy中只能存储一种数据
8. vec = np.array([ 1, 2, 3])
9. print(vec)
10. # vec[0] = "LA"
11. # vec[0] = 666
12. print(vec)
13. print(np.zeros( 5))
14. print(np.ones( 5))
15. print(np.full( 5, 666))
16. print(vec)
17. print( "size = ", vec.size)
18. print( "size = ", len(vec))
19. print(vec[ 0])
20. print(vec[ -1])
21. print(vec[ 0:2])
22. print(type(vec[ 0:2]))
23. #点乘
24. vec2 = np.array([ 4, 5, 6])
25. print( "{} + {} = {}".format(vec, vec2, vec + vec2))
26. print( "{} - {} = {}".format(vec, vec2, vec - vec2))
27. print( "{} * {} = {}".format(2, vec, 2 * vec))
28. print( "{} * {} = {}".format(vec, 2, vec * 2))
29. print( "{} * {} = {}".format(vec, vec2, vec * vec2))
30. print( "{}.dot({})= {}".format(vec, vec2, vec.dot(vec2)))
31. #求模
32. print(np.linalg.norm(vec))
33. print(vec/ np.linalg.norm(vec))
34. print(np.linalg.norm(vec/ np.linalg.norm(vec)))
35. #为什么输出nan
36. zero3 = np.zeros( 3)
37. print(zero3 /np.linalg.norm(zero3))

4矩阵

4.2实现矩阵

Matrix.py

1. from .Vector import Vector
2. class Matrix:
3. #list2d二维数组
4. def __init__(self, list2d):
5. self._values = [row[:] for row in list2d]
6. def __repr__(self):
7. return "Matrix({})".format(self._values)
8. __str__ = __repr__
9. def shape(self):
10. return len(self._values),len(self._values[0])
11. def row_num(self):
12. return self.shape()[0]
13. def col_num(self):
14. return self.shape()[1]
15. def size(self):
16. r, c = self.shape()
17. return r * c
18. __len__ = row_num
19. def __getitem__(self, pos):
20. r, c =pos
21. return self._values[r][c]
22. #第index个行向量
23. def row_vector(self, index):
24. return Vector(self._values[index])
25. def col_vector(self, index):
26. return Vector([row[index] for row in self._values])

main_matrix.py

1. from playLA.Matrix import Matrix
2. if __name__ == "__main__":
3. matrix = Matrix([[ 1, 2],[3, 4]])
4. print(matrix)
5. print( "matrix.shape = {}".format(matrix.shape()))
6. print( "matrix.size = {}".format(matrix.size()))
7. print( "matrix.len = {}".format(len(matrix)))
8. print( "matrix[0][0]= {}".format(matrix[0, 0]))
9. print( "{}".format(matrix.row_vector(0)))
10. print( "{}".format(matrix.col_vector(0)))

4.4 实现矩阵的基本计算

Matrix.py

1. def __add__(self, another):
2. assert self.shape() == another.shape(),"ERROR in shape"
3. return Matrix([[a + b for a, b in zip(self.row_vector(i), another.row_vector(i))]for i in range(self.row_num())])
4. def __sub__(self, another):
5. assert self.shape() == another.shape(),"ERROR in shape"
6. return Matrix([[a - b for a, b in zip(self.row_vector(i), another.row_vector(i))]for i in range(self.row_num())])
7. def __mul__(self, k):
8. return Matrix([[e*k for e in self.row_vector(i)] for i in range(self.row_num())])
9. def __rmul__(self, k):
10. return self * k
11. #数量除法
12. def __truediv__(self, k):
13. return (1/k) * self
14. def __pos__(self):
15. return 1 * self
16. def __neg__(self):
17. return -1 * self
18. @classmethod
19. def zero(cls, r, c):
20. return cls([[0]*c for _ in range(r)])

main_matrix.py

1. matrix2 = Matrix([[ 5, 6], [7, 8]])
2. print( "add: {}".format(matrix + matrix2))
3. print( "sub: {}".format(matrix - matrix2))
4. print( "mul: {}".format(matrix * 2))
5. print( "rmul: {}".format(2 * matrix))
6. print( "zero_2_3:{}".format(Matrix.zero(2, 3)))

