The system is homogeneous. Homogeneous systems of linear equations

The linear system is called homogeneous , if all its free terms are equal to 0.

In matrix form, a homogeneous system is written:
.

Homogeneous system (2) is always consistent . Obviously, the set of numbers
,
, …,
satisfies every equation of the system. Solution
called zero or trivial decision. Thus, a homogeneous system always has a zero solution.

Under what conditions will the homogeneous system (2) have non-zero (non-trivial) solutions?

Theorem 1.3 Homogeneous system (2) has non-zero solutions if and only if the rank r its main matrix fewer unknowns n .

System (2) – uncertain
.

Corollary 1. If the number of equations m homogeneous system has fewer variables
, then the system is uncertain and has many non-zero solutions.

Corollary 2. Square homogeneous system
has non-zero solutions if and when the main matrix of this system degenerate, i.e. determinant
.

Otherwise, if the determinant
, a square homogeneous system has the only thing zero solution
.

Let the rank of the system (2)
that is, system (2) has non-trivial solutions.

Let And - particular solutions of this system, i.e.
And
.

Properties of solutions of a homogeneous system

Really, .

Really, .

Combining properties 1) and 2), we can say that if

…,
- solutions of a homogeneous system (2), then any linear combination of them is also its solution. Here
- arbitrary real numbers.

Can be found
linearly independent partial solutions homogeneous system (2), with the help of which you can obtain any other particular solution of this system, i.e. obtain a general solution to system (2).

Definition 2.2 Totality
linearly independent partial solutions

…,
homogeneous system (2) such that each solution of system (2) can be represented as a linear combination of them is called fundamental system of solutions (FSR) of a homogeneous system (2).

Let

…,
is a fundamental system of solutions, then the general solution of the homogeneous system (2) can be represented as:

Where

.

Comment. To obtain the FSR, you need to find private solutions

…,
, giving one free variable the value “1” in turn, and all other free variables the value “0”.

We get ,, …,- FSR.

Example. Find the general solution and fundamental system of solutions of the homogeneous system of equations:

Solution. Let's write down the extended matrix of the system, having previously put the last equation of the system in first place, and bring it to a stepwise form. Since the right-hand sides of the equations do not change as a result of elementary transformations, remaining zero, the column

may not be written out.

̴
̴
̴

System rank where
- number of variables. The system is uncertain and has many solutions.

Basic minor for variables
non-zero:
choose
as basic variables, the rest
- free variables (take any real values).

The last matrix in the chain corresponds to a stepwise system of equations:

(3)

Let's express the basic variables
through free variables
(reverse of the Gaussian method).

From the last equation we express :
and substitute it into the first equation. We'll get it. Let us open the brackets, give similar ones and express :
.

Believing
,
,
, Where
, let's write

- general solution of the system.

Let's find a fundamental system of solutions

,,.

Then the general solution of the homogeneous system can be written as:

Comment. The FSR could have been found in another way, without first finding a general solution to the system. To do this, the resulting step system (3) had to be solved three times, assuming for :
; For :
; For :
.

Homogeneous systems of linear algebraic equations

As part of the lessons Gaussian method And Incompatible systems/systems with a common solution we considered inhomogeneous systems of linear equations, Where free member(which is usually on the right) at least one from the equations was different from zero.
And now, after a good warm-up with matrix rank, we will continue to polish the technique elementary transformations on homogeneous system of linear equations.
Based on the first paragraphs, the material may seem boring and mediocre, but this impression is deceptive. In addition to further development of techniques, there will be a lot of new information, so please try not to neglect the examples in this article.

What is a homogeneous system of linear equations?

The answer suggests itself. A system of linear equations is homogeneous if the free term everyone equation of the system is zero. For example:

It is absolutely clear that a homogeneous system is always consistent, that is, it always has a solution. And, first of all, what catches your eye is the so-called trivial solution . Trivial, for those who do not understand the meaning of the adjective at all, means without a show-off. Not academically, of course, but intelligibly =) ...Why beat around the bush, let's find out if this system has any other solutions:

Example 1

Solution: to solve a homogeneous system it is necessary to write system matrix and with the help of elementary transformations bring it to a stepwise form. Please note that here there is no need to write down the vertical bar and the zero column of free terms - after all, no matter what you do with zeros, they will remain zeros:

(1) The first line was added to the second line, multiplied by –2. The first line was added to the third line, multiplied by –3.

