Python学线性代数（1）向量相关话题

作者: __method__ | 来源:发表于2020-07-09 17:06 被阅读0次

Python学线性代数（1）向量相关话题
Python学线性代数（2）矩阵相关话题
线性代数基础
机器学习（八）最小二乘法
线性代数的本质（笔记1）
20180316周五～技术收获
线性代数——(1)向量
人工智能数学知识
线性相关、生成子空间、范数
【Python】关于数据归一化MinMaxScaler

实现我们自己的向量类

Vector.py

class Vector:

    def __init__(self, lst):
        self._values = lst

    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):
        return "Vector({})".format(self._values)

    def __str__(self):
        return "({})".format(", ".join(str(e) for e in self._values))

建立main_vector.py测试：

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]))

05-Implement-Vector-Operations

class Vector:

    def __init__(self, lst):
        self._values = list(lst)

    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"""
        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):
        return "Vector({})".format(self._values)

    def __str__(self):
        return "({})".format(", ".join(str(e) for e in self._values))

建立main_vector.py测试：

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))

08-Implementation-of-Zero-Vector

class Vector:

    def __init__(self, lst):
        self._values = list(lst)

    @classmethod
    def zero(cls, dim):
        """返回一个dim维的零向量"""
        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, 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"""
        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):
        return "Vector({})".format(self._values)

    def __str__(self):
        return "({})".format(", ".join(str(e) for e in self._values))

测试

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))

关于向量的更多话题

02-Normalization-Implementation

_globals.py

EPSILON = 1e-8

init.py

from ._globals import EPSILON

Vector.py

import math
from ._globals import EPSILON


class Vector:

    def __init__(self, lst):
        self._values = list(lst)

    @classmethod
    def zero(cls, dim):
        """返回一个dim维的零向量"""
        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, 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 norm(self):
        """返回向量的模"""
        return math.sqrt(sum(e**2 for e in self))

    def normalize(self):
        """返回向量的单位向量"""
        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 __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"""
        return self * k

    def __truediv__(self, k):
        """返回数量除法的结果向量：self / k"""
        return (1 / k) * self

    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):
        return "Vector({})".format(self._values)

    def __str__(self):
        return "({})".format(", ".join(str(e) for e in self._values))

05-Implementations-of-Dot-Product

import math
from ._globals import EPSILON


class Vector:

    def __init__(self, lst):
        self._values = list(lst)

    @classmethod
    def zero(cls, dim):
        """返回一个dim维的零向量"""
        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, 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 norm(self):
        """返回向量的模"""
        return math.sqrt(sum(e**2 for e in self))

    def normalize(self):
        """返回向量的单位向量"""
        if self.norm() < EPSILON:
            raise ZeroDivisionError("Normalize error! norm is zero.")
        return Vector(self._values) / self.norm()

    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 __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"""
        return self * k

    def __truediv__(self, k):
        """返回数量除法的结果向量：self / k"""
        return (1 / k) * self

    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):
        return "Vector({})".format(self._values)

    def __str__(self):
        return "({})".format(", ".join(str(e) for e in self._values))

测试

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(vec.dot(vec2))

07-Vectors-in-Numpy

main_numpy_vector.py

import numpy as np

if __name__ == "__main__":

    print(np.__version__)

    # np.array 基础
    lst = [1, 2, 3]
    lst[0] = "Linear Algebra"
    print(lst)

    vec = np.array([1, 2, 3])
    print(vec)
    # vec[0] = "Linear Algebra"
    # vec[0] = 666
    # print(vec)

    # np.array的创建
    print(np.zeros(5))
    print(np.ones(5))
    print(np.full(5, 666))

    # 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]))

    # 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("{} * {} = {}".format(vec, vec2, vec * vec2))
    print("{}.dot({}) = {}".format(vec, vec2, vec.dot(vec2)))

    print(np.linalg.norm(vec))
    print(vec / np.linalg.norm(vec))
    print(np.linalg.norm(vec / np.linalg.norm(vec)))

    # zero3 = np.zeros(3)
    # print(zero3 / np.linalg.norm(zero3))

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