线性代数 linear algebra
2.3 实现属于我们自己的向量
Vector.py
class Vector:
def __init__(self, lst):
self._values = lst
#return len
def __len__(self):
return len(self._values)
#return index th item
def __getitem__(self, index):
return self._values[index]
#direct use call this method
def __repr__(self):
return "Vector({})".format(self._values)
#print call this method
def __str__(self):
return "({})".format(", ".join(str(e) for e in self._values))
main_vector.py
import sys
import numpy
import scipy
from playLA.Vector import Vector
if __name__ == "__main__":
vec = Vector([5, 2])
print(vec)
print(len(vec))
print("vec[0] = {}, vec[1] = {}".format(vec[0], vec[1]))2.5 实现向量的基本运算
Vector.py
class Vector:
def __init__(self, lst):
self._values = lst
#return len
def __len__(self):
return len(self._values)
#return index th item
def __getitem__(self, index):
return self._values[index]
#direct use call this method
def __repr__(self):
return "Vector({})".format(self._values)
#print call this method
def __str__(self):
return "({})".format(", ".join(str(e) for e in self._values))
#vector add method
def __add__(self, another):
assert len(self) == len(another),"lenth not same"
# return Vector([a + b for a, b in zip(self._values, another._values)])
return Vector([a + b for a, b in zip(self, another)])
#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
#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
def __iter__(self):
return self._values.__iter__()
#sub
def __sub__(self, another):
# return Vector([a + b for a, b in zip(self._values, another._values)])
return Vector([a - b for a, b in zip(self, another)])
#self * k
def __mul__(self, k):
return Vector([k * e for e in self])
# k * self
def __rmul__(self, k):
return Vector([k * e for e in self])
#取正
def __pos__(self):
return 1 * self
#取反
def __neg__(self):
return -1 * self
main_vector.py
import sys
import numpy
import scipy
from playLA.Vector import Vector
if __name__ == "__main__":
vec = Vector([5, 2])
print(vec)
print(len(vec))
print("vec[0] = {}, vec[1] = {}".format(vec[0], vec[1]))
vec2 = Vector([3, 1])
print("{} + {} = {}".format(vec, vec2, vec + vec2))
print("{} - {} = {}".format(vec, vec2, vec - vec2))
print("{} * {} = {}".format(vec, 3, vec * 3))
print("{} * {} = {}".format(3, vec, vec * 3))
print("-{} = {}".format(vec, -vec))
print("+{} = {}".format(vec, +vec))
2.8 实现0向量
Vector.py
@classmethod
def zero(cls, dim):
return cls([0] * dim)main_vector.py
zero2 = Vector.zero(2)
print(zero2)
print("{} + {} = {}".format(vec, zero2, vec + zero2))3.2实现向量规范
Vector.py
# self / k
def __truediv__(self, k):
return Vector((1 / k) * self)
#模
def norm(self):
return math.sqrt(sum(e**2 for e in self))
#归一化
def normalize(self):
if self.norm() < EPSILON:
raise ZeroDivisionError("Normalize error! norm is zero.")
return Vector(self._values)/self.norm()main_vector.py
print("normalize vec is ({})".format(vec.normalize()))
print(vec.normalize().norm())
try :
zero2.normalize()
except ZeroDivisionError:
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
def dot(self, another):
assert len(self) == len(another), "Error in dot product. Length of vectors must be same."
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
import numpy as np
if __name__ == "__main__":
print(np.__version__)
lst = [1, 2, 3]
lst[0] = "LA"
print(lst)
#numpy中只能存储一种数据
vec = np.array([1, 2, 3])
print(vec)
# vec[0] = "LA"
# vec[0] = 666
print(vec)
print(np.zeros(5))
print(np.ones(5))
print(np.full(5, 666))
print(vec)
print("size = ", vec.size)
print("size = ", len(vec))
print(vec[0])
print(vec[-1])
print(vec[0:2])
print(type(vec[0:2]))
#点乘
vec2 = np.array([4, 5, 6])
print("{} + {} = {}".format(vec, vec2, vec + vec2))
print("{} - {} = {}".format(vec, vec2, vec - vec2))
print("{} * {} = {}".format(2, vec, 2 * vec))
print("{} * {} = {}".format(vec, 2, vec * 2))
print("{} * {} = {}".format(vec, vec2, vec * vec2))
print("{}.dot({})= {}".format(vec, vec2, vec.dot(vec2)))
#求模
print(np.linalg.norm(vec))
print(vec/ np.linalg.norm(vec))
print(np.linalg.norm(vec/ np.linalg.norm(vec)))
#为什么输出nan
zero3 = np.zeros(3)
print(zero3 /np.linalg.norm(zero3))4矩阵
4.2实现矩阵
Matrix.py
from .Vector import Vector
class Matrix:
#list2d二维数组
def __init__(self, list2d):
