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# Linear equations

## Introduction

The system of linear equations is a system of equations in which the unknowns of each equation are all first-order (for example, a system of first-order equations with two elements). The study of linear equations in China was at least 1500 years earlier than Europe, and it was recorded in the equation chapter of "Nine Chapters of Arithmetic" in the early AD.

Linear equations are widely used. The well-known linear programming problem is the discussion of linear equations with certain constraints on the solution.

## Definition

xj is the unknown quantity, aij is the coefficient, and bi is the constant term.

It is called the coefficient matrix and augmented matrix. If x1=c1, x2=c2,..., xn=cn are substituted into the given equations and all the equations are established, then (c1, c2,..., cn) is called a solution. If c1, c2,..., cn are not all 0, then (c1, c2,..., cn) is said to be a non-zero solution. If the constant terms are all 0, it is called a homogeneous linear equation system, which always has zero solutions (0, 0, ..., 0). Two systems of equations, if their number of unknowns are the same and their solution sets are equal, are called the same solution equation system. The main issues discussed in linear equations are:

①When does a system of equations have a solution?

②There are solutions to the number of equations.

③Solve the system of solvable equations and determine the structure of the solution. These problems are all solved satisfactorily: the given equation system has a solution, then rank (A) = rank (augmented matrix); if rank (A) = rank = r, then r = n, there is a unique solution; r Solve by elimination method.

When a system of non-homogeneous linear equations has a solution, the only necessary and sufficient condition for the solution is that the corresponding homogeneous linear equation system has zero solutions; the necessary and sufficient condition for the infinite number of solutions is the corresponding homogeneous linear equation system There are non-zero solutions. But on the contrary, when the derived group of non-homogeneous linear equations has only zero solutions and non-zero solutions, it is not necessarily that the original equations have unique or infinite solutions. In fact, the equations may not have, that is, they may not have Solution.

Clem's rule (see Determinant) gives a formula for the solution of a special type of linear equations. The solution set of any homogeneous equation system with n unknowns constitutes a subspace of the n-dimensional space.

## Method of solution

① Cramer’s rule. There are two premises for using Cramer’s rule to solve a system of equations. One is that the number of equations must be equal to the number of unknowns, and the other is The determinant of the coefficient matrix must not be equal to zero. Using Cramer's rule to solve a system of equations is actually equivalent to solving a system of linear equations by the inverse matrix method. It establishes the relationship between the solution of the system of linear equations and its coefficients and constants. Formula, its workload is often very large, so Cramer’s law is often used for theoretical proofs and rarely used for specific solutions. ②Matrix elimination method. The augmented matrix of the linear equation group is transformed into a row-simplified ladder-shaped matrix through the elementary transformation of rows, and then the linear equation group with the row-simplified ladder-shaped matrix as the augmented matrix and The original system of equations has the same solution. When the equation system has a solution, the unknown quantity corresponding to the unit column vector is taken as the non-free unknown quantity, and the remaining unknown quantities are taken as the free unknown quantity, and the solution of the linear equation group can be found.

Regarding the unknown quantity is a linear equation, its general form is

where x1, x2,... , xn represents the unknown quantity, αij(1≤i≤m,1≤jn) is called the coefficient of equation ⑴, and bi (1≤i≤m) is called the constant term. The coefficients and constant terms are arbitrary complex numbers or elements of a certain domain.

When the constant terms b1, b2,..., bn are all equal to zero, then the equation system ⑴ It is a homogeneous linear equation system.

The m-rowncolumn matrix

formed by the coefficients of the equation group ⑴ is called the coefficient matrix of the equation group ⑴. Add a column consisting of constant items in A to obtain an m-row n+1 column matrix

called the augmented matrix of the equation system ⑴ .

If in the equation system ⑴, replace the unknown quantity x1 with a set of complex numbers or elements of the field F с1, с2,..., сn, x2,...,xn, the two ends of each equation are equal, then с1, с2,..., сn are called a solution of equations ⑴.

As for the linear equations, there are the following main results.

①The necessary and sufficient condition for the linear equations to have a solution is that the coefficient matrix A and the augmented matrix have the same rank.

②In the case of A and both have the same rank r>0, A has one rThe order sub-formula D is not equal to zero. Let

then the equation system ⑴ is the same as the equation system containing only the first r equations Solution. The first r equations can be rewritten as

The general solution formula of equations ⑵ is x1=D1/ D,x2=D2/D,...,xr= Dr/D,　⑶

where Dj(j=1, 2,...,r) is the one obtained by replacing the column j of D with the column on the right end of the equation system ⑵ r hierarchical determinant, that is,

so x1,x2,...,xr can be The remaining unknown quantities xr+1,xr+2,...,xn are expressed linearly, xr+1,xr+2,...,xn are called free unknowns.

When r<n, give a set of values ​​of free unknown quantity arbitrarily, and x can be obtained by ⑶ The value of i>1,x2,...,xr is one solution of equation system ⑴, and there is more than one solution of equation system ⑴. When r=n, the equation system ⑵ does not contain free unknowns, and the only solution of equation system ⑴ is given by ⑶. When m=n=r, formula (3) is called Cramer's rule.

Linear equations are the simplest and most important type of algebraic equations. A large number of scientific and technological problems are often boiled down to solving linear equations. Therefore, the numerical solution of linear equations occupies an important position in computational mathematics.