线性方程组中英文翻译

应数121 陈珍妮 12453101

英文原稿：

System of Linear equations

As we all know，Linear equations are important components of linear algebra, and in real life, there is a wide range of production applications，and it plays an important role in electronic engineering, software development, personnel management, transportation, etc. There are different ways in different types of linear equations, mainly Cramer's rule, matrix elimination method.

A general system of m linear equations with n unknowns can be written as

Here are the unknowns, are the coefficients of the system, and are the constant terms. Matrix equation

The vector equation is equivalent to a matrix equation of the form

，where A is an m×n matrix, x is a column vector with n entries, and b is a column vector with m entries.

The number of vectors in a basis is now expressed as the rank of the matrix.

The main methods：

（1）Elimination of variables

The simplest method for solving a system of linear equations is to repeatedly eliminate variables. This method can be described as follows:

1.In the first equation, solve for one of the variables in terms

of the others.

2.Substitute this expression into the remaining equations. This

yields a system of equations with one fewer equation and one fewer unknown.

3.Continue until you have reduced the system to a single linear

equation.

4.Solve this equation, and then back-substitute until the entire

solution is found.

For example, consider the following system:

Solving the first equation for x gives x= 5 + 2z? 3y, and plugging this into the second and third equation yields

Solving the first of these equations for y yields y= 2 + 3z, and plugging this into the second equation yields z= 2. We now have:

Substituting z= 2 into the second equation gives y= 8, and substituting z= 2 and y= 8 into the first equation yields x= ?15. Therefore, the solution is (x, y, z) = (?15, 8, 2).

（2）Row reduction

In row reduction, the linear system is represented as an augmented matrix:

This matrix is then modified using elementary row operations until it reaches reduced row echelon form. There are three types of elementary row operations:

Type 1: Swap the positions of two rows.

Type 2: Multiply a row by a nonzero scalar.

Type 3: Add to one row a scalar multiple of another.

Because these operations are reversible, the augmented matrix produced always represents a linear system that is equivalent to the original.

The following computation shows Gauss-Jordan elimination applied to the matrix above:

The last matrix is in reduced row echelon form, and represents the system x= ?15, y= 8, z= 2.

（3）Cramer's rule

Cramer's rule is an explicit formula for the solution of a system of linear equations, with each variable given by a quotient of

two determinants. For example, the solution to the system

is given by

For each variable, the denominator is the determinant of the matrix of coefficients, while the numerator is the determinant of a matrix in which one column has been replaced by the vector of constant terms.

Though Cramer's rule is important theoretically, it has little practical value for large matrices, since the computation of large determinants is somewhat cumbersome. Further, Cramer's rule has very poor numerical properties, making it unsuitable for solving even small systems, unless the operations are performed in rational arithmetic with unbounded precision.

（4）Matrix solution

If the equation system is expressed in the matrix form , the entire solution set can also be expressed in matrix form. If the matrix A is square (has m rows and n=m columns) and has full rank (all m rows are independent), then the system has a unique solution

given by where is the inverse of A. More generally, regardless of whether m=n or not and regardless of the rank of A, all solutions (if any exist) are given using the Moore-Penrose

pseudo-inverse of A, denoted , as follows:

，where is a vector of free parameters that ranges over all possible n×1 vectors. A necessary and sufficient condition for any solution(s) to exist is that the potential solution obtained using satisfy — that

is,that If this condition does not hold, the equation system is inconsistent and has no solution. If the condition holds, the system is consistent and at least one solution exists. For example, in the above-mentioned case in which A is square and of full

rank, simply equals and the general solution equation simplifies

to as

previously stated, where has completely dropped out of the solution, leaving only a single solution.

中文翻译：

线性方程组

众所周知，线性方程组是线性代数的重要组成部分，它在现实生活中有广泛的生产应用，并且在电子工程、软件开发、人事管理、运输等也扮演重要的角色。不同类型的线性方程组有不同的方法，主要方法有克莱姆法则，矩阵消元法。

由n个未知数、m个线性方程组成的方程组一般形式可以写为

这里是未知数，是方程组的系数，

是常数项。

矩阵方程

向量方程相当于一个形如的矩阵方程，其中 A 是 m × n 矩阵，x 是n 项的一个列向量与和 b 是一个m项的一个列向量。

在向量基底上的数量现在被表示为矩阵的秩。

主要方法：

（1）消元法

用于求解线性方程系统的最简单的方法是反复消除变量。该方法可以描述为如下：

1、在第一个方程，其中一个变量用其他变量表示。

2、把该表达式代入剩余的方程。这就产生少一个方程和少一个未知数的方程组。

3、继续，直到你的系统减少为一个线性方程

4、解这个方程，然后返回代入，直到整个答案找到。

例如，考虑下面的方程组：

解决第一个方程对于x给出x= 5+2 Z - 3Y，并把它代入第二和第三方程产生

解决第一个这些方程对于Y 产生Y=2+3Z，并把它代入到这第二个方程产生Z =2。我们现在有

代Z = 2到第二方程给出：Y =8，而把Z = 2和y= 8代入到第一方程产生x= -15。因此，该答案是（x，Y，Z）=（-15，8,2）。

（2）行变换

在行变换中，线性方程组被表示为增广矩阵

然后，这个矩阵用初等行变换修改，直到达到最简行阶梯形式的。这样有三种类型的初等行操作：

1、类型1：交换两行的位置

2、类型2：用一个非零标量乘以一排。

3、类型3：把另一个标量的倍数添加到一行上。

因为这些操作是可逆的，产生的增广矩阵总是表示的线性方程组，所以它等于最初的。

下面的计算示出了高斯消去法应用到上面的矩阵：

最后一个矩阵是最简行阶梯形式，并代表方程组x=-15，Y =8，Z =2。

（3）克莱姆法则

对于线性方程组的解来说，克莱姆公式是一个明确的公式，即由两个决定因素的商得出的每个变量。例如，将该答案在方程组

中由下式

得出。

对于每个变量，分母是系数矩阵的行列式，而分子是一个矩阵的行列式，其中一列是已被取代的常数项向量。虽然克莱姆法则是很重要的理论，但它对于大型矩阵的实用价值不大，因为大决定因素的计算是有点麻烦的。此外，克莱姆法则具有非常差的数值特性，使得它不适合解决哪怕是很小的方程组，除非操作都是在理性的计算和无限精度下进行。

（4）矩阵方法

如果该方程组表示在矩阵形式中，整个解集也可以表示为矩阵形式。如果矩阵A是正方形（有m行n列）那么它就是满秩的（全部m行是独立的），

然后该方程组可以由给出一个唯一的解，其中是A的逆矩阵。

更一般地，无论M是否等于 n，也不管A的秩是多少，所有的答案（如果有的话）都是通过Moore-Penrose逆矩阵得出的，记为，答案如下:

,其中w是自由参数的向量，其范围在所有可能的n

×1向量。对于任何解决方案都存在的一个充分必要条件是用满足得到的一个潜在的答案 - 也就是如果这个条件不成立，该方程组不一致，也没有解。如果该条件成立，则方程组是一致的，并至少有一

个解存在。例如，在上述案例中，A是方形并且满秩的，简单地等于，一般的解方程可以简化为

，

如前所述，已经完全退出了答案，只留下一个单一的答案。