4.8实现矩阵乘法

Matrix.py

main_matrix.py

Matrix.py

1. def dot(self, another):
2. if isinstance(another, Vector):
3. assert self.col_num() == len(another), "error in shape"
4. return Vector([self.row_vector(i).dot(another) for i in range(self.row_num())])
5. if isinstance(another, Matrix):
6. assert self.col_num() == another.row_num(),"error in shape"
7. return Matrix([self.row_vector(i).dot(another.col_vector(j)) for j in range(another.col_num())] for i in range(self.row_num()))

main_matrix.py

1. T = Matrix([[ 1.5, 0], [0, 2]])
2. p = Vector([ 5, 3])
3. print( "T.dot(p)= {}".format(T.dot(p)))
4. P = Matrix([[ 0, 4, 5], [0, 0, 3]])
5. print( "T.dot(P)={}".format(T.dot(P)))

4.11 实现矩阵转置和Numpy中的矩阵

main_numpy_matrix.py

1. import numpy as np
2. if __name__ == "__main__":
3. #创建矩阵
4. A = np.array([[ 1, 2], [3, 4]])
5. print(A)
6. #矩阵属性
7. print(A.shape)
8. print(A.T)
9. #获取矩阵元素
10. print(A[ 1, 1])
11. print(A[ 0])
12. print(A[:, 0])
13. print(A[ 1, :])
14. #矩阵的基本运算
15. B = np.array([[ 5, 6], [7, 8]])
16. print(A + B)
17. print(A - B)
18. print( 10 * A)
19. print(A * 10)
20. print(A * B)
21. print(A.dot(B))

5 矩阵进阶

5.3 矩阵变换

main_matrix_transformation.py

1. import math
2. import matplotlib.pyplot as plt
3. from playLA.Matrix import Matrix
4. from playLA.Vector import Vector
5. if __name__ == "__main__":
6. points = [[ 0, 0], [0, 5], [3, 5], [3, 4], [1, 4],
7. [ 1, 3], [2, 3], [2, 2], [1, 2], [1, 0]]
8. x = [point[ 0] for point in points]
9. y = [point[ 1] for point in points]
10. plt.figure(figsize=( 5, 5))
11. plt.xlim( -10, 10)
12. plt.ylim( -10, 10)
13. plt.plot(x, y)
14. # plt.show()
15. P = Matrix(points)
16. # T = Matrix([[2, 0], [0, 1.5]])#x扩大2倍，y扩大1.5倍
17. # T = Matrix([[1, 0], [0, -1]])#关于X轴对称
18. # T = Matrix([[-1, 0], [0, 1]])#关于X轴对称
19. # T = Matrix([[-1, 0], [0, -1]])#关于原点对称
20. # T = Matrix([[1, 0.5], [0, 1]])
21. # T = Matrix([[1, 0], [0.5, 1]])
22. theta = math.pi / 3
23. #旋转theta角度
24. T = Matrix([[math.cos(theta), math.sin(theta)], [-math.sin(theta), math.cos(theta)]])
25. P2 = T.dot(P.T())
26. plt.plot([P2.col_vector(i)[ 0] for i in range(P2.col_num())],[P2.col_vector(i)[1] for i in range(P2.col_num())])
27. plt.show()

5.6实现单位矩阵和numpy中的逆矩阵

Matrix.py

1. #单位矩阵
2. @classmethod
3. def identity(cls, n):
4. m = [[ 0]*n for _ in range(n)]
5. for i in range(n):
6. m[i][i] = 1
7. return cls(m)

main_matrix.py

1. I = Matrix.identity( 2)
2. print(I)
3. print( "A.dot(I) = {}".format(matrix.dot(I)))
4. print( "I.dot(A) = {}".format(I.dot(matrix)))

main_numpy_matrix.py

1. #numpy中的逆矩阵
2. invA = np.linalg.inv(A)
3. print(invA)
4. print(A.dot(invA))
5. print(invA.dot(A))
6. C = np.array([[ 1,2]])
7. print(np.linalg.inv(C))

5.8用矩阵表示空间

 

x轴就是(0，1)y轴就是(-1，0)

6 线性系统

6.4实现高斯-约旦消元法

https://blog.csdn.net/weixin_40709094/article/details/105602775