Screw movement. Addition of translational motions of a rigid body
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.
Linear system homogeneous equations always compatible, because 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 less than number unknown n, i.e. r < n.
□
one). 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 be 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 be 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.
So 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 common solution the corresponding system of linear homogeneous equations (1) and an arbitrary particular solution of this system (1).
Example. Solve the system
Decision. 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 fundamental system solutions.
Decision. 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:
SLAU solution_ matrix method
Solution SLAU_Cramer's method
Solution SLAE_Gauss method
・Solution packages math problems Mathematica: search for analytical and numerical solution of systems of linear equations
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.: graduate 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.
Homogeneous systems of linear algebraic equations
Within the lessons Gauss 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 of 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 the homogeneous system of linear equations.
According to the first paragraphs, the material may seem boring and ordinary, but this impression is deceptive. In addition to further development of technical methods, there will be many 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 system equation is zero. For example:
It is quite clear that homogeneous system is always consistent, that is, it always has a solution. And, first of all, the so-called trivial decision 
. Trivial, for those who do not understand the meaning of the adjective at all, means bespontovoe. Not academically, of course, but intelligibly =) ... Why beat around the bush, let's find out if this system has any other solutions:
Example 1
Decision: to solve a homogeneous system it is necessary to write system matrix and with the help of elementary transformations bring it to a stepped form. Note that there is no need to write down the vertical bar and zero column of free members here - after all, whatever you do with zeros, they will remain zero: 
(1) The first row was added to the second row, 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 row by 3 doesn't make much sense.
As a result of elementary transformations, an equivalent homogeneous system is obtained 
, and, applying the reverse move 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 trivial solution, if system matrix rank(in this case, 3) is equal to the number of variables (in this case, 3 pcs.).
We warm up and tune our radio to a wave of elementary transformations:
Example 2
Solve a homogeneous system of linear equations 
From the article How to find the rank of a matrix? we recall the rational method of incidentally reducing the numbers of the matrix. Otherwise, you will have to butcher large, and often biting fish. An example of an assignment 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 matrix of the system linearly dependent. And then the appearance of a general solution is inevitable:
Example 3
Solve a homogeneous system of linear equations 
Decision: we write the matrix of the system and, using elementary transformations, we bring it to a step form. The first action is aimed not only at obtaining a single value, but also at reducing the numbers in the first column:
(1) The third row was added to the first row, 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 has been removed. Honestly, did not customize the solution - it happened. If you perform transformations in a template, then linear dependence lines would show up a little later.
(3) To the third line, add the second line, multiplied by 3.
(4) The sign of the first line has been changed.
As a result of elementary transformations, an equivalent system is obtained: 
The algorithm works exactly the same as for heterogeneous systems. Variables "sitting on the steps" are the main ones, the variable that did not get the "steps" is free.
We express the basic variables in terms of the free variable: 
Answer: common decision: 
The trivial solution is included in general formula, and there is no need to write it separately.
The verification is also carried out according to the usual scheme: the resulting general solution must be substituted into left side each equation of the system and get a legitimate zero for all substitutions.
This could be quietly ended, but the solution of a homogeneous system of equations often needs to be represented in vector form via fundamental decision system. Please temporarily forget about analytical geometry, since now we will talk about vectors in the general algebraic sense, which I slightly opened in an article about matrix rank. Terminology is not necessary to shade, everything is quite simple.
Kaluga Branch of the Federal State Budgetary Educational Institution of Higher Professional Education
Moscow State Technical University named after N.E. Bauman"
(KF MSTU named after N.E. Bauman)
Vlaikov N.D.
Solution of homogeneous SLAE
Guidelines for conducting exercises
on the course of analytical geometry
Kaluga 2011
Lesson objectives page 4
Lesson plan page 4
Required theoretical information p.5
Practical part p.10
Control of the development of the material covered p.13
Homework page 14
Number of hours: 2
Lesson objectives:
To systematize the received theoretical knowledge about the types of SLAE and ways to solve them.
Get skills in solving homogeneous SLAEs.
Lesson plan:
Briefly state the theoretical material.
Solve a homogeneous SLAE.
Find a fundamental system of solutions for a homogeneous SLAE.
Find a particular solution of the homogeneous SLAE.
Formulate an algorithm for solving a homogeneous SLAE.
Check your current homework.
Carry out verification work.
Introduce the topic of the next seminar.
Submit current homework.
Necessary theoretical information.
Matrix rank.
Def. The rank of a matrix is the number that is equal to the maximum order among its non-zero minors. The rank of a matrix is denoted by .
If a square matrix is nondegenerate, then the rank is equal to its order. If a square matrix is degenerate, then its rank is less than its order.
