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 Vector

if __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

    @classmethod
    def 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 * self
        self本身就是一个列表
        """
        return self * k

    def __pos__(self):
        """返回向量取正的结果向量"""
        return 1 * self

    def __neg__(self):
        """返回向量取负的结果向量"""
        return -1 * self

    def __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 Vector

if __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 np

if __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))