Homogeneous systems. Fundamental decision system (specific example)

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 only 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 manner, 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.

Homogeneous system of linear equations over a field

DEFINITION. A fundamental system of solutions to a system of equations (1) is a non-empty linearly independent system of its solutions, the linear span of which coincides with the set of all solutions to system (1).

Note that a homogeneous system of linear equations that has only a zero solution does not have a fundamental system of solutions.

PROPOSAL 3.11. Any two fundamental systems of solutions to a homogeneous system of linear equations consist of the same number of solutions.

Proof. In fact, any two fundamental systems of solutions to the homogeneous system of equations (1) are equivalent and linearly independent. Therefore, by Proposition 1.12, their ranks are equal. Consequently, the number of solutions included in one fundamental system is equal to the number of solutions included in any other fundamental system of solutions.

If the main matrix A of the homogeneous system of equations (1) is zero, then any vector from is a solution to system (1); in this case, any set of linearly independent vectors from is a fundamental system of solutions. If the column rank of matrix A is equal to , then system (1) has only one solution - zero; therefore, in this case, the system of equations (1) does not have a fundamental system of solutions.

THEOREM 3.12. If the rank of the main matrix of a homogeneous system of linear equations (1) is less than the number of variables , then system (1) has a fundamental solution system consisting of solutions.

Proof. If the rank of the main matrix A of the homogeneous system (1) is equal to zero or , then it was shown above that the theorem is true. Therefore, below it is assumed that Assuming , we will assume that the first columns of matrix A are linearly independent. In this case, matrix A is rowwise equivalent to a reduced stepwise matrix, and system (1) is equivalent to the following reduced stepwise system of equations:

It is easy to check that any system of values ​​of free variables of system (2) corresponds to one and only one solution to system (2) and, therefore, to system (1). In particular, only the zero solution of system (2) and system (1) corresponds to a system of zero values.

In system (2) we will assign one of the free variables a value equal to 1, and the remaining variables - zero values. As a result, we obtain solutions to the system of equations (2), which we write in the form of rows of the following matrix C:

The row system of this matrix is ​​linearly independent. Indeed, for any scalars from the equality

equality follows

and, therefore, equality

Let us prove that the linear span of the system of rows of the matrix C coincides with the set of all solutions to system (1).

Arbitrary solution of system (1). Then the vector

is also a solution to system (1), and

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 - 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 every 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 a 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: search 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.

The linear equation is called homogeneous, if its free term is equal to zero, and inhomogeneous otherwise. A system consisting of homogeneous equations is called homogeneous and has the general form:

It is obvious that every homogeneous system is consistent and has a zero (trivial) solution. Therefore, when applied to homogeneous systems of linear equations, one often has to look for an answer to the question of the existence of nonzero solutions. The answer to this question can be formulated as the following theorem.

Theorem . A homogeneous system of linear equations has a nonzero solution if and only if its rank is less than the number of unknowns .

Proof: Let us assume that a system whose rank is equal has a non-zero solution. Obviously it does not exceed . In case the system has a unique solution. Since a system of homogeneous linear equations always has a zero solution, then the zero solution will be this unique solution. Thus, non-zero solutions are possible only for .

Corollary 1 : A homogeneous system of equations, in which the number of equations is less than the number of unknowns, always has a non-zero solution.

Proof: If a system of equations has , then the rank of the system does not exceed the number of equations, i.e. . Thus, the condition is satisfied and, therefore, the system has a non-zero solution.

Corollary 2 : A homogeneous system of equations with unknowns has a nonzero solution if and only if its determinant is zero.

Proof: Let us assume that a system of linear homogeneous equations, the matrix of which with the determinant , has a nonzero solution. Then, according to the proven theorem, and this means that the matrix is ​​singular, i.e. .

Homogeneous system of linear algebraic equations.

A system of m linear equations with n variables is called a system of linear homogeneous equations if all free terms are equal to 0. A system of linear homogeneous equations is always consistent, because it always has at least a zero solution. A system of linear homogeneous equations has a non-zero solution if and only if the rank of its matrix of coefficients for variables is less than the number of variables, i.e. for rank A (n. Any linear combination

Lin system solutions. homogeneous. ur-ii is also a solution to this system.

A system of linear independent solutions e1, e2,...,еk is called fundamental if each solution of the system is a linear combination of solutions. Theorem: if the rank r of the matrix of coefficients for the variables of a system of linear homogeneous equations is less than the number of variables n, then every fundamental system of solutions to the system consists of n-r solutions. Therefore, the general solution of the linear system. one-day ur-th has the form: c1e1+c2e2+...+skek, where e1, e2,..., ek – any fundamental system of solutions, c1, c2,..., ck – arbitrary numbers and k=n-r. The general solution of a system of m linear equations with n variables is equal to the sum

of the general solution of the system corresponding to it is homogeneous. linear equations and an arbitrary particular solution of this system.

7. Linear spaces. Subspaces. Basis, dimension. Linear shell. Linear space is called n-dimensional, if it contains a system of linearly independent vectors, and any system of a larger number of vectors is linearly dependent. The number is called dimension (number of dimensions) linear space and is denoted by . In other words, the dimension of a space is the maximum number of linearly independent vectors of this space. If such a number exists, then the space is called finite-dimensional. If, for any natural number n, there is a system in space consisting of linearly independent vectors, then such a space is called infinite-dimensional (written: ). In what follows, unless otherwise stated, finite-dimensional spaces will be considered.

