The linear equation is called homogeneous if its intercept is zero, and inhomogeneous otherwise. A system consisting of homogeneous equations is called homogeneous and has the general form:
Obviously, any homogeneous system is consistent and has a zero (trivial) solution. Therefore, for homogeneous systems 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: Suppose a system whose rank is equal has a nonzero solution. Obviously, does not exceed . In the case the system has a unique solution. Since the system of homogeneous linear equations always has a zero solution, it is precisely the zero solution that will be this unique solution. Thus, nonzero 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 nonzero solution.
Proof: If the 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 nonzero solution.
Consequence 2 : A homogeneous system of equations with unknowns has a nonzero solution if and only if its determinant is zero.
Proof: Suppose a system of linear homogeneous equations whose matrix with determinant has a nonzero solution. Then, according to the proved theorem, , which means that the matrix is degenerate, i.e. .
Kronecker-Capelli theorem: The SLE is consistent if and only if the rank of the matrix of the system is equal to the rank of the extended matrix of this system. A system ur-th is called compatible if it has at least one solution.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 compatible, because it always has at least a zero solution. A system of linear homogeneous equations has a nonzero solution if and only if the rank of its matrix of coefficients at variables is less than the number of variables, i.e. for rank A (n. Any linear combination
solutions of the system of lines. homogeneous ur-ii is also a solution to this system.
A system of linearly independent solutions e1, e2,…,ek 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 at the variables of the system of linear homogeneous equations is less than the number of variables n, then any fundamental system of solutions of the system consists of n-r solutions. Therefore, the general solution of the system of lines. single ur-th has the form: c1e1+c2e2+…+ckek, where e1, e2,…, ek is any fundamental system of solutions, c1, c2,…,ck are arbitrary numbers and k=n-r. Common decision system of m linear equations with n variables is equal to the sum
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 more vectors is linearly dependent. The number is called dimension (number of measurements) linear space and is denoted by . In other words, the dimension of a space is the maximum number of linearly independent vectors in that space. If such a number exists, then the space is said to be finite-dimensional. If for any natural number n in space there is a system consisting of linearly independent vectors, then such a space is called infinite-dimensional (write: ). In what follows, unless otherwise stated, finite-dimensional spaces will be considered.
The basis of an n-dimensional linear space is an ordered set of linearly independent vectors ( basis vectors).
Theorem 8.1 on the expansion of a vector in terms of a basis. If is a 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 a unique way, i.e. coefficients are uniquely determined. In other words, any space vector can be expanded in a basis and, moreover, in a unique way.
Indeed, the dimension of the space is . The system of vectors is linearly independent (this is the basis). After joining any vector to the basis, we get a linearly dependent system (since this system consists of vectors in an n-dimensional space). By the property of 7 linearly dependent and linearly independent vectors, we obtain the conclusion of the theorem.
The linear system is called homogeneous if all its free terms are 0.
IN matrix form homogeneous system is written: 
.
The homogeneous system (2) is always consistent
. It is obvious that 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 nonzero (nontrivial) 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) - indefinite 
.
Consequence 1.
If the number of equations m
homogeneous system is less than the number of variables 
, then the system is indefinite and has a set of nonzero solutions.
Consequence 2.
Square homogeneous system 
has nonzero solutions if and if the main matrix of this system 
is degenerate, i.e. determinant 
.
Otherwise, if the determinant 
, the square homogeneous system has the only thing
zero solution
.
Let the rank of system (2) 
i.e., system (2) has nontrivial solutions.
Let 
And 
- particular solutions of this system, i.e. 
And 
.
Properties of Solutions to a Homogeneous System
Really, .
Really, .
Combining properties 1) and 2), we can say that if 
…,
- solutions of the homogeneous system (2), then any linear combination of them is also its solution. Here 
are arbitrary real numbers.
Can be found 
linearly independent particular solutions
homogeneous system (2), which can be used to obtain any other particular solution of this system, i.e. obtain the general solution of system (2).
Definition 2.2
Aggregate 
linearly independent particular solutions 
…,
homogeneous system (2) such that each solution of system (2) can be represented as a linear combination of them is called fundamental decision system
(FSR) of homogeneous system (2).
Let 
…,
is the fundamental system of solutions, then the general solution of the homogeneous system (2) can be represented as:
Where 
.
