The solution of a system of linear equations is called a set. Formulas connecting the coordinates of vectors in the old and new bases. First multiplication and division, then addition and subtraction

Systems of linear equations. Lecture 6.

Systems of linear equations.

Basic concepts.

View system

called system - linear equations with unknowns.

The numbers , , are called system coefficients.

The numbers are called free members of the system, – system variables. Matrix

called main matrix of the system, and the matrix

– extended matrix system. Matrices - columns

And correspondingly matrices of free terms and unknowns of the system. Then in matrix form the system of equations can be written as . System solution is called the values of variables, upon substitution of which, all equations of the system turn into correct numerical equalities. Any solution to the system can be represented as a matrix-column. Then the matrix equality is true.

The system of equations is called joint if it has at least one solution and incompatible if there is no solution.

Solving a system of linear equations means finding out whether it is consistent and, if so, finding its general solution.

The system is called homogeneous if all its free terms are equal to zero. A homogeneous system is always consistent, since it has a solution

Kronecker–Copelli theorem.

The answer to the question of the existence of solutions to linear systems and their uniqueness allows us to obtain the following result, which can be formulated in the form of the following statements regarding a system of linear equations with unknowns

(1)

Theorem 2. System of linear equations (1) is consistent if and only if the rank of the main matrix is equal to the rank of the extended matrix (.

Theorem 3. If the rank of the main matrix of a simultaneous system of linear equations is equal to the number of unknowns, then the system has a unique solution.

Theorem 4. If the rank of the main matrix of a joint system is less than the number of unknowns, then the system has an infinite number of solutions.

Rules for solving systems.

3. Find the expression of the main variables in terms of free ones and obtain the general solution of the system.

4. By assigning arbitrary values to free variables, all values of the main variables are obtained.

Methods for solving systems of linear equations.

Inverse matrix method.

and , i.e. the system has a unique solution. Let's write the system in matrix form

Where , , .

Let's multiply both sides of the matrix equation on the left by the matrix

Since , we obtain , from which we obtain the equality for finding the unknowns

Example 27. Solve a system of linear equations using the inverse matrix method

Solution. Let us denote by the main matrix of the system

Let, then we find the solution using the formula.

Let's calculate.

Since , then the system has a unique solution. Let's find all algebraic complements

, ,

Thus

Let's check

The inverse matrix was found correctly. From here, using the formula, we find the matrix of variables.

Comparing the values of the matrices, we get the answer: .

Cramer's method.

Let a system of linear equations with unknowns be given

and , i.e. the system has a unique solution. Let us write the solution of the system in matrix form or

Let's denote

. . . . . . . . . . . . . . ,

Thus, we obtain formulas for finding the values of unknowns, which are called Cramer formulas.

Example 28. Solve the following system of linear equations using the Cramer method .

Solution. Let's find the determinant of the main matrix of the system

Since , then the system has a unique solution.

Let's find the remaining determinants for Cramer's formulas

Using Cramer's formulas we find the values of the variables

Gauss method.

The method consists of sequential elimination of variables.

Let a system of linear equations with unknowns be given.

The Gaussian solution process consists of two stages:

At the first stage, the extended matrix of the system is reduced, using elementary transformations, to a stepwise form

where , to which the system corresponds

After this the variables are considered free and are transferred to the right side in each equation.

At the second stage, the variable is expressed from the last equation, and the resulting value is substituted into the equation. From this equation

the variable is expressed. This process continues until the first equation. The result is an expression of the main variables through free variables .

Example 29. Solve the following system using the Gauss method

Solution. Let's write out the extended matrix of the system and bring it to stepwise form

Because greater than the number of unknowns, then the system is consistent and has an infinite number of solutions. Let's write the system for the step matrix

The determinant of the extended matrix of this system, composed of the first three columns, is not equal to zero, so we consider it to be basic. Variables

They will be basic and the variable will be free. Let’s move it in all equations to the left side

From the last equation we express

Substituting this value into the penultimate second equation, we get

where . Substituting the values of the variables and into the first equation, we find . Let's write the answer in the following form

To study a system of linear agebraic equations (SLAEs) for consistency means to find out whether this system has solutions or does not have them. Well, if there are solutions, then indicate how many there are.