The basis of an n-dimensional linear space is an ordered collection of linearly independent vectors ( basis vectors).

Theorem 8.1 on the expansion of a vector in terms of a basis. If is the basis of an n-dimensional linear space, then any vector can be represented as a linear combination of basis vectors:

V=v1*e1+v2*e2+…+vn+en
and, moreover, in the only way, i.e. the coefficients are determined uniquely. In other words, any vector of space can be expanded into a basis and, moreover, in a unique way.

Indeed, the dimension of space is . The system of vectors is linearly independent (this is a basis). After adding any vector to the basis, we obtain a linearly dependent system (since this system consists of vectors of n-dimensional space). Using the property of 7 linearly dependent and linearly independent vectors, we obtain the conclusion of the theorem.

Let M 0 – set of solutions to a homogeneous system (4) of linear equations.

Definition 6.12. Vectors With 1 ,With 2 , …, with p, which are solutions of a homogeneous system of linear equations are called fundamental set of solutions(abbreviated FNR), if

1) vectors With 1 ,With 2 , …, with p linearly independent (i.e., none of them can be expressed in terms of the others);

2) any other solution to a homogeneous system of linear equations can be expressed in terms of solutions With 1 ,With 2 , …, with p.

Note that if With 1 ,With 2 , …, with p– any f.n.r., then the expression k 1× With 1 + k 2× With 2 + … + k p× with p you can describe the whole set M 0 solutions to system (4), so it is called general view of the system solution (4).

Theorem 6.6. Any indeterminate homogeneous system of linear equations has a fundamental set of solutions.

The way to find the fundamental set of solutions is as follows:

Find a general solution to a homogeneous system of linear equations;

Build ( n – r) partial solutions of this system, while the values ​​of the free unknowns must form an identity matrix;

Write down the general form of the solution included in M 0 .

Example 6.5. Find a fundamental set of solutions to the following system:

Solution. Let's find a general solution to this system.

~ ~ ~ ~ Þ Þ Þ There are five unknowns in this system ( n= 5), of which there are two main unknowns ( r= 2), there are three free unknowns ( n – r), that is, the fundamental solution set contains three solution vectors. Let's build them. We have x 1 and x 3 – main unknowns, x 2 , x 4 , x 5 – free unknowns

Values ​​of free unknowns x 2 , x 4 , x 5 form the identity matrix E third order. Got that vectors With 1 ,With 2 , With 3 form f.n.r. of this system. Then the set of solutions of this homogeneous system will be M 0 = {k 1× With 1 + k 2× With 2 + k 3× With 3 , k 1 , k 2 , k 3 О R).

Let us now find out the conditions for the existence of nonzero solutions of a homogeneous system of linear equations, in other words, the conditions for the existence of a fundamental set of solutions.

A homogeneous system of linear equations has non-zero solutions, that is, it is uncertain if

1) the rank of the main matrix of the system is less than the number of unknowns;

2) in a homogeneous system of linear equations, the number of equations is less than the number of unknowns;

3) if in a homogeneous system of linear equations the number of equations is equal to the number of unknowns, and the determinant of the main matrix is ​​equal to zero (i.e. | A| = 0).

Example 6.6. At what parameter value a homogeneous system of linear equations has non-zero solutions?

Solution. Let's compose the main matrix of this system and find its determinant: = = 1×(–1) 1+1 × = – A– 4. The determinant of this matrix is ​​equal to zero at a = –4.

Answer: –4.

7. Arithmetic n-dimensional vector space

Basic Concepts

In previous sections we have already encountered the concept of a set of real numbers arranged in a certain order. This is a row matrix (or column matrix) and a solution to a system of linear equations with n unknown. This information can be summarized.

Definition 7.1. n-dimensional arithmetic vector called an ordered set of n real numbers.

Means A= (a 1 , a 2 , …, a n), where a iО R, i = 1, 2, …, n– general view of the vector. Number n called dimension vectors, and numbers a i are called his coordinates.

For example: A= (1, –8, 7, 4, ) – five-dimensional vector.

All set n-dimensional vectors are usually denoted as Rn.

Definition 7.2. Two vectors A= (a 1 , a 2 , …, a n) And b= (b 1 , b 2 , …, b n) of the same dimension equal if and only if their corresponding coordinates are equal, i.e. a 1 = b 1 , a 2 = b 2 , …, a n= b n.

Definition 7.3.Amount two n-dimensional vectors A= (a 1 , a 2 , …, a n) And b= (b 1 , b 2 , …, b n) is called a vector a + b= (a 1 + b 1, a 2 + b 2, …, a n+b n).

Definition 7.4. The work real number k to vector A= (a 1 , a 2 , …, a n) is called a vector k× A = (k×a 1, k×a 2 , …, k×a n)

Definition 7.5. Vector O= (0, 0, …, 0) is called zero(or null vector).

It is easy to verify that the actions (operations) of adding vectors and multiplying them by a real number have the following properties: " a, b, c Î Rn, " k, lО R:

1) a + b = b + a;

2) a + (b+ c) = (a + b) + c;

3) a + O = a;

4) a+ (–a) = O;

5) 1× a = a, 1 О R;

6) k×( l× a) = l×( k× a) = (l× k)× a;

7) (k + l)× a = k× a + l× a;

8) k×( a + b) = k× a + k× b.

Definition 7.6. A bunch of Rn with the operations of adding vectors and multiplying them by a real number given on it is called arithmetic n-dimensional vector space.