一、代码仓库

https://github.com/Chufeng-Jiang/Python-Linear-Algebra-for-Beginner/tree/main

二、向量的基本运算

2.1 加法

2.2 数量乘法

2.3 向量运算的基本性质

2.4 零向量

2.5 向量的长度

2.6 单位向量

单位向量叫做 u hat

2.7 点乘/内积：两个向量的乘法 --答案是一个标量

三、手写Vector代码

3.1 在控制台测试__repr__和__str__方法

3.2 创建实例测试代码

from playLA.Vector import Vectorif __name__ == "__main__":vec = Vector([5, 2])print(vec)print("len(vec) = {}".format(len(vec)))print("vec[0] = {}, vec[1] = {}".format(vec[0], vec[1]))

3.3 完整代码

Vector.py

import math
from ._globals import EPSILON
class Vector:def __init__(self, lst):"""__init__ 代表类的构造函数双下划线开头的变量 例如_values,代表类的私有成员lst是个引用，list(lst)将值复制一遍，防止用户修改值"""self._values = list(lst)def dot(self, another):"""向量点乘，返回结果标量"""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))def norm(self):"""返回向量的模"""return math.sqrt(sum(e**2 for e in self))def normalize(self):"""归一化，规范化返回向量的单位向量此处设计到了除法: def __truediv__(self, k):"""if self.norm() < EPSILON:raise ZeroDivisionError("Normalize error! norm is zero.")return Vector(self._values) / self.norm()# return 1 / self.norm() * Vector(self._values)# return Vector([e / self.norm() for e in self])def __truediv__(self, k):"""返回数量除法的结果向量：self / k"""return (1 / k) * self@classmethoddef zero(cls, dim):"""返回一个dim维的零向量@classmethod 修饰符对应的函数不需要实例化，不需要 self 参数，但第一个参数需要是表示自身类的cls参数，可以来调用类的属性，类的方法，实例化对象等。"""return cls([0] * dim)def __add__(self, another):"""向量加法，返回结果向量"""assert len(self) == len(another), \"Error in adding. Length of vectors must be 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)])def __sub__(self, another):"""向量减法，返回结果向量"""assert len(self) == len(another), \"Error in subtracting. Length of vectors must be same."return Vector([a - b for a, b in zip(self, another)])def __mul__(self, k):"""返回数量乘法的结果向量：self * k"""return Vector([k * e for e in self])def __rmul__(self, k):"""返回数量乘法的结果向量：k * selfself本身就是一个列表"""return self * kdef __pos__(self):"""返回向量取正的结果向量"""return 1 * selfdef __neg__(self):"""返回向量取负的结果向量"""return -1 * selfdef __iter__(self):"""返回向量的迭代器"""return self._values.__iter__()def __getitem__(self, index):"""取向量的第index个元素"""return self._values[index]def __len__(self):"""返回向量长度（有多少个元素）"""return len(self._values)def __repr__(self):"""打印显示：Vector([5, 2])"""return "Vector({})".format(self._values)def __str__(self):"""打印显示：(5, 2)"""return "({})".format(", ".join(str(e) for e in self._values))

_globals.py

# 包中的变量，但是对包外不可见，因此使用“_”开头
EPSILON = 1e-8

main_vector.py

from playLA.Vector import Vectorif __name__ == "__main__":vec = Vector([5, 2])print(vec)print("len(vec) = {}".format(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, 3 * vec))print("+{} = {}".format(vec, +vec))print("-{} = {}".format(vec, -vec))zero2 = Vector.zero(2)print(zero2)print("{} + {} = {}".format(vec, zero2, vec + zero2))print("norm({}) = {}".format(vec, vec.norm()))print("norm({}) = {}".format(vec2, vec2.norm()))print("norm({}) = {}".format(zero2, zero2.norm()))print("normalize {} is {}".format(vec, vec.normalize()))print(vec.normalize().norm())print("normalize {} is {}".format(vec2, vec2.normalize()))print(vec2.normalize().norm())try:zero2.normalize()except ZeroDivisionError:print("Cannot normalize zero vector {}.".format(zero2))print("========点乘：========")print(vec.dot(vec2))

main_numpy_vector.py

import numpy as npif __name__ == "__main__":print(np.__version__)# np.array 基础print("========np.array 基础========")lst = [1, 2, 3]lst[0] = "Linear Algebra"print(lst)print("========vec = np.array([1, 2, 3])========")vec = np.array([1, 2, 3])print(vec)# vec[0] = "Linear Algebra"# vec[0] = 666# print(vec)print("========np.array的创建========")# np.array的创建print(np.zeros(5))print(np.ones(5))print(np.full(5, 666))print("========np.array的基本属性========")# np.array的基本属性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]))print("========np.array的基本运算========")# np.array的基本运算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("没有数学意义的乘法：{} * {} = {}".format(vec, vec2, vec * vec2))print("{}.dot({}) = {}".format(vec, vec2, vec.dot(vec2)))print("========求模========")print(np.linalg.norm(vec))print("========归一化========")print(vec / np.linalg.norm(vec))print("========单位向量========")print(np.linalg.norm(vec / np.linalg.norm(vec)))print("========零向量会报错========")zero3 = np.zeros(3)print(zero3 / np.linalg.norm(zero3))