The rank of a diagonal matrix is equal to the number of its non-zero diagonal elements.
Theor. When a matrix is transposed, its rank does not change, i.e. 
.
Theor. The rank of a matrix does not change under elementary transformations of its rows and columns.
Basis minor theorem.
Def. Minor 
matrices 
is called basic if two conditions are met:
a) it is not equal to zero;
b) its order is equal to the rank of the matrix 
.
Matrix 
may have several basis minors.
Rows and columns of a matrix 
, in which the chosen basic minor is located, are called basic.
Theor. Basis minor theorem. Basic rows (columns) of a matrix 
corresponding to any of its basic minor 
, are linearly independent. Any rows (columns) of a matrix 
, not included in 
, are linear combinations of basic rows (columns).
Theor. For any matrix, its rank is equal to the maximum number of its linearly independent rows (columns).
Matrix rank calculation. Method of elementary transformations.
With the help of elementary row transformations, any matrix can be reduced to a stepped form. The rank of a step matrix is equal to the number of non-zero rows. The base element in it is the minor located at the intersection of non-zero rows with columns corresponding to the first non-zero elements on the left in each of the rows.
SLAU. Basic definitions.
Def. System 
(15.1)
Numbers 
are called SLAE coefficients. Numbers 
are called free terms of the equations.
The record of SLAE in the form (15.1) is called coordinate.
Def. A SLAE is said to be homogeneous if 
. Otherwise, it is called heterogeneous.
Def. SLAE solution is such a set of values of unknowns, when substituting which each equation of the system turns into an identity. Any specific SLAE solution is also called its particular solution.
Solving SLAE means solving two problems:
Find out if the SLAE has solutions;
Find all solutions if they exist.
Def. A SLAE is called joint if it has at least one solution. Otherwise, it is called inconsistent.
Def. If SLAE (15.1) has a solution, and, moreover, unique, then it is called definite, and if the solution is not unique, then indefinite.
Def. If in equation (15.1) 
,SLAE is called square.
Forms of recording SLAU.
In addition to the coordinate form (15.1), SLAE records often use other representations of it.
(15.2)
The ratio is called the vector form of the SLAE.
If we take the product of matrices as a basis, then SLAE (15.1) can be written as follows:
(15.3)
or 
.
The record of SLAE (15.1) in the form (15.3) is called matrix.
Homogeneous SLAE.
homogeneous system 
linear algebraic equations with 
unknown is a system of the form
Homogeneous SLAEs are always consistent, since there is always a zero solution.
Criterion for the existence of a nonzero solution. For a homogeneous square SLAE to have a nonzero solution, it is necessary and sufficient that its matrix be degenerate.
Theor. If columns 
,
,
…,
are solutions of a homogeneous SLAE, then any linear combination of them is also a solution to this system.
Consequence. If a homogeneous SLAE has a non-zero solution, then it has an infinite number of solutions.
It is natural to try to find such solutions 
,
,
…,
systems so that any other solution can be represented as a linear combination of them and, moreover, in a unique way.
Def. Any set of 
linearly independent columns 
,
,
…,
, which are solutions of the homogeneous SLAE 
, where 
is the number of unknowns, and 
is the rank of its matrix 
, is called the fundamental system of solutions of this homogeneous SLAE.
In the study and solution of homogeneous systems of linear equations in the matrix of the system, we will fix the basic minor. The basis minor will correspond to the basis columns and hence the basis unknowns. The remaining unknowns will be called free.
Theor. On the structure of the general solution of a homogeneous SLAE. If a 
,
,
…,
- an arbitrary fundamental system of solutions of a homogeneous SLAE 
, then any of its solutions can be represented in the form
Where 
,
…,
- some constants.
That. the general solution of the homogeneous SLAE has the form
Practical part.
Consider possible sets of solutions for the following types of SLAE and their graphical interpretation.
;
;
.
Consider the possibility of solving these systems using Cramer's formulas and the matrix method.
Describe the essence of the Gauss method.
Solve the following tasks.
Example 1. Solve a homogeneous SLAE. Find FSR.
.
Let us write down the matrix of the system and reduce it to a stepped form.
.
the system will have infinitely many solutions. The FSR will consist of 
columns.
Let's discard the zero lines and write the system again:
.
We will consider the basic minor standing in the upper left corner. That. 
are the basic unknowns, and 
- free. Express 
through free 
:
;
Let's put 
.
Finally we have:
- the coordinate form of the answer, or
- matrix form of the answer, or
- vector form of the answer (vector - the columns are the columns of the FSR).
Algorithm for solving a homogeneous SLAE.