We will need information from the topic "System of linear algebraic equations. Basic terms. Matrix form of notation". In particular, concepts such as system matrix and extended system matrix are needed, since the formulation of the Kronecker-Capelli theorem is based on them. As usual, we will denote the system matrix by the letter $A$, and the extended matrix of the system by the letter $\widetilde(A)$.

Kronecker-Capelli theorem

A system of linear algebraic equations is consistent if and only if the rank of the system matrix is equal to the rank of the extended matrix of the system, i.e. $\rang A=\rang\widetilde(A)$.

Let me remind you that a system is called joint if it has at least one solution. The Kronecker-Capelli theorem says this: if $\rang A=\rang\widetilde(A)$, then there is a solution; if $\rang A\neq\rang\widetilde(A)$, then this SLAE has no solutions (inconsistent). The answer to the question about the number of these solutions is given by a corollary of the Kronecker-Capelli theorem. In the formulation of the corollary, the letter $n$ is used, which is equal to the number of variables of the given SLAE.

Corollary to the Kronecker-Capelli theorem

If $\rang A\neq\rang\widetilde(A)$, then the SLAE is inconsistent (has no solutions).
If $\rang A=\rang\widetilde(A)< n$, то СЛАУ является неопределённой (имеет бесконечное количество решений).
If $\rang A=\rang\widetilde(A) = n$, then the SLAE is definite (has exactly one solution).

Please note that the formulated theorem and its corollary do not indicate how to find a solution to the SLAE. With their help, you can only find out whether these solutions exist or not, and if they exist, then how many.

Example No. 1

Explore SLAE $ \left \(\begin(aligned) & -3x_1+9x_2-7x_3=17;\\ & -x_1+2x_2-4x_3=9;\\ & 4x_1-2x_2+19x_3=-42. \end(aligned )\right.$ for compatibility. If the SLAE is compatible, indicate the number of solutions.

To find out the existence of solutions to a given SLAE, we use the Kronecker-Capelli theorem. We will need the matrix of the system $A$ and the extended matrix of the system $\widetilde(A)$, we will write them:

$$ A=\left(\begin(array) (ccc) -3 & 9 & -7 \\ -1 & 2 & -4 \\ 4 & -2 & 19 \end(array) \right);\; \widetilde(A)=\left(\begin(array) (ccc|c) -3 & 9 &-7 & 17 \\ -1 & 2 & -4 & 9\\ 4 & -2 & 19 & -42 \end(array) \right). $$

We need to find $\rang A$ and $\rang\widetilde(A)$. There are many ways to do this, some of which are listed in the Matrix Rank section. Typically, two methods are used to study such systems: “Calculating the rank of a matrix by definition” or “Calculating the rank of a matrix by the method of elementary transformations”.

Method number 1. Computing ranks by definition.

According to the definition, rank is the highest order of the minors of a matrix, among which there is at least one that is different from zero. Usually, the study begins with first-order minors, but here it is more convenient to immediately begin calculating the third-order minor of the matrix $A$. The third-order minor elements are located at the intersection of three rows and three columns of the matrix in question. Since the matrix $A$ contains only 3 rows and 3 columns, the third order minor of the matrix $A$ is the determinant of the matrix $A$, i.e. $\Delta A$. To calculate the determinant, we apply formula No. 2 from the topic “Formulas for calculating determinants of the second and third orders”:

$$ \Delta A=\left| \begin(array) (ccc) -3 & 9 & -7 \\ -1 & 2 & -4 \\ 4 & -2 & 19 \end(array) \right|=-21. $$

So, there is a third order minor of the matrix $A$, which is not equal to zero. It is impossible to construct a fourth-order minor, since it requires 4 rows and 4 columns, and the matrix $A$ has only 3 rows and 3 columns. So, the highest order of the minors of the matrix $A$, among which there is at least one that is not equal to zero, is equal to 3. Therefore, $\rang A=3$.