self._values = [row[:] for row in list2d]
def __repr__(self):
return "Matrix({})".format(self._values)
__str__ = __repr__
def shape(self):
return len(self._values),len(self._values[0])
def row_num(self):
return self.shape()[0]
def col_num(self):
return self.shape()[1]
def size(self):
r, c = self.shape()
return r * c
__len__ = row_num
def __getitem__(self, pos):
r, c =pos
return self._values[r][c]
#第index个行向量
def row_vector(self, index):
return Vector(self._values[index])
def col_vector(self, index):
return Vector([row[index] for row in self._values])
main_matrix.py
from playLA.Matrix import Matrix
if __name__ == "__main__":
matrix = Matrix([[1, 2],[3, 4]])
print(matrix)
print("matrix.shape = {}".format(matrix.shape()))
print("matrix.size = {}".format(matrix.size()))
print("matrix.len = {}".format(len(matrix)))
print("matrix[0][0]= {}".format(matrix[0, 0]))
print("{}".format(matrix.row_vector(0)))
print("{}".format(matrix.col_vector(0)))4.4 实现矩阵的基本计算
Matrix.py
def __add__(self, another):
assert self.shape() == another.shape(),"ERROR in shape"
return Matrix([[a + b for a, b in zip(self.row_vector(i), another.row_vector(i))]for i in range(self.row_num())])
def __sub__(self, another):
assert self.shape() == another.shape(),"ERROR in shape"
return Matrix([[a - b for a, b in zip(self.row_vector(i), another.row_vector(i))]for i in range(self.row_num())])
def __mul__(self, k):
return Matrix([[e*k for e in self.row_vector(i)] for i in range(self.row_num())])
def __rmul__(self, k):
return self * k
#数量除法
def __truediv__(self, k):
return (1/k) * self
def __pos__(self):
return 1 * self
def __neg__(self):
return -1 * self
@classmethod
def zero(cls, r, c):
return cls([[0]*c for _ in range(r)])main_matrix.py
matrix2 = Matrix([[5, 6], [7, 8]])
print("add: {}".format(matrix + matrix2))
print("sub: {}".format(matrix - matrix2))
print("mul: {}".format(matrix * 2))
print("rmul: {}".format(2 * matrix))
print("zero_2_3:{}".format(Matrix.zero(2, 3)))4.8实现矩阵乘法
Matrix.py
main_matrix.py
Matrix.py
def dot(self, another):
if isinstance(another, Vector):
assert self.col_num() == len(another), "error in shape"
return Vector([self.row_vector(i).dot(another) for i in range(self.row_num())])
if isinstance(another, Matrix):
assert self.col_num() == another.row_num(),"error in shape"
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
T = Matrix([[1.5, 0], [0, 2]])
p = Vector([5, 3])
print("T.dot(p)= {}".format(T.dot(p)))
P = Matrix([[0, 4, 5], [0, 0, 3]])
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
import numpy as np
if __name__ == "__main__":
#创建矩阵
A = np.array([[1, 2], [3, 4]])
print(A)
#矩阵属性
print(A.shape)
print(A.T)
#获取矩阵元素
print(A[1, 1])
print(A[0])
print(A[:, 0])
print(A[1, :])
#矩阵的基本运算
B = np.array([[5, 6], [7, 8]])
print(A + B)
print(A - B)
print(10 * A)
print(A * 10)
print(A * B)
print(A.dot(B))
5 矩阵进阶
5.3 矩阵变换
main_matrix_transformation.py
import math
import matplotlib.pyplot as plt
from playLA.Matrix import Matrix
from playLA.Vector import Vector
if __name__ == "__main__":
points = [[0, 0], [0, 5], [3, 5], [3, 4], [1, 4],
[1, 3], [2, 3], [2, 2], [1, 2], [1, 0]]
x = [point[0] for point in points]
y = [point[1] for point in points]
plt.figure(figsize=(5, 5))
plt.xlim(-10, 10)
plt.ylim(-10, 10)
plt.plot(x, y)
# plt.show()
P = Matrix(points)
# T = Matrix([[2, 0], [0, 1.5]])#x扩大2倍，y扩大1.5倍
# T = Matrix([[1, 0], [0, -1]])#关于X轴对称
# T = Matrix([[-1, 0], [0, 1]])#关于X轴对称
# T = Matrix([[-1, 0], [0, -1]])#关于原点对称
# T = Matrix([[1, 0.5], [0, 1]])
# T = Matrix([[1, 0], [0.5, 1]])
theta = math.pi / 3
#旋转theta角度
T = Matrix([[math.cos(theta), math.sin(theta)], [-math.sin(theta), math.cos(theta)]])
P2 = T.dot(P.T())
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())])
plt.show()
5.6实现单位矩阵和numpy中的逆矩阵
Matrix.py
#单位矩阵
@classmethod
def identity(cls, n):
m = [[0]*n for _ in range(n)]
for i in range(n):
m[i][i] = 1
return cls(m)main_matrix.py
I = Matrix.identity(2)
print(I)
print("A.dot(I) = {}".format(matrix.dot(I)))
print("I.dot(A) = {}".format(I.dot(matrix)))main_numpy_matrix.py
#numpy中的逆矩阵
invA = np.linalg.inv(A)
print(invA)
print(A.dot(invA))
print(invA.dot(A))
C = np.array([[1,2]])
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