Comment.
To get the FSR, you need to find private solutions 
…,
, giving in turn to any one free variable the value "1", and to all other free variables - the value "0".
Get 
,,
…,
- FSR.
Example. Find the general solution and the fundamental system of solutions of the homogeneous system of equations:
Solution. Let us write down the extended matrix of the system, first putting the last equation of the system in the first place, and reduce 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 indefinite and has many solutions.
Basis minor with variables 
different from zero: 
choose 
as basic variables, the rest 
- free variables (take any real values).
The last matrix in the chain corresponds to the stepwise system of equations:
(3)
Express the basic variables 
through free variables 
(the reverse course of the Gauss method).
From the last equation we express 
:
and substitute into the first equation. We will receive. We open the brackets, give similar ones and express 
:
.
Assuming 
,
,
, Where 
, write
is the 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 be found in another way, without first finding the general solution of the system. To do this, the resulting step system (3) had to be solved three times, assuming for 
:
; For 
:
; For 
:
.
System m linear equations c n unknown is called 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.
The 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 nonzero solutions.
Theorem 1. A system of linear homogeneous equations has nonzero solutions if and only if the rank of its main matrix r fewer unknowns n, i.e. r < n.
□
1). Let the system of linear homogeneous equations have a nonzero solution. Since the rank cannot exceed the size of the matrix, it is obvious that r ≤ n. Let r = n. Then one of the minors of size n n different from zero. Therefore, the corresponding system of linear equations has a unique solution: 
, , . Hence, there are no solutions other than trivial ones. So, if there is a non-trivial solution, then r < n.
2). Let r < n. Then a homogeneous system, being consistent, is indefinite. Hence, it has an infinite number of solutions, i.e. also 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 nonzero solutions if and only if its determinant is equal to zero: = 0.
□ If system (2) has a non-zero solution, then = 0. For at , the system has only a unique 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. also has non-zero solutions. ■
Denote the solution of system (1) X 1 = k 1 , X 2 = k 2 , …, x n = k n as a string 
.
Solutions to a system of linear homogeneous equations have the following properties:
1.
If the string is a solution to system (1), then the string is also 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).
You can check the validity of these properties by directly substituting them into the equations of the system.
It follows from the formulated properties 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 the matrix of coefficients for the variables of the system of linear homogeneous equations (1) is less than the number of variables n, then any fundamental system of solutions to system (1) consists of n–r solutions.
That's why common decision system of linear homogeneous equations (1) has the form:
Where e 1 , e 2 , …, e r is any fundamental system of solutions to system (9), With 1 , With 2 , …, with p- arbitrary numbers, R = n–r.
Theorem 4. General system solution 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 a 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 express x =4z- y. From where we get an infinite set of solutions: (4 z- y, y, z) 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, we assign values to the free variables: first y = 1, z= 0, then y = 0, z= 1. We obtain particular 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:
Solving SLAE_matrix method
Solution SLAU_Cramer's method
Solution SLAE_Gauss method
· Packages for solving mathematical problems Mathematica: search for analytical and numerical solution of systems of linear equations
Control questions:
1. Define a linear equation
2. What kind of system does m linear equations with n unknown?
3. What is called the solution of systems of linear equations?
4. What systems are called equivalent?
5. What system is called incompatible?
6. What system is called joint?
7. What system is called defined?
8. What 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 by 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 the Gauss method for solving systems of linear equations
19. What systems can be solved using 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. Fridman. Ed. N.Sh. Kremer. - M.: UNITI, 2005. - 471 p.
2. General course of higher mathematics for economists: Textbook. / Ed. IN AND. Ermakov. -M.: INFRA-M, 2006. - 655 p.
3. Collection of problems in higher mathematics for economists: Tutorial/ Under the editorship of V.I. Ermakov. M.: INFRA-M, 2006. - 574 p.
4. V. E. Gmurman, Guide to Problem Solving in Probability Theory and Magmatic Statistics. - M.: Higher School, 2005. - 400 p.
5. Gmurman. VE Theory of Probability and Mathematical Statistics. - M.: Higher school, 2005.
6. Danko P.E., Popov A.G., Kozhevnikova T.Ya. Higher mathematics in exercises and tasks. Part 1, 2. - M .: Onyx 21st century: World and education, 2005. - 304 p. Part 1; – 416 p. Part 2
7. Mathematics in Economics: Textbook: In 2 hours / A.S. Solodovnikov, V.A. Babaitsev, 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.