We also need to find $\rang\widetilde(A)$. Let's look at the structure of the matrix $\widetilde(A)$. Up to the line in the matrix $\widetilde(A)$ there are elements of the matrix $A$, and we found out that $\Delta A\neq 0$. Consequently, the matrix $\widetilde(A)$ has a third-order minor, which is not equal to zero. We cannot construct fourth-order minors of the matrix $\widetilde(A)$, so we conclude: $\rang\widetilde(A)=3$.

Since $\rang A=\rang\widetilde(A)$, then according to the Kronecker-Capelli theorem the system is consistent, i.e. has a solution (at least one). To indicate the number of solutions, we take into account that our SLAE contains 3 unknowns: $x_1$, $x_2$ and $x_3$. Since the number of unknowns is $n=3$, we conclude: $\rang A=\rang\widetilde(A)=n$, therefore, according to the corollary of the Kronecker-Capelli theorem, the system is definite, i.e. has a unique solution.

The problem is solved. What disadvantages and advantages does this method have? First, let's talk about the advantages. Firstly, we only needed to find one determinant. After this, we immediately made a conclusion about the number of solutions. Typically, standard standard calculations give systems of equations that contain three unknowns and have a unique solution. For such systems, this method is very convenient, because we know in advance that there is a solution (otherwise the example would not have been in the standard calculation). Those. All we have to do is show the existence of a solution in the fastest way. Secondly, the calculated value of the determinant of the system matrix (i.e. $\Delta A$) will be useful later: when we begin to solve a given system using the Cramer method or using the inverse matrix.

However, the method of calculating the rank is by definition undesirable to use if the matrix of the system $A$ is rectangular. In this case, it is better to use the second method, which will be discussed below. In addition, if $\Delta A=0$, then we cannot say anything about the number of solutions of a given inhomogeneous SLAE. Maybe the SLAE has an infinite number of solutions, or maybe none. If $\Delta A=0$, then additional research is required, which is often cumbersome.

To summarize what has been said, I note that the first method is good for those SLAEs whose system matrix is square. Moreover, the SLAE itself contains three or four unknowns and is taken from standard standard calculations or tests.

Method No. 2. Calculation of rank by the method of elementary transformations.

This method is described in detail in the corresponding topic. We will begin to calculate the rank of the matrix $\widetilde(A)$. Why matrices $\widetilde(A)$ and not $A$? The fact is that the matrix $A$ is part of the matrix $\widetilde(A)$, therefore, by calculating the rank of the matrix $\widetilde(A)$ we will simultaneously find the rank of the matrix $A$.

\begin(aligned) &\widetilde(A) =\left(\begin(array) (ccc|c) -3 & 9 &-7 & 17 \\ -1 & 2 & -4 & 9\\ 4 & - 2 & 19 & -42 \end(array) \right) \rightarrow \left|\text(swap the first and second lines)\right| \rightarrow \\ &\rightarrow \left(\begin(array) (ccc|c) -1 & 2 & -4 & 9 \\ -3 & 9 &-7 & 17\\ 4 & -2 & 19 & - 42 \end(array) \right) \begin(array) (l) \phantom(0) \\ r_2-3r_1\\ r_3+4r_1 \end(array) \rightarrow \left(\begin(array) (ccc| c) -1 & 2 & -4 & 9 \\ 0 & 3 &5 & -10\\ 0 & 6 & 3 & -6 \end(array) \right) \begin(array) (l) \phantom(0 ) \\ \phantom(0)\\ r_3-2r_2 \end(array)\rightarrow\\ &\rightarrow \left(\begin(array) (ccc|c) -1 & 2 & -4 & 9 \\ 0 & 3 &5 & -10\\ 0 & 0 & -7 & 14 \end(array) \right) \end(aligned)

We have reduced the matrix $\widetilde(A)$ to echelon form. The resulting echelon matrix has three non-zero rows, so its rank is 3. Consequently, the rank of the matrix $\widetilde(A)$ is equal to 3, i.e. $\rang\widetilde(A)=3$. When making transformations with the elements of the matrix $\widetilde(A)$, we simultaneously transformed the elements of the matrix $A$ located up to the line. The matrix $A$ is also reduced to echelon form: $\left(\begin(array) (ccc) -1 & 2 & -4 \\ 0 & 3 &5 \\ 0 & 0 & -7 \end(array) \right )$. Conclusion: the rank of matrix $A$ is also 3, i.e. $\rang A=3$.