(2) The second line was added to the third line, multiplied by –1.

Dividing the third line by 3 doesn't make much sense.

As a result of elementary transformations, an equivalent homogeneous system is obtained , and, using the inverse of the Gaussian method, it is easy to verify that the solution is unique.

Answer:

Let us formulate an obvious criterion: a homogeneous system of linear equations has just a trivial solution, If system matrix rank(in this case 3) is equal to the number of variables (in this case – 3 pieces).

Let's warm up and tune our radio to the wave of elementary transformations:

Example 2

Solve a homogeneous system of linear equations

From the article How to find the rank of a matrix? Let us recall the rational technique of simultaneously decreasing the matrix numbers. Otherwise, you will have to cut large, and often biting fish. An approximate example of a task at the end of the lesson.

Zeros are good and convenient, but in practice the case is much more common when the rows of the system matrix linearly dependent. And then the emergence of a general solution is inevitable:

Example 3

Solve a homogeneous system of linear equations

Solution: let's write down the matrix of the system and, using elementary transformations, bring it to a stepwise form. The first action is aimed not only at obtaining a single value, but also at decreasing the numbers in the first column:

(1) A third line was added to the first line, multiplied by –1. The third line was added to the second line, multiplied by –2. At the top left I got a unit with a “minus”, which is often much more convenient for further transformations.

(2) The first two lines are the same, one of them was deleted. Honestly, I didn’t push the solution - it turned out that way. If you perform transformations in a template way, then linear dependence lines would have been revealed a little later.

(3) The second line was added to the third line, multiplied by 3.

(4) The sign of the first line was changed.

As a result of elementary transformations, an equivalent system was obtained:

The algorithm works exactly the same as for heterogeneous systems. The variables “sitting on the steps” are the main ones, the variable that did not get a “step” is free.

Let's express the basic variables through a free variable:

Answer: common decision:

The trivial solution is included in the general formula, and it is unnecessary to write it down separately.

The check is also carried out according to the usual scheme: the resulting general solution must be substituted into the left side of each equation of the system and a legal zero must be obtained for all substitutions.

It would be possible to finish this quietly and peacefully, but the solution to a homogeneous system of equations often needs to be represented in vector form by using fundamental system of solutions. Please forget about it for now analytical geometry, since now we will talk about vectors in the general algebraic sense, which I opened a little in the article about matrix rank. There is no need to gloss over the terminology, everything is quite simple.

Systems of linear homogeneous equations- has the form ∑a k i x i = 0. where m > n or m A homogeneous system of linear equations is always consistent, since rangA = rangB. It obviously has a solution consisting of zeros, which is called trivial.

Purpose of the service. The online calculator is designed to find a non-trivial and fundamental solution to the SLAE. The resulting solution is saved in a Word file (see example solution).

Instructions. Select matrix dimension:

Properties of systems of linear homogeneous equations

In order for the system to have non-trivial solutions, it is necessary and sufficient that the rank of its matrix be less than the number of unknowns.

Theorem. A system in the case m=n has a nontrivial solution if and only if the determinant of this system is equal to zero.

Theorem. Any linear combination of solutions to a system is also a solution to that system.
Definition. The set of solutions to a system of linear homogeneous equations is called fundamental system of solutions, if this set consists of linearly independent solutions and any solution to the system is a linear combination of these solutions.

Theorem. If the rank r of the system matrix is ​​less than the number n of unknowns, then there exists a fundamental system of solutions consisting of (n-r) solutions.

