Application of decoupling systems of linear algebras using the matrix method. Applications of linear equation systems: the decoupling method The linear equation system of algebra is called
System of linear algebra levels. Basic terms Matrix form of notation.
Values of the system of linear levels of algebra. Version of the system. Classification of systems.
Pid system of linear algebra levels(SLAU) toil on the system
Parameters aij name coefficients, and bi – free members Slough. Sometimes, in order to emphasize the number of ranks and unknown ones, say so “m×n system of linear ranks” - thereby indicating that SLAU will take revenge on m ranks and n unknown ones.
If all the free terms are bi=0, then the SLAE is called one-of-a-kind. If among the great members I would like to call one, subtracted from zero, SLAU heterogeneous.
Resolved SLAU(1) I would give up the pure of numbers (α1, α2, ..., αn), Yakshcho Eleemeni pyknosti, pirotstavali in the assigned order of the non-solidomich X1, X2, ..., XN, to rewind the tanger of the Rivnynnya Slau for the total.
Whether the SLAU is the same, I want one solution: zero(In other terminology, it’s obvious), then. x1=x2=…=xn=0.
As SLAU (1) may want one solution, it is called sleeping room, since there is no decision - crazy. As a whole SLAU has one solution, it is called singing, because the decision is impersonal – unsigned.
Matrix form for writing systems of linear algebra.
A matrix can be associated with the skin SLAU; Moreover, the SLAE itself can be written as a matrix equation. For SLAE (1) we consider the following matrices:
Matrix A is called matrix of the system. The elements of this matrix are the coefficients of the given SLAE.
The matrix A˜ is called expanded matrix of the system. They must be added to the matrix of the stovpc system in order to replace the free members b1, b2, ..., bm. This should be topped with vertical rice for softness.
Matrix B is called matrix of free members, and the matrix X - matrix of the unknown.
Vikoristically introduced higher value, SLAE (1) can be written in the form of matrix equation: A⋅X=B.
Note
The matrices associated with the system can be written in different ways: everything is stored in the order in which the variables are passed and the levels of the analyzed SLAE. But in any case, the order of following the unknown tasks of the SLAU may, however, be the same
Kronecker-Capelli theorem. Investigation of systems of linear comparisons for madness.
Kronecker-Capelli theorem
The system of linear ranks of algebra is complete and only if the rank of the system matrix is equal to the rank of the extended matrix of the system, then. rangA=rangA˜.
The system is called a complete system because there is only one solution possible. The Kronecker-Capelli theorem says: if rangA=rangA˜, then the solution is correct; if rangA≠rangA˜, then the given SLAU has no solution (impossible). The answer to nutrition about the number of these solutions is given by a corollary to the Kronecker-Cappelli theorem. The formula of the corollary contains the letter n, as there are many different given SLAEs.
Corollary to the Kronecker-Capelli theorem
If rangA≠rangA˜, then the SLAU is absurd (there is no solution).
Yakscho rangA=rangA˜ If rangA=rangA˜=n, then SLAU is a song (there is one solution).
Please note that the theorem is formulated and therefore does not indicate how to know the solution of the SLAE. With this help, you can only understand that there is no solution, and how much there is to understand.
Methods of development of SLAU
Cramer method
Cramer's method of assignments for the development of systems of linear algebraic equations (SLAE), in which the primary matrix of the system has a variable output of zero. Naturally, it should be taken into account that the matrix is a square system (the concept of the origin only applies to square matrices). The essence of Cramer's method can be expressed in three points:
Fold the primary matrix of the system (also called the primary system), and reconvert so that it is not equal to zero, then. Δ≠0.
For the skin change xi, it is necessary to fold the origin X i, subtracting from the origin and replacing the i-th column with the column of the free terms of the given SLAU.
Find out the values of the unknowns using the formula xi = Δ X i /Δ
Unlinking systems of linear algebra ranks using an additional gate matrix.
The solution of systems of linear algebraic equations (SLAEs) using an additional gate matrix (this method is also called the matrix method or the gate matrix method) requires prior awareness of such concepts as the matrix form of notation of SLAEs. The method of the return matrix of values for the enhancement of those systems of linear algebra levels, in which the primary matrix of the system has a subordinate input of zero. Of course, it is important to remember that the matrix of the system is square (the original concept applies only to square matrices). The essence of the gate matrix method can be expressed in three points:
Write down three matrices: the matrix of the system A, the matrix of unknowns X, the matrix of arbitrary terms B.
Find the gate matrix A-1.
Vikorist's jealousy X=A -1 ⋅B to achieve the release of the given SLAE.
Gauss method. Application of decoupling systems of linear algebras using the Gaussian method.
The Gauss method is one of the most effective and simplest methods of systems of linear algebras(SLAU): both homogeneous and heterogeneous. Briefly speaking, the essence of this method lies in the consistent exclusion of the unknown.