Since $\rang A=\rang\widetilde(A)$, then according to the Kronecker-Capelli theorem the system is consistent, i.e. has a solution. To indicate the number of solutions, we take into account that our SLAE contains 3 unknowns: $x_1$, $x_2$ and $x_3$. Since the number of unknowns is $n=3$, we conclude: $\rang A=\rang\widetilde(A)=n$, therefore, according to the corollary of the Kronecker-Capelli theorem, the system is defined, i.e. has a unique solution.

What are the advantages of the second method? The main advantage is its versatility. It doesn't matter to us whether the matrix of the system is square or not. In addition, we actually carried out forward transformations of the Gaussian method. There are only a couple of steps left, and we could obtain a solution to this SLAE. To be honest, I like the second method more than the first, but the choice is a matter of taste.

Answer: The given SLAE is consistent and defined.

Example No. 2

Explore SLAE $ \left\( \begin(aligned) & x_1-x_2+2x_3=-1;\\ & -x_1+2x_2-3x_3=3;\\ & 2x_1-x_2+3x_3=2;\\ & 3x_1- 2x_2+5x_3=1;\\ & 2x_1-3x_2+5x_3=-4. \end(aligned) \right.$ for compatibility.

We will find the ranks of the system matrix and the extended system matrix using the method of elementary transformations. Extended system matrix: $\widetilde(A)=\left(\begin(array) (ccc|c) 1 & -1 & 2 & -1\\ -1 & 2 & -3 & 3 \\ 2 & -1 & 3 & 2 \\ 3 & -2 & 5 & 1 \\ 2 & -3 & 5 & -4 \end(array) \right)$. Let's find the required ranks by transforming the extended matrix of the system:

$$ \left(\begin(array) (ccc|c) 1 & -1 & 2 & -1\\ -1 & 2 & -3 & 3 \\ 2 & -3 & 5 & -4 \\ 3 & -2 & 5 & 1 \\ 2 & -1 & 3 & 2 \end(array) \right) \begin(array) (l) \phantom(0)\\r_2+r_1\\r_3-2r_1\\ r_4 -3r_1\\r_5-2r_1\end(array)\rightarrow \left(\begin(array) (ccc|c) 1 & -1 & 2 & -1\\ 0 & 1 & -1 & 2 \\ 0 & -1 & 1 & -2 \\ 0 & 1 & -1 & 4 \\ 0 & 1 & -1 & 4 \end(array) \right) \begin(array) (l) \phantom(0)\\ \phantom(0)\\r_3-r_2\\ r_4-r_2\\r_5+r_2\end(array)\rightarrow\\ $$ $$ \rightarrow\left(\begin(array) (ccc|c) 1 & -1 & 2 & -1\\ 0 & 1 & -1 & 2 \\ 0 & 0 & 0 & 2 \\ 0 & 0 & 0 & 2 \\ 0 & 0 & 0 & 0 \end(array) \ right) \begin(array) (l) \phantom(0)\\\phantom(0)\\\phantom(0)\\ r_4-r_3\\\phantom(0)\end(array)\rightarrow \left (\begin(array) (ccc|c) 1 & -1 & 2 & -1\\ 0 & 1 & -1 & 2 \\ 0 & 0 & 0 & 2 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end(array) \right) $$

The extended matrix of the system is reduced to a stepwise form. The rank of an echelon matrix is equal to the number of its nonzero rows, so $\rang\widetilde(A)=3$. The matrix $A$ (up to the line) is also reduced to echelon form, and its rank is 2, $\rang(A)=2$.