Find the FSR and the general solution of the following systems:
№2.225(4.39)
. Answer: 
№2.223(2.37)
. Answer:
№2.227(2.41)
. Answer: 
Solve the homogeneous SLAE:
. Answer: 
Solve the homogeneous SLAE:
. Answer: 
Introducing the topic of the next seminar.
Solution of systems of linear inhomogeneous equations.
Monitoring the development of the material covered.
Test work 3 - 5 minutes. 4 students with odd numbers in the magazine participate, starting with #10
|
Run actions:
|
Run actions:
|
|
|
Calculate the determinant:
|
;
;
|
Run actions:
|
Run actions:
|
|
Find the matrix inverse of a given one:
|
Calculate the determinant:
|
undefined
Homework:
1. Solve problems:
№ 2.224, 2.226, 2.228, 2.230, 2.231, 2.232.
2. Work out lectures on the topics:
Systems of linear algebraic equations (SLAE). Coordinate, matrix and vector notation. Criterion Kronecker - Capelli compatibility SLAE. Inhomogeneous SLAE. Criterion for the existence of a nonzero solution of a homogeneous SLAE. Properties of solutions of a homogeneous SLAE. Fundamental system of solutions of a homogeneous SLAE, a theorem on its existence. Normal fundamental system of solutions. Theorem on the structure of the general solution of a homogeneous SLAE. Theorem on the structure of the general solution of the inhomogeneous SLAE.
A homogeneous system is always consistent and has a trivial solution 
. For a nontrivial solution to exist, it is necessary that the rank of the matrix 
was less than the number of unknowns:
.
Fundamental decision system
homogeneous system 
call the system of solutions in the form of column vectors 
, which correspond to the canonical basis, i.e. basis in which arbitrary constants 
are alternately set equal to one, while the rest are set to zero.
Then the general solution of the homogeneous system has the form:
where 
are arbitrary constants. In other words, the general solution is a linear combination of the fundamental system of solutions.
Thus, the basic solutions can be obtained from the general solution if the free unknowns are alternately given the value of unity, assuming all others equal to zero.
Example. Let's find a solution to the system
We accept , then we get the solution in the form:
Let us now construct a fundamental system of solutions:
.
The general solution can be written as:
Solutions to a system of homogeneous linear equations have the following properties:
In other words, any linear combination of solutions to a homogeneous system is again a solution.
Solution of systems of linear equations by the Gauss method
Solving systems of linear equations has been of interest to mathematicians for several centuries. The first results were obtained in the XVIII century. In 1750, G. Kramer (1704–1752) published his works on the determinants of square matrices and proposed an algorithm for finding the inverse matrix. In 1809, Gauss outlined a new solution method known as the elimination method.
The Gauss method, or the method of successive elimination of unknowns, consists in the fact that, with the help of elementary transformations, the system of equations is reduced to an equivalent system of a stepped (or triangular) form. Such systems allow you to consistently find all the unknowns in a certain order.
Suppose that in system (1) 
(which is always possible).
(1)
Multiplying the first equation in turn by the so-called suitable numbers
and adding the result of multiplication with the corresponding equations of the system, we get an equivalent system in which all equations, except for the first one, will have no unknown X 1
(2)
We now multiply the second equation of system (2) by appropriate numbers, assuming that 
,
and adding it to the lower ones, we eliminate the variable 
of all equations, starting with the third.
Continuing this process, after 
steps we get:
(3)
If at least one of the numbers 
is not equal to zero, then the corresponding equality is inconsistent and system (1) is inconsistent. Conversely, for any joint number system 
are equal to zero. Number 
is nothing but the rank of the system matrix (1).
The transition from system (1) to (3) is called in a straight line Gaussian method, and finding unknowns from (3) - backwards .
Comment : It is more convenient to perform transformations not with the equations themselves, but with the extended matrix of the system (1).
Example. Let's find a solution to the system
.
Let's write the augmented matrix of the system:
.
Let's add to the lines 2,3,4 the first, multiplied by (-2), (-3), (-2) respectively:
.
Let's swap rows 2 and 3, then in the resulting matrix add row 2 to row 4, multiplied by 
:
.
Add to line 4 line 3 multiplied by 
:
.
It's obvious that 
, hence the system is consistent. From the resulting system of equations
we find the solution by reverse substitution:
,
,
,
.
Example 2 Find system solution:
.
It is obvious that the system is inconsistent, because 
, a 
.
Advantages of the Gauss method :
Less time consuming than Cramer's method.
Unambiguously establishes the compatibility of the system and allows you to find a solution.
Gives the ability to determine the rank of any matrices.
