向量空间中的零向量_线性代数_day4

概念

对于任意一个向量，都存在一个向量，满足：

对于任意一个向量，都存在一个一个向量，满足：

证明零向量定理

所以要证明,得出结论

python代码实现零向量

class Vector:
    def __init__(self, my_list):
        self._values = my_list

    @classmethod
    def zero(cls, dim):
        """返回一个dim维的零向量"""
        return cls([0]*dim)

    def __add__(self, other):
        """向量的加法，返回结果向量"""
        assert len(self) == len(other), \
            "向量的长度错误，向量之间长度必须是相等的"
        return Vector([a + b for a, b in zip(self, other)])

    def __sub__(self, other):
        """向量的减法， 返回结果向量"""
        assert len(self) == len(other), \
            "向量的长度错误，向量之间长度必须是相等的"
        return Vector([a - b for a, b in zip(self, other)])

    def __mul__(self, other):
        """返回数量乘法的结果向量， 只定义了self * other"""
        return Vector([other * e for e in self])

    def __rmul__(self, other):
        """返回向量的右乘方法， 只定义了 other * self"""
        return Vector([other * e for e in self])

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

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

    def __iter__(self):
        """返回向量的迭代器"""
        return self._values.__iter__()

    def __getitem__(self, item):
        """取向量的第index元素"""
        return self._values[item]

    def __len__(self):
        """返回向量的长度"""
        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))


if __name__ == '__main__':
    vec = Vector([5, 2])
    print(vec)
    print(len(vec))

    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))

    # 创建一个二维的0向量
    zero2 = Vector.zero(2)
    print("{} + {} = {}".format(vec, zero2, vec + zero2))   ###########

# 输出结果
(5, 2)
2
(5, 2) + (3, 1) = (8, 3)
(5, 2) - (3, 1) = (2, 1)
(5, 2) * 3 = (15, 6)
3 * (5, 2) = (15, 6)
+(5, 2) = (5, 2)
-(5, 2) = (-5, -2)
(5, 2) + (0, 0) = (5, 2)   ############