Since $\rang A\neq\rang\widetilde(A)$, then according to the Kronecker-Capelli theorem the system is inconsistent (i.e., has no solutions).

Answer: The system is inconsistent.

Example No. 3

Explore SLAE $ \left\( \begin(aligned) & 2x_1+7x_3-5x_4+11x_5=42;\\ & x_1-2x_2+3x_3+2x_5=17;\\ & -3x_1+9x_2-11x_3-7x_5=-64 ;\\ & -5x_1+17x_2-16x_3-5x_4-4x_5=-90;\\ & 7x_1-17x_2+23x_3+15x_5=132. \end(aligned) \right.$ for compatibility.

We bring the extended matrix of the system to a stepwise form:

$$ \left(\begin(array)(ccccc|c) 2 & 0 & 7 & -5 & 11 & 42\\ 1 & -2 & 3 & 0 & 2 & 17 \\ -3 & 9 & -11 & 0 & -7 & -64 \\ -5 & 17 & -16 & -5 & -4 & -90 \\ 7 & -17 & 23 & 0 & 15 & 132 \end(array) \right) \overset (r_1\leftrightarrow(r_3))(\rightarrow) $$ $$ \rightarrow\left(\begin(array)(ccccc|c) 1 & -2 & 3 & 0 & 2 & 17\\ 2 & 0 & 7 & -5 & 11 & 42\\ -3 & 9 & -11 & 0 & -7 & -64\\ -5 & 17 & -16 & -5 & -4 & -90\\ 7 & -17 & 23 & 0 & 15 & 132 \end(array) \right) \begin(array) (l) \phantom(0)\\ r_2-2r_1 \\r_3+3r_1 \\ r_4+5r_1 \\ r_5-7r_1 \end( array) \rightarrow \left(\begin(array)(ccccc|c) 1 & -2 & 3 & 0 & 2 & 17\\ 0 & 4 & 1 & -5 & 7 & 8\\ 0 & 3 & - 2 & 0 & -1 & -13\\ 0 & 7 & -1 & -5 & 6 & -5 \\ 0 & -3 & 2 & 0 & 1 & 13 \end(array) \right) \begin( array) (l) \phantom(0)\\ \phantom(0)\\4r_3+3r_2 \\ 4r_4-7r_2 \\ 4r_5+3r_2 \end(array) \rightarrow $$ $$ \rightarrow\left(\begin (array)(ccccc|c) 1 & -2 & 3 & 0 & 2 & 17\\ 0 & 4 & 1 & -5 & 7 & 8\\ 0 & 0 & -11 & 15 & -25 & -76 \\ 0 & 0 & -11 & 15 & -25 & -76 \\ 0 & 0 & 11 & -15 & 25 & 76 \end(array) \right) \begin(array) (l) \phantom(0 )\\ \phantom(0)\\\phantom(0) \\ r_4-r_3 \\ r_5+r_2 \end(array) \rightarrow \left(\begin(array)(ccccc|c) 1 & -2 & 3 & 0 & 2 & 17\\ 0 & 4 & 1 & -5 & 7 & 8\\ 0 & 0 & -11 & 15 & -25 & -76\\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end(array) \right) $$

We have brought the extended matrix of the system and the matrix of the system itself to a stepwise form. The rank of the extended matrix of the system is equal to three, the rank of the matrix of the system is also equal to three. Since the system contains $n=5$ unknowns, i.e. $\rang\widetilde(A)=\rang(A)\lt(n)$, then according to the corollary of the Kronecker-Capelli theorem, this system is indeterminate, i.e. has an infinite number of solutions.

Answer: The system is uncertain.

In the second part, we will analyze examples that are often included in standard calculations or tests in higher mathematics: consistency research and solution of SLAE depending on the values of the parameters included in it.

Solving systems of linear algebraic equations is one of the main problems of linear algebra. This problem has important applied significance in solving scientific and technical problems; in addition, it is auxiliary in the implementation of many algorithms in computational mathematics, mathematical physics, and processing the results of experimental research.

A system of linear algebraic equations is called a system of equations of the form: (1)

Where – unknown; - free members.

Solving a system of equations(1) call any collection of numbers that, when placed in system (1) in place of the unknowns converts all equations of the system into correct numerical equalities.

The system of equations is called joint, if it has at least one solution, and incompatible, if it has no solutions.

The simultaneous system of equations is called certain, if it has one unique solution, and uncertain, if it has at least two different solutions.

The two systems of equations are called equivalent or equivalent, if they have the same set of solutions.

System (1) is called homogeneous, if the free terms are zero:

A homogeneous system is always consistent - it has a solution (perhaps not the only one).

If in system (1), then we have the system n linear equations with n unknown: where – unknown; – coefficients for unknowns, - free members.

A linear system may have a single solution, infinitely many solutions, or no solution at all.

Consider a system of two linear equations with two unknowns

If then the system has a unique solution;

if then the system has no solutions;

if then the system has an infinite number of solutions.

Example. The system has a unique solution to a pair of numbers

The system has an infinite number of solutions. For example, solutions to a given system are pairs of numbers, etc.

The system has no solutions, since the difference of two numbers cannot take two different values.

Definition. Second order determinant called an expression of the form:

The determinant is designated by the symbol D.

Numbers A 11, …, A 22 are called elements of the determinant.

Diagonal formed by elements A 11 ; A 22 are called main diagonal formed by elements A 12 ; A 21 − side

Thus, the second-order determinant is equal to the difference between the products of the elements of the main and secondary diagonals.

Note that the answer is a number.

Example. Let's calculate the determinants:

Consider a system of two linear equations with two unknowns: where X 1, X 2 – unknown; A 11 , …, A 22 – coefficients for unknowns, b 1 ,b 2 – free members.

If a system of two equations with two unknowns has a unique solution, then it can be found using second-order determinants.

Definition. A determinant made up of coefficients for unknowns is called system determinant: D= .

The columns of the determinant D contain the coefficients, respectively, for X 1 and at , X 2. Let's introduce two additional qualifier, which are obtained from the determinant of the system by replacing one of the columns with a column of free terms: D 1 = D 2 = .

Theorem 14(Kramer, for the case n=2). If the determinant D of the system is different from zero (D¹0), then the system has a unique solution, which is found using the formulas:

These formulas are called Cramer's formulas.

Example. Let's solve the system using Cramer's rule:

Solution. Let's find the numbers

Answer.

Definition. Third order determinant called an expression of the form:

Elements A 11; A 22 ; A 33 – form the main diagonal.

Numbers A 13; A 22 ; A 31 – form a side diagonal.

The entry with a plus includes: the product of elements on the main diagonal, the remaining two terms are the product of elements located at the vertices of triangles with bases parallel to the main diagonal. The minus terms are formed according to the same scheme with respect to the secondary diagonal.

Example. Let's calculate the determinants:

Consider a system of three linear equations with three unknowns: where – unknown; – coefficients for unknowns, - free members.

In the case of a unique solution, a system of 3 linear equations with three unknowns can be solved using 3rd order determinants.

The determinant of system D has the form:

Let us introduce three additional determinants:

Theorem 15(Kramer, for the case n=3). If the determinant D of the system is different from zero, then the system has a unique solution, which is found using Cramer’s formulas:

Example. Let's solve the system using Cramer's rule.

Solution. Let's find the numbers

Let's use Cramer's formulas and find the solution to the original system:

Answer.

Note that Cramer's theorem is applicable when the number of equations is equal to the number of unknowns and when the determinant of the system D is nonzero.

If the determinant of a system is equal to zero, then in this case the system can either have no solutions or have an infinite number of solutions. These cases are studied separately.

Let us note only one case. If the determinant of the system is equal to zero (D=0), and at least one of the additional determinants is different from zero, then the system has no solutions, that is, it is inconsistent.

Cramer's theorem can be generalized to the system n linear equations with n unknown: where – unknown; – coefficients for unknowns, - free members.

If the determinant of a system of linear equations with unknowns, then the only solution to the system is found using Cramer’s formulas:

An additional determinant is obtained from the determinant D if it contains a column of coefficients for the unknown x i replace with a column of free members.