Algorithm for solving systems of linear homogeneous equations

1. Finding the rank of the matrix.
2. We select the basic minor. We distinguish dependent (basic) and free unknowns.
3. We cross out those equations of the system whose coefficients are not included in the basis minor, since they are consequences of the others (according to the theorem on the basis minor).
4. We move the terms of the equations containing free unknowns to the right side. As a result, we obtain a system of r equations with r unknowns, equivalent to the given one, the determinant of which is nonzero.
5. We solve the resulting system by eliminating unknowns. We find relationships expressing dependent variables through free ones.
6. If the rank of the matrix is ​​not equal to the number of variables, then we find the fundamental solution of the system.
7. In the case rang = n we have a trivial solution.

Example. Find the basis of the system of vectors (a 1, a 2,...,a m), rank and express the vectors based on the base. If a 1 =(0,0,1,-1), and 2 =(1,1,2,0), and 3 =(1,1,1,1), and 4 =(3,2,1 ,4), and 5 =(2,1,0,3).
Let's write down the main matrix of the system:

Multiply the 3rd line by (-3). Let's add the 4th line to the 3rd:

0	0	1	-1
0	0	-1	1
0	-1	-2	1
3	2	1	4
2	1	0	3

Multiply the 4th line by (-2). Let's multiply the 5th line by (3). Let's add the 5th line to the 4th:
Let's add the 2nd line to the 1st:
Let's find the rank of the matrix.
The system with the coefficients of this matrix is ​​equivalent to the original system and has the form:
- x 3 = - x 4
- x 2 - 2x 3 = - x 4
2x 1 + x 2 = - 3x 4
Using the method of eliminating unknowns, we find a nontrivial solution:
We obtained relations expressing the dependent variables x 1 , x 2 , x 3 through the free ones x 4 , that is, we found a general solution:
x 3 = x 4
x 2 = - x 4
x 1 = - x 4

System m linear equations c n called unknowns system of linear homogeneous equations if all free terms are equal to zero. Such a system looks like:

Where and ij (i = 1, 2, …, m; j = 1, 2, …, n) - given numbers; x i– unknown.

A system of linear homogeneous equations is always consistent, since r(A) = r(). It always has at least zero ( trivial) solution (0; 0; …; 0).

Let us consider under what conditions homogeneous systems have non-zero solutions.

Theorem 1. A system of linear homogeneous equations has nonzero solutions if and only if the rank of its main matrix is r fewer unknowns n, i.e. r < n.

□ 1). Let a system of linear homogeneous equations have a nonzero solution. Since the rank cannot exceed the size of the matrix, then, obviously, r ≤ n. Let r = n. Then one of the minor sizes n n different from zero. Therefore, the corresponding system of linear equations has a unique solution: . . . This means that there are no other solutions other than trivial ones. So, if there is a non-trivial solution, then r < n.

2). Let r < n. Then the homogeneous system, being consistent, is uncertain. This means that it has an infinite number of solutions, i.e. has non-zero solutions. ■

Consider a homogeneous system n linear equations c n unknown:

(2)

Theorem 2. Homogeneous system n linear equations c n unknowns (2) has non-zero solutions if and only if its determinant is equal to zero: = 0.

□ If system (2) has a non-zero solution, then = 0. Because when the system has only a single zero solution. If = 0, then the rank r the main matrix of the system is less than the number of unknowns, i.e. r < n. And, therefore, the system has an infinite number of solutions, i.e. has non-zero solutions. ■

Let us denote the solution of system (1) X 1 = k 1 , X 2 = k 2 , …, x n = k n as a string .

Solutions of a system of linear homogeneous equations have the following properties:

1. If the line is a solution to system (1), then the line is a solution to system (1).

2. If the lines and are solutions of system (1), then for any values With 1 and With 2 their linear combination is also a solution to system (1).

The validity of these properties can be verified by directly substituting them into the equations of the system.

From the formulated properties it follows that any linear combination of solutions to a system of linear homogeneous equations is also a solution to this system.

System of linearly independent solutions e 1 , e 2 , …, e r called fundamental, if each solution of system (1) is a linear combination of these solutions e 1 , e 2 , …, e r.

Theorem 3. If rank r matrices of coefficients for variables of the system of linear homogeneous equations (1) are less than the number of variables n, then any fundamental system of solutions to system (1) consists of n–r decisions.

That's why common decision system of linear homogeneous equations (1) has the form:

Where e 1 , e 2 , …, e r– any fundamental system of solutions to system (9), With 1 , With 2 , …, with p– arbitrary numbers, R = n–r.

Theorem 4. General solution of the system m linear equations c n unknowns is equal to the sum of the general solution of the corresponding system of linear homogeneous equations (1) and an arbitrary particular solution of this system (1).

Example. Solve the system

Solution. For this system m = n= 3. Determinant

by Theorem 2, the system has only a trivial solution: x = y = z = 0.

Example. 1) Find general and particular solutions of the system

2) Find the fundamental system of solutions.

Solution. 1) For this system m = n= 3. Determinant

by Theorem 2, the system has nonzero solutions.

Since there is only one independent equation in the system

x + y – 4z = 0,

then from it we will express x =4z- y. Where do we get an infinite number of solutions: (4 z- y, y, z) – this is the general solution of the system.

At z= 1, y= -1, we get one particular solution: (5, -1, 1). Putting z= 3, y= 2, we get the second particular solution: (10, 2, 3), etc.

2) In the general solution (4 z- y, y, z) variables y And z are free, and the variable X- dependent on them. In order to find the fundamental system of solutions, let’s assign values ​​to the free variables: first y = 1, z= 0, then y = 0, z= 1. We obtain partial solutions (-1, 1, 0), (4, 0, 1), which form the fundamental system of solutions.

Illustrations:

Rice. 1 Classification of systems of linear equations

Rice. 2 Study of systems of linear equations

Presentations:

· Solution SLAE_matrix method

· Solution of SLAE_Cramer method

· Solution SLAE_Gauss method

· Packages for solving mathematical problems Mathematica, MathCad: searching for analytical and numerical solutions to systems of linear equations

Control questions:

1. Define a linear equation

2. What type of system does it look like? m linear equations with n unknown?

3. What is called solving systems of linear equations?

4. What systems are called equivalent?

5. Which system is called incompatible?

6. What system is called joint?

7. Which system is called definite?

8. Which system is called indefinite

9. List the elementary transformations of systems of linear equations

10. List the elementary transformations of matrices

11. Formulate a theorem on the application of elementary transformations to a system of linear equations

12. What systems can be solved using the matrix method?

13. What systems can be solved by Cramer's method?

14. What systems can be solved by the Gauss method?

15. List 3 possible cases that arise when solving systems of linear equations using the Gauss method

16. Describe the matrix method for solving systems of linear equations

17. Describe Cramer’s method for solving systems of linear equations

18. Describe Gauss’s method for solving systems of linear equations

19. What systems can be solved using an inverse matrix?

20. List 3 possible cases that arise when solving systems of linear equations using the Cramer method

Literature:

1. Higher mathematics for economists: Textbook for universities / N.Sh. Kremer, B.A. Putko, I.M. Trishin, M.N. Friedman. Ed. N.Sh. Kremer. – M.: UNITY, 2005. – 471 p.

2. General course of higher mathematics for economists: Textbook. / Ed. IN AND. Ermakova. –M.: INFRA-M, 2006. – 655 p.

3. Collection of problems in higher mathematics for economists: Textbook / Edited by V.I. Ermakova. M.: INFRA-M, 2006. – 574 p.

4. Gmurman V. E. Guide to solving problems in probability theory and magmatic statistics. - M.: Higher School, 2005. – 400 p.

5. Gmurman. V.E Probability theory and mathematical statistics. - M.: Higher School, 2005.

6. Danko P.E., Popov A.G., Kozhevnikova T.Ya. Higher mathematics in exercises and problems. Part 1, 2. – M.: Onyx 21st century: Peace and Education, 2005. – 304 p. Part 1; – 416 p. Part 2.

7. Mathematics in economics: Textbook: In 2 parts / A.S. Solodovnikov, V.A. Babaytsev, A.V. Brailov, I.G. Shandara. – M.: Finance and Statistics, 2006.

8. Shipachev V.S. Higher mathematics: Textbook for students. universities - M.: Higher School, 2007. - 479 p.

Related information.

Let's consider homogeneous system m linear equations with n variables:

(15)

A system of homogeneous linear equations is always consistent, because it always has a zero (trivial) solution (0,0,…,0).

If in system (15) m=n and , then the system has only a zero solution, which follows from Cramer’s theorem and formulas.

Theorem 1. Homogeneous system (15) has a nontrivial solution if and only if the rank of its matrix is ​​less than the number of variables, i.e. . r(A)< n.

Proof. The existence of a nontrivial solution to system (15) is equivalent to a linear dependence of the columns of the system matrix (i.e., there are numbers x 1, x 2,...,x n, not all equal to zero, such that equalities (15) are true).

According to the basis minor theorem, the columns of a matrix are linearly dependent  when not all columns of this matrix are basic, i.e.  when the order r of the basis minor of the matrix is ​​less than the number n of its columns. Etc.

Consequence. A square homogeneous system has non-trivial solutions  when |A|=0.

Theorem 2. If columns x (1), x (2),..., x (s) are solutions to a homogeneous system AX = 0, then any linear combination of them is also a solution to this system.

Proof. Consider any combination of solutions:

Then AX=A()===0. etc.

Corollary 1. If a homogeneous system has a nontrivial solution, then it has infinitely many solutions.

That. it is necessary to find such solutions x (1), x (2),..., x (s) of the system Ax = 0, so that any other solution of this system is represented in the form of their linear combination and, moreover, in a unique way.

Definition. The system k=n-r (n is the number of unknowns in the system, r=rg A) of linearly independent solutions x (1), x (2),…, x (k) of the system Ах=0 is called fundamental system of solutions this system.

Theorem 3. Let a homogeneous system Ах=0 with n unknowns and r=rg A be given. Then there is a set of k=n-r solutions x (1), x (2),…, x (k) of this system, forming a fundamental system of solutions.

Proof. Without loss of generality, we can assume that the basis minor of the matrix A is located in the upper left corner. Then, by the basis minor theorem, the remaining rows of matrix A are linear combinations of the basis rows. This means that if the values ​​x 1, x 2,…, x n satisfy the first r equations, i.e. equations corresponding to the rows of the basis minor), then they also satisfy other equations. Consequently, the set of solutions to the system will not change if we discard all equations starting from the (r+1)th one. We get the system:

Let us move the free unknowns x r +1 , x r +2 ,…, x n to the right side, and leave the basic ones x 1 , x 2 ,…, x r on the left:

(16)

Because in this case all b i =0, then instead of the formulas

c j =(M j (b i)-c r +1 M j (a i , r +1)-…-c n M j (a in)) j=1,2,…,r ((13), we get:

c j =-(c r +1 M j (a i , r +1)-…-c n M j (a in)) j=1,2,…,r (13)

If we set the free unknowns x r +1 , x r +2 ,…, x n to arbitrary values, then with respect to the basic unknowns we obtain a square SLAE with a non-singular matrix for which there is a unique solution. Thus, any solution of a homogeneous SLAE is uniquely determined by the values ​​of the free unknowns x r +1, x r +2,…, x n. Consider the following k=n-r series of values ​​of free unknowns:

1, =0, ….,=0,

1, =0, ….,=0, (17)

………………………………………………

1, =0, ….,=0,

(The series number is indicated by a superscript in parentheses, and the series of values ​​are written in the form of columns. In each series =1 if i=j and =0 if ij.

The i-th series of values ​​of free unknowns uniquely correspond to the values ​​of ,,...,basic unknowns. The values ​​of the free and basic unknowns together give solutions to system (17).

Let us show that the columns e i =,i=1,2,…,k (18)

form a fundamental system of solutions.

Because These columns, by construction, are solutions to the homogeneous system Ax=0 and their number is equal to k, then it remains to prove the linear independence of solutions (16). Let there be a linear combination of solutions e 1 , e 2 ,…, e k(x (1) , x (2) ,…, x (k)), equal to the zero column:

 1 e 1 +  2 e 2 +…+  k e k ( 1 X (1) + 2 X(2) +…+ k X(k) = 0)

Then the left side of this equality is a column whose components with numbers r+1,r+2,…,n are equal to zero. But the (r+1)th component is equal to  1 1+ 2 0+…+ k 0= 1 . Similarly, the (r+2)th component is equal to  2 ,…, the kth component is equal to  k. Therefore  1 =  2 = …= k =0, which means linear independence of solutions e 1 , e 2 ,…, e k ( x (1) , x (2) ,…, x (k)).