Similar information.
Even at school, each of us studied equations and, for sure, systems of equations. But not many people know that there are several ways to solve them. Today we will analyze in detail all the methods for solving a system of linear algebraic equations, which consist of more than two equalities.
Story
Today it is known that the art of solving equations and their systems originated in ancient Babylon and Egypt. However, equalities in their usual form appeared after the emergence of the equal sign "=", which was introduced in 1556 by the English mathematician Record. By the way, this sign was chosen for a reason: it means two parallel equal segments. Indeed, there is no better example of equality.
The founder of modern letter designations of unknowns and signs of degrees is a French mathematician. However, his designations differed significantly from today's. For example, he denoted the square of an unknown number with the letter Q (lat. "quadratus"), and the cube with the letter C (lat. "cubus"). These notations seem awkward now, but back then it was the most understandable way to write systems of linear algebraic equations.
However, a drawback in the then methods of solution was that mathematicians considered only positive roots. Perhaps this is due to the fact that negative values did not have any practical application. One way or another, it was the Italian mathematicians Niccolo Tartaglia, Gerolamo Cardano and Rafael Bombelli who were the first to consider negative roots in the 16th century. And the modern view, the main solution method (through the discriminant) was created only in the 17th century thanks to the work of Descartes and Newton.
In the mid-18th century, the Swiss mathematician Gabriel Cramer found a new way to make solving systems of linear equations easier. This method was subsequently named after him and to this day we use it. But we will talk about Cramer's method a little later, but for now we will discuss linear equations and methods for solving them separately from the system.
Linear equations
Linear equations are the simplest equalities with variable(s). They are classified as algebraic. write in general form as follows: a 1 * x 1 + a 2 * x 2 + ... and n * x n \u003d b. We will need their representation in this form when compiling systems and matrices further.
Systems of linear algebraic equations
The definition of this term is as follows: it is a set of equations that have common unknowns and a common solution. As a rule, at school, everything was solved by systems with two or even three equations. But there are systems with four or more components. Let's first figure out how to write them down so that it is convenient to solve them later. First, systems of linear algebraic equations will look better if all variables are written as x with the appropriate index: 1,2,3, and so on. Secondly, all equations should be brought to the canonical form: a 1 * x 1 + a 2 * x 2 + ... a n * x n =b.
After all these actions, we can begin to talk about how to find a solution to systems of linear equations. Matrices are very useful for this.
matrices
A matrix is a table that consists of rows and columns, and at their intersection are its elements. These can either be specific values or variables. Most often, to designate elements, subscripts are placed under them (for example, a 11 or a 23). The first index means the row number and the second one the column number. Over matrices, as well as over any other mathematical element, you can perform various operations. Thus, you can:
2) Multiply a matrix by some number or vector.
3) Transpose: turn matrix rows into columns and columns into rows.
4) Multiply matrices if the number of rows of one of them is equal to the number of columns of the other.
We will discuss all these techniques in more detail, as they will be useful to us in the future. Subtracting and adding matrices is very easy. Since we take matrices of the same size, each element of one table corresponds to each element of another. Thus, we add (subtract) these two elements (it is important that they are in the same places in their matrices). When multiplying a matrix by a number or vector, you simply need to multiply each element of the matrix by that number (or vector). Transposition is a very interesting process. It's very interesting sometimes to see him in real life, for example, when you change the orientation of your tablet or phone. The icons on the desktop are a matrix, and when you change the position, it transposes and becomes wider, but decreases in height.
Let's analyze such a process as Although it will not be useful to us, it will still be useful to know it. You can multiply two matrices only if the number of columns in one table is equal to the number of rows in the other. Now let's take the elements of a row of one matrix and the elements of the corresponding column of another. We multiply them by each other and then add them (that is, for example, the product of the elements a 11 and a 12 by b 12 and b 22 will be equal to: a 11 * b 12 + a 12 * b 22). Thus, one element of the table is obtained, and it is filled further by a similar method.
Now we can begin to consider how the system of linear equations is solved.