Conversions allowed in the Gaussian method:
Change the place of two rows;
Multiplication of all elements of the row by a number that is not equal to zero.
Adding to elements of one row similar elements of another row, multiplied by any multiplier.
After recreating the row, all elements are reduced to zero.
Sunday rows that repeat.
Apart from the remaining two points: rows that are repeated can be eliminated at any stage of the solution using the Gaussian method - of course, leaving out one of them. For example, if rows No. 2, No. 5, No. 6 are repeated, you can delete one of them - for example, row No. 5. In this case, rows No. 2 and No. 6 will be removed.
Zero rows are taken from the expanded matrix of the system in the world of their appearance.
Solving linear algebra systems (SLAUs) is undoubtedly the most important topic in the linear algebra course. A large number of tasks from all branches of mathematics are reduced to the highest levels of systems of linear levels. These officials explain the reason for the creation of this statute. Material of statistics and structuring so that you can help
- choose the optimal method for improving your system of linear algebra,
- apply the theory to the chosen method,
- adjust your system of linear rankings by looking at the report on the decisions of characteristic applications and tasks.
A short description of the statistics.
From now on, all the necessary meanings are given, and we will introduce the meanings.
Next, we will look at methods for unraveling systems of linear equations in algebra, in which the number of equations is equal to the number of unknown variables and which can lead to a single solution. Firstly, based on Cramer's method, in another way, we will show the matrix method of untying such systems of ranks, and thirdly, we will analyze the Gaussian method (the method of sequential inclusion of unknown variables). To consolidate the theory of obov'yazkovo, the SLAU is based on different methods.
After this, we will move on to the top systems of linear levels of algebra of the literal form, in which the number of levels does not coincide with the number of unknown variables, or the main matrix of the system is a virogen. We formulate the Kronecker–Capelli theorem, which allows us to establish the complexity of the SLAE. Let us analyze the solutions of systems (at different strengths) using the additional concept of a basic matrix minor. We will also look at the Gaussian method and describe the solutions to the applications.
Obviously, we focus on the structure of the formal solution of homogeneous and heterogeneous systems of linear levels of algebra. Let us understand the fundamental solution system and show how the secret solution of the SLAE is written behind the additional vectors of the fundamental solution system. For the best understanding, let's take a look at a bunch of butts.
Finally, let's look at the level systems, which are reduced to linear ones, and various tasks are created, and at the same time the most important ones are triggered by SLAUs.
Navigation on the page.
Meaning, understanding, meaning.
We will look at systems with p linear levels of algebra with n unknown variables (p may be related to n) in the form
Unknown variables, - coefficients (active and complex numbers), - free members (also active and complex numbers).
This form of SLAE record is called coordinate.
U matrix form The record system is visible,
de 
- the main matrix of the system, - the matrix of the unknown changes, - the matrix of the free members.
If we add to matrix A the (n+1)-th column of the matrix-combination of its terms, then we can reject this name expanded the matrix systems of linear rows Designate the expanded matrix to be designated by the letter T, and the set of extended members is reinforced with a vertical line from other columns, so that 
Solutions to the system of linear algebras call the set the value of unknown variables, which wraps all the equalities of the system in the same way. The matrix relationship for the given value of unknown variables also transforms into identity.
Since the system of equalization may want one solution, it is called sleeping room.
Since the system of equalization has no solutions, it is called crazy.
As SLAU has a single solution, it is called singing; if there is more than one decision, then - unsigned.
As free members of all levels of the system reach zero 
, then the system is called one-of-a-kind, in another case - heterogeneous.
Unraveling of elementary systems of linear algebra.
Since the number of levels of the system is equal to the number of unknown variables and the origin of the main matrix is not equal to zero, then such SLAEs will be called elementary. Such systems are equal to a single solution, and in each homogeneous system all unknown variables equal zero.
We started teaching such SLAUs at the secondary school. With their leaders, they took one level, determined one unknown change through others and put them in the line that was lost, then took the advance of the equal, determined the approach of the unknown a minute and was substituted in other equals and so on. Or they used the method of adding, so they added two or more layers in order to turn off the actions of invisible changes. Let us not dwell on these methods, as they really are modifications of the Gaussian method.
The main methods for linking elementary systems of linear levels are the Cramer method, the matrix method and the Gauss method. Let's figure it out.
Verification of linear ranking systems using Cramer's method.
Please let us develop a system of linear levels of algebra 
in which number is equal to the number of unknown variables and the source of the main matrix of the system is removed from zero, then .
Nehai is the originator of the main matrix of the system, and 
- primary matrix, which comes out with A replacement 1st, 2nd, …, nth The line is consistent with the hundred of free members: 
For such values, unknown variables are calculated using formulas using Cramer’s method. 
. This is how to solve the system of linear algebras using Cramer's method.
butt.
Cramer's method 
.
Decision.
