线性代数 linear algebra

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))
16.  
     
17.  
     

main_vector.py

1.  
   import sys
2.  
   import numpy
3.  
   import scipy
4.  
   from playLA.Vector import Vector
5.  
    
6.  
   if __name__ == "__main__":
7.  
   vec = Vector([5, 2])
8.  
   print(vec)
9.  
   print(len(vec))
10.  
    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.  
    
6.  
   if __name__ == "__main__":
7.  
   vec = Vector([5, 2])
8.  
   print(vec)
9.  
   print(len(vec))
10.  
    print("vec[0] = {}, vec[1] = {}".format(vec[0], vec[1]))
11.  
    vec2 = Vector([3, 1])
12.  
    print("{} + {} = {}".format(vec, vec2, vec + vec2))
13.  
    print("{} - {} = {}".format(vec, vec2, vec - vec2))
14.  
    print("{} * {} = {}".format(vec, 3, vec * 3))
15.  
    print("{} * {} = {}".format(3, vec, vec * 3))
16.  
    print("-{} = {}".format(vec, -vec))
17.  
    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.  
    
5.  
   lst = [1, 2, 3]
6.  
   lst[0] = "LA"
7.  
   print(lst)
8.  
   #numpy中只能存储一种数据
9.  
   vec = np.array([1, 2, 3])
10.  
    print(vec)
11.  
    # vec[0] = "LA"
12.  
    # vec[0] = 666
13.  
    print(vec)
14.  
    print(np.zeros(5))
15.  
    print(np.ones(5))
16.  
    print(np.full(5, 666))
17.  
     
18.  
    print(vec)
19.  
    print("size = ", vec.size)
20.  
    print("size = ", len(vec))
21.  
    print(vec[0])
22.  
    print(vec[-1])
23.  
    print(vec[0:2])
24.  
    print(type(vec[0:2]))
25.  
    #点乘
26.  
    vec2 = np.array([4, 5, 6])
27.  
    print("{} + {} = {}".format(vec, vec2, vec + vec2))
28.  
    print("{} - {} = {}".format(vec, vec2, vec - vec2))
29.  
    print("{} * {} = {}".format(2, vec, 2 * vec))
30.  
    print("{} * {} = {}".format(vec, 2, vec * 2))
31.  
    print("{} * {} = {}".format(vec, vec2, vec * vec2))
32.  
    print("{}.dot({})= {}".format(vec, vec2, vec.dot(vec2)))
33.  
    #求模
34.  
    print(np.linalg.norm(vec))
35.  
    print(vec/ np.linalg.norm(vec))
36.  
    print(np.linalg.norm(vec/ np.linalg.norm(vec)))
37.  
    #为什么输出nan
38.  
    zero3 = np.zeros(3)
39.  
    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.  
    
3.  
   if __name__ == "__main__":
4.  
   #创建矩阵
5.  
   A = np.array([[1, 2], [3, 4]])
6.  
   print(A)
7.  
   #矩阵属性
8.  
   print(A.shape)
9.  
   print(A.T)
10.  
    #获取矩阵元素
11.  
    print(A[1, 1])
12.  
    print(A[0])
13.  
    print(A[:, 0])
14.  
    print(A[1, :])
15.  
    #矩阵的基本运算
16.  
    B = np.array([[5, 6], [7, 8]])
17.  
    print(A + B)
18.  
    print(A - B)
19.  
    print(10 * A)
20.  
    print(A * 10)
21.  
    print(A * B)
22.  
    print(A.dot(B))
23.  
     
24.  
     

 
 

5 矩阵进阶

5.3 矩阵变换

main_matrix_transformation.py

1.  
   import math
2.  
    
3.  
   import matplotlib.pyplot as plt
4.  
   from playLA.Matrix import Matrix
5.  
   from playLA.Vector import Vector
6.  
   if __name__ == "__main__":
7.  
   points = [[0, 0], [0, 5], [3, 5], [3, 4], [1, 4],
8.  
   [1, 3], [2, 3], [2, 2], [1, 2], [1, 0]]
9.  
   x = [point[0] for point in points]
10.  
    y = [point[1] for point in points]
11.  
    plt.figure(figsize=(5, 5))
12.  
    plt.xlim(-10, 10)
13.  
    plt.ylim(-10, 10)
14.  
    plt.plot(x, y)
15.  
    # plt.show()
16.  
    P = Matrix(points)
17.  
    # T = Matrix([[2, 0], [0, 1.5]])#x扩大2倍，y扩大1.5倍
18.  
    # T = Matrix([[1, 0], [0, -1]])#关于X轴对称
19.  
    # T = Matrix([[-1, 0], [0, 1]])#关于X轴对称
20.  
    # T = Matrix([[-1, 0], [0, -1]])#关于原点对称
21.  
    # T = Matrix([[1, 0.5], [0, 1]])
22.  
    # T = Matrix([[1, 0], [0.5, 1]])
23.  
    theta = math.pi / 3
24.  
    #旋转theta角度
25.  
    T = Matrix([[math.cos(theta), math.sin(theta)], [-math.sin(theta), math.cos(theta)]])
26.  
    P2 = T.dot(P.T())
27.  
    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())])
28.  
    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.  
    
7.  
   C = np.array([[1,2]])
8.  
   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