Gauss method
This topic starts at school. We know well the concept of "system of two linear equations" and know how to solve them. But what if the number of equations is more than two? This will help us
Of course, this method is convenient to use if you make a matrix out of the system. But you can not transform it and solve it in its pure form.
So, how is the system of linear Gaussian equations solved by this method? By the way, although this method is named after him, it was discovered in ancient times. Gauss proposes the following: to carry out operations with equations in order to eventually reduce the entire set to a stepped form. That is, it is necessary that from top to bottom (if placed correctly) from the first equation to the last, one unknown decreases. In other words, we need to make sure that we get, say, three equations: in the first - three unknowns, in the second - two, in the third - one. Then from the last equation we find the first unknown, substitute its value into the second or first equation, and then find the remaining two variables.
Cramer method
To master this method, it is vital to master the skills of addition, subtraction of matrices, and you also need to be able to find determinants. Therefore, if you do all this poorly or do not know how at all, you will have to learn and practice.
What is the essence of this method, and how to make it so that a system of linear Cramer equations is obtained? Everything is very simple. We have to construct a matrix from numerical (almost always) coefficients of a system of linear algebraic equations. To do this, we simply take the numbers in front of the unknowns and put them in the table in the order they are written in the system. If the number is preceded by a "-" sign, then we write down a negative coefficient. So, we have compiled the first matrix of the coefficients of the unknowns, not including the numbers after the equal signs (naturally, the equation should be reduced to the canonical form, when only the number is on the right, and all the unknowns with coefficients are on the left). Then you need to create several more matrices - one for each variable. To do this, in the first matrix, in turn, we replace each column with coefficients with a column of numbers after the equal sign. Thus, we obtain several matrices and then find their determinants.
After we have found the determinants, the matter is small. We have an initial matrix, and there are several resulting matrices that correspond to different variables. To get the solutions of the system, we divide the determinant of the resulting table by the determinant of the initial table. The resulting number is the value of one of the variables. Similarly, we find all the unknowns.
Other Methods
There are several more methods for obtaining a solution to systems of linear equations. For example, the so-called Gauss-Jordan method, which is used to find solutions to the system quadratic equations and is also related to the use of matrices. There is also a Jacobi method for solving a system of linear algebraic equations. It is the easiest to adapt to a computer and is used in computer technology.
Difficult cases
Complexity usually arises when the number of equations is less than the number of variables. Then we can say for sure that either the system is inconsistent (that is, it has no roots), or the number of its solutions tends to infinity. If we have the second case, then we need to write down the general solution of the system of linear equations. It will contain at least one variable.
Conclusion
Here we come to the end. Let's summarize: we have analyzed what a system and a matrix are, learned how to find a general solution to a system of linear equations. In addition, other options were considered. We found out how a system of linear equations is solved: the Gauss method and We talked about difficult cases and other ways of finding solutions.
In fact, this topic is much more extensive, and if you want to better understand it, then we advise you to read more specialized literature.
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 certainly has a solution consisting of zeros, which is called trivial.Service assignment. 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 solution example).
Instruction. Select the dimension of the matrix:
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. The system in the case m=n has a non-trivial 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 decision system if this collection consists of linearly independent solutions and any solution of 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 is a fundamental system of solutions consisting of (n-r) solutions.
Algorithm for solving systems of linear homogeneous equations
- Find the rank of the matrix.
- We select the basic minor. We select dependent (basic) and free unknowns.
- We cross out those equations of the system whose coefficients were not included in the basis minor, since they are consequences of the rest (according to the basic minor theorem).
- The terms of the equations containing free unknowns will be transferred 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 different from zero.
- We solve the resulting system by eliminating the unknowns. We find relations expressing dependent variables in terms of free ones.
- If the rank of the matrix is not equal to the number of variables, then we find the fundamental solution of the system.
- In the case of 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 in terms of 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).
We write the main matrix of the system:
Multiply the 3rd row 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 row by (-2). Multiply the 5th row by (3). Let's add the 5th line to the 4th:
Let's add the 2nd line to the 1st:
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
2x1 + x2 = - 3x4
By the method of elimination of unknowns, we find a non-trivial solution:
We got relations expressing dependent variables x 1, x 2, x 3 through free x 4, that is, we found a general solution:
x 3 = x 4
x 2 = - x 4
x 1 = - x 4
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