The main matrix of the system looks like this: 
. This is calculable (if necessary, see the article): 
Since the source of the main matrix of the system is removed from zero, the system has a single solution, which can be found by Cramer’s method.
Composite and quantifiable necessary subsidiaries 
(the primary is removed by replacing the first column in matrix A with a stack of free members, the primary - replacing the other stack with a stack of free members, replacing the third column of matrix A with a stack of free members): 
Known unknown changes behind the formulas 
:
Subject:
The main drawback of Cramer's method (which can be called a drawback) is the difficulty of calculating the results when the number of levels in the system is more than three.
The development of linear systems of algebra using the matrix method (with the help of a gate matrix).
The system of linear algebraic equations is given in matrix form, where the matrix A has dimensions n by n and its primary value is zero.
Fragments, then matrix A is a reverse matrix, then a reversal matrix is created. Once we multiply the offending parts of jealousy by the left hand, we obtain a formula for finding a matrix of unknown variables. This is how we found solutions to systems of linear levels of algebra using the matrix method.
butt.
Untie the system of linear ranks in a matrix way.
Decision.
Let's rewrite the system of ranks in the matrix form: 
So yak
then the SLAE can be calculated using the matrix method. Using the additional gate matrix, the solution to the system's value can be found as follows: 
.
We will reverse the matrix after the additional matrix with the addition of the algebra of elements of matrix A (if necessary, use the article): 
Lost counting - the matrix of unknown variables, multiplying the return matrix 
to the matrix of independent members (if necessary, look at the article): 
Subject:
or in other entries x 1 = 4, x 2 = 0, x 3 = -1.
The main problem in solving systems of linear algebra equations using the matrix method lies in the difficulty of finding a gate matrix, especially for square matrices of order higher than the third.
Verification of linear systems using the Gaussian method.
Let us know the solution of a system with n linear levels and n unknown changes 
The source of the main matrix is identical to zero.
The essence of the Gauss method lies with the sequential switching off of unknown variables: x 1 of all levels of the system is switched on, starting from the other, then x 2 of all levels are switched on, starting from the third, and so on until the remaining Everything will be deprived of only unknown change x n. This process of reorganizing the system to consistently turn off unknown variables is called directly following the Gaussian method. After completing the direct run of the Gauss method, the remaining balance is calculated as x n, for the additional value, x n-1 is calculated from the remaining balance, and so on from the first balance, x 1 is calculated. The process of calculating unknown variables during the collapse from the remaining alignment of the system to the first is called reversal of the Gaussian method.
We will briefly describe the algorithm for turning off unknown variables.
It is important that what we can achieve by rearranging the levels of the system. Including an unknown change x 1 from all levels of the system, starting from another. For which to the next level of the system we add first, multiplied by , to the third level we add first, multiplied by , and so on, until the nth level we add first, multiplied by . The system of ranks after such changes will never be seen 
de , a 
.
We would have reached the same result if we had determined x 1 through other unknown changes in the first-level systems and subtracted the results from all other equalities. In this way, the value x 1 is turned off from all levels, starting from the other.
Further, the situation is similar, except with a partly removed system, which is assigned to the baby 
For which, up to the third level of the system, we add another, multiplied by , up to the fourth level, we add another, multiplied by , and so on, until the nth level, we add another, multiplied by . The system of ranks after such changes will never be seen 
de , a 
. Thus, the value x 2 is turned off from all levels, starting from the third.
Then we proceed until the unknown is turned off x 3, in which case the situation is similar to that of the part of the system assigned to the little one 
So we continue the direct approach to the Gaussian method and the system never fails to appear 
At this point, the reversal of the Gaussian method begins: we calculate x n from the remaining equalization as, using the additional value of x n, we find x n-1 from the remaining equalization, and so on, we find x 1 from the first equalization.
butt.
Untie the system of linear ranks Gauss method.
Decision.
Including unknown change x 1 from another and third level of the system. For which to both parts of the other and the third level we add similar parts of the first level, multiplied by and likewise: 
Now from the third level we turn off x 2, adding to the second the left and right parts of the left and right parts of the other level, multiplied by: 
Once the direct progress of the Gaussian method is completed, the reversal move begins.
The remaining level of the derived level system is known x 3: 
We can take away the jealousy from others.
From the first, we find an unknown quantity that has been lost, and this completes the reversal of the Gaussian method.
Subject:
X 1 = 4, x 2 = 0, x 3 = -1.
Virus of linear systems of algebra in a non-formal form.
In the halal case, the number of levels of the system p does not match the number of unknown variables n:
Such SLAUs can be a solution, a single solution, or an infinitely rich solution. This applies to the systems of levels, the main matrix of which is square and virogen.
Kronecker - Capelli theorem.
First you need to unravel the system of linear ranks, it is necessary to establish their strength. The answer to nutrition, if the SLAU is healthy, and if it is absurd, gives Kronecker - Capