编写程序定义向量类 Vector，重载运算符+、-、*和=，实现向量之间的加、减、乘、赋值运算；重载 运算符>>、<<实现向量的输入、输出功能。

以下是向量类 Vector 的定义及重载运算符的实现：

#include <iostream>
#include <vector>
using namespace std;

class Vector {
public:
    Vector() {}
    Vector(const vector<double>& v) : vec(v) {}

    Vector operator+(const Vector& rhs) const {
        if (vec.size() != rhs.vec.size()) {
            throw "Sizes don't match";
        }
        vector<double> res(vec.size());
        for (int i = 0; i < vec.size(); ++i) {
            res[i] = vec[i] + rhs.vec[i];
        }
        return Vector(res);
    }

    Vector operator-(const Vector& rhs) const {
        if (vec.size() != rhs.vec.size()) {
            throw "Sizes don't match";
        }
        vector<double> res(vec.size());
        for (int i = 0; i < vec.size(); ++i) {
            res[i] = vec[i] - rhs.vec[i];
        }
        return Vector(res);
    }

    Vector operator*(double k) const {
        vector<double> res(vec.size());
        for (int i = 0; i < vec.size(); ++i) {
            res[i] = vec[i] * k;
        }
        return Vector(res);
    }

    Vector& operator=(const Vector& rhs) {
        vec = rhs.vec;
        return *this;
    }

    friend istream& operator>>(istream& is, Vector& v);
    friend ostream& operator<<(ostream& os, const Vector& v);

private:
    vector<double> vec;
};

istream& operator>>(istream& is, Vector& v) {
    int n;
    is >> n;
    vector<double> tmp(n);
    for (int i = 0; i < n; ++i) {
        is >> tmp[i];
    }
    v = Vector(tmp);
    return is;
}

ostream& operator<<(ostream& os, const Vector& v) {
    os << "[ ";
    for (int i = 0; i < v.vec.size(); ++i) {
        os << v.vec[i] << " ";
    }
    os << "]";
    return os;
}

int main() {
    Vector v1, v2;
    cout << "Enter the first vector: ";
    cin >> v1;
    cout << "Enter the second vector: ";
    cin >> v2;
    cout << "v1 = " << v1 << endl;
    cout << "v2 = " << v2 << endl;
    cout << "v1 + v2 = " << v1 + v2 << endl;
    cout << "v1 - v2 = " << v1 - v2 << endl;
    cout << "2 * v1 = " << 2 * v1 << endl;
    cout << "v1 = v2 = " << (v1 = v2) << endl;
    return 0;
}

在上面的程序中，向量类 Vector 中重载了运算符 +、-、* 和 =，并且定义了友元函数重载运算符 >> 和 <<。其中，运算符 + 和 - 分别实现了向量的加法和减法，需要先判断两个向量的维数是否相同；运算符 * 实现了向量与标量的乘法；运算符 = 实现了向量的赋值操作。友元函数重载运算符 >> 实现了向量的输入，需要先输入向量的维数，再输入向量中的每个元素；友元函数重载运算符 << 实现了向量的输出，将向量的每个元素用空格隔开并放在方括号内。在 main 函数中，先输入两个向量，然后分别进行加、减、乘、赋值操作，并输出结果。