Note that the determinants D, D 1 , … , D n have order n.

Gauss method for solving systems of linear equations

One of the most common methods for solving systems of linear algebraic equations is the method of sequential elimination of unknowns −Gauss method. This method is a generalization of the substitution method and consists of sequentially eliminating unknowns until one equation with one unknown remains.

The method is based on some transformations of a system of linear equations, which results in a system equivalent to the original system. The method algorithm consists of two stages.

The first stage is called straight ahead Gauss method. It consists of sequentially eliminating unknowns from equations. To do this, in the first step, divide the first equation of the system by (otherwise, rearrange the equations of the system). Designate the coefficients of the resulting reduced equation, multiply it by the coefficient and subtract it from the second equation of the system, thereby eliminating it from the second equation (zeroing the coefficient).

Do the same with the remaining equations and obtain a new system, in all equations of which, starting from the second, the coefficients for , contain only zeros. Obviously, the resulting new system will be equivalent to the original system.

If the new coefficients, for , are not all equal to zero, they can be excluded in the same way from the third and subsequent equations. Continuing this operation for the following unknowns, the system is brought to the so-called triangular form:

Here the symbols indicate the numerical coefficients and free terms that have changed as a result of transformations.

From the last equation of the system, the remaining unknowns are determined in a unique way, and then by sequential substitution.

Comment. Sometimes, as a result of transformations, in any of the equations all the coefficients and the right-hand side turn to zero, that is, the equation turns into the identity 0=0. By eliminating such an equation from the system, the number of equations is reduced compared to the number of unknowns. Such a system cannot have a single solution.

If, in the process of applying the Gauss method, any equation turns into an equality of the form 0 = 1 (the coefficients for the unknowns turn to 0, and the right-hand side takes on a non-zero value), then the original system has no solution, since such an equality is false for any values unknown.

Consider a system of three linear equations with three unknowns:

Where – unknown; – coefficients for unknowns, - free members. , substituting what was found

Solution. Applying the Gaussian method to this system, we obtain

Where does the last equality fail for any values of the unknowns, therefore, the system has no solution.

Answer. The system has no solutions.

Note that the previously discussed Cramer method can be used to solve only those systems in which the number of equations coincides with the number of unknowns, and the determinant of the system must be non-zero. The Gauss method is more universal and suitable for systems with any number of equations.

A system of linear equations is a union of n linear equations, each containing k variables. It is written like this:

Many, when encountering higher algebra for the first time, mistakenly believe that the number of equations must necessarily coincide with the number of variables. In school algebra this usually happens, but for higher algebra this is, generally speaking, not true.

The solution to a system of equations is a sequence of numbers (k 1, k 2, ..., k n), which is the solution to each equation of the system, i.e. when substituting into this equation instead of the variables x 1, x 2, ..., x n gives the correct numerical equality.

Accordingly, solving a system of equations means finding the set of all its solutions or proving that this set is empty. Since the number of equations and the number of unknowns may not coincide, three cases are possible:

The system is inconsistent, i.e. the set of all solutions is empty. A rather rare case that is easily detected no matter what method is used to solve the system.
The system is joint and determined, i.e. has exactly one solution. The classic version, well known since school.
The system is consistent and undefined, i.e. has infinitely many solutions. This is the toughest option. It is not enough to indicate that “the system has an infinite set of solutions” - it is necessary to describe how this set is structured.

A variable x i is called allowed if it is included in only one equation of the system, and with a coefficient of 1. In other words, in other equations the coefficient of the variable x i must be equal to zero.

If we select one allowed variable in each equation, we obtain a set of allowed variables for the entire system of equations. The system itself, written in this form, will also be called resolved. Generally speaking, one and the same original system can be reduced to different permitted ones, but for now we are not concerned about this. Here are examples of permitted systems:

Both systems are resolved with respect to the variables x 1 , x 3 and x 4 . However, with the same success it can be argued that the second system is resolved with respect to x 1, x 3 and x 5. It is enough to rewrite the very last equation in the form x 5 = x 4.