The constructed fundamental system of solutions (18) is called normal. By virtue of formula (13), it has the following form:

(20)

Corollary 2. Let e 1 , e 2 ,…, e k-normal fundamental system of solutions of a homogeneous system, then the set of all solutions can be described by the formula:

x=c 1 e 1 +s 2 e 2 +…+с k e k (21)

where с 1,с 2,…,с k – take arbitrary values.

Proof. By Theorem 2, column (19) is a solution to the homogeneous system Ax=0. It remains to prove that any solution to this system can be represented in the form (17). Consider the column X=y r +1 e 1 +…+y n e k. This column coincides with the column y in elements with numbers r+1,...,n and is a solution to (16). Therefore the columns X And at coincide, because solutions of system (16) are uniquely determined by the set of values ​​of its free unknowns x r +1 ,…,x n , and the columns at And X these sets are the same. Hence, at=X= y r +1 e 1 +…+y n e k, i.e. solution at is a linear combination of columns e 1 ,…,y n normal FSR. Etc.

The proven statement is true not only for a normal FSR, but also for an arbitrary FSR of a homogeneous SLAE.

X=c 1 X 1 + c 2 X 2 +…+s n - r X n - r - common decision systems of linear homogeneous equations

Where X 1, X 2,…, X n - r – any fundamental system of solutions,

c 1 ,c 2 ,…,c n - r are arbitrary numbers.

Example. (p. 78)

Let us establish a connection between the solutions of the inhomogeneous SLAE (1) and the corresponding homogeneous SLAE (15)

Theorem 4. The sum of any solution to the inhomogeneous system (1) and the corresponding homogeneous system (15) is a solution to system (1).

Proof. If c 1 ,…,c n is a solution to system (1), and d 1 ,…,d n is a solution to system (15), then substituting the unknown numbers c into any (for example, i-th) equation of system (1) 1 +d 1 ,…,c n +d n , we get:

B i +0=b i h.t.d.

Theorem 5. The difference between two arbitrary solutions of the inhomogeneous system (1) is a solution to the homogeneous system (15).

Proof. If c 1 ,…,c n and c 1 ,…,c n are solutions of system (1), then substituting the unknown numbers c into any (for example, i-th) equation of system (1) 1 -с 1 ,…,c n -с n , we get:

B i -b i =0 p.t.d.

From the proven theorems it follows that the general solution of a system of m linear homogeneous equations with n variables is equal to the sum of the general solution of the corresponding system of homogeneous linear equations (15) and an arbitrary number of a particular solution of this system (15).

X neod. =X total one +X frequent more than once (22)

As a particular solution to an inhomogeneous system, it is natural to take the solution that is obtained if in the formulas c j =(M j (b i)-c r +1 M j (a i, r +1)-…-c n M j (a in)) j=1,2,…,r ((13) set all numbers c r +1 ,…,c n equal to zero, i.e.

X 0 =(,…,,0,0,…,0) (23)

Adding this particular solution to the general solution X=c 1 X 1 + c 2 X 2 +…+s n - r X n - r corresponding homogeneous system, we obtain:

X neod. =X 0 +C 1 X 1 +C 2 X 2 +…+S n - r X n - r (24)

Consider a system of two equations with two variables:

in which at least one of the coefficients a ij 0.

To solve, we eliminate x 2 by multiplying the first equation by a 22, and the second by (-a 12) and adding them: Eliminate x 1 by multiplying the first equation by (-a 21), and the second by a 11 and adding them: The expression in parentheses is the determinant

Having designated ,, then the system will take the form:, i.e., if, then the system has a unique solution:,.

If Δ=0, and (or), then the system is inconsistent, because reduced to the form If Δ=Δ 1 =Δ 2 =0, then the system is uncertain, because reduced to form