Now let's consider a more general case. Let us have k variables in total, of which r are allowed. Then two cases are possible:

The number of allowed variables r is equal to the total number of variables k: r = k. We obtain a system of k equations in which r = k allowed variables. Such a system is joint and definite, because x 1 = b 1, x 2 = b 2, ..., x k = b k;
The number of allowed variables r is less than the total number of variables k: r< k . Остальные (k − r ) переменных называются свободными - они могут принимать любые значения, из которых легко вычисляются разрешенные переменные.

So, in the above systems, the variables x 2, x 5, x 6 (for the first system) and x 2, x 5 (for the second) are free. The case when there are free variables is better formulated as a theorem:

Please note: this is a very important point! Depending on how you write the resulting system, the same variable can be either allowed or free. Most higher mathematics tutors recommend writing out variables in lexicographic order, i.e. ascending index. However, you are under no obligation to follow this advice.

Theorem. If in a system of n equations the variables x 1, x 2, ..., x r are allowed, and x r + 1, x r + 2, ..., x k are free, then:

If we set the values of the free variables (x r + 1 = t r + 1, x r + 2 = t r + 2, ..., x k = t k), and then find the values x 1, x 2, ..., x r, we get one of decisions.

If in two solutions the values of free variables coincide, then the values of allowed variables also coincide, i.e. solutions are equal.

What is the meaning of this theorem? To obtain all solutions to a resolved system of equations, it is enough to isolate the free variables. Then, assigning different values to the free variables, we will obtain ready-made solutions. That's all - in this way you can get all the solutions of the system. There are no other solutions.

Conclusion: the resolved system of equations is always consistent. If the number of equations in a resolved system is equal to the number of variables, the system will be definite; if less, it will be indefinite.

And everything would be fine, but the question arises: how to obtain a resolved one from the original system of equations? For this there is

WITH n unknown is a system of the form:

Where a ij And b i (i=1,…,m; b=1,…,n)- some known numbers, and x 1 ,…,x n- unknown numbers. In the designation of coefficients a ij index i determines the number of the equation, and the second j- the number of the unknown at which this coefficient is located.

Homogeneous system - when all free terms of the system are equal to zero ( b 1 = b 2 = … = b m = 0), the opposite situation is heterogeneous system.

Square system - when the number m equations equals the number n unknown.

System solution- totality n numbers c 1, c 2, …, c n, such that substitution of all c i instead of x i into a system turns all its equations into identities.

Joint system - when the system has at least 1 solution, and non-cooperative system when the system has no solutions.

A joint system of this type (as given above, let it be (1)) can have one or more solutions.

Solutions c 1 (1) , c 2 (1) , …, c n (1) And c 1 (2) , c 2 (2) , …, c n (2) joint systems of type (1) will be various, when even 1 of the equalities is not satisfied:

c 1 (1) = c 1 (2) , c 2 (1) = c 2 (2) , …, c n (1) = c n (2) .

A joint system of type (1) will be certain when she has only one solution; when a system has at least 2 different solutions, it becomes underdetermined. When there are more equations than unknowns, the system is redefined.

The coefficients for the unknowns are written as a matrix:

It is called matrix of the system.

The numbers that appear on the right sides of the equations are b 1 ,…,b m are free members.

Totality n numbers c 1 ,…,c n is a solution to this system when all equations of the system become equal after substituting numbers in them c 1 ,…,c n instead of the corresponding unknowns x 1 ,…,x n.

When solving a system of linear equations, 3 options may arise:

1. The system has only one solution.

2. The system has an infinite number of solutions. For example, . The solution to this system will be all pairs of numbers that differ in sign.

3. The system has no solutions. For example. . if a solution existed, then x 1 + x 2 would be equal to 0 and 1 at the same time.

Methods for solving systems of linear equations.

Direct methods give an algorithm by which the exact solution is found SLAU(systems of linear algebraic equations). And if the accuracy had been absolute, they would have found it. A real electrical computer, of course, operates with an error, so the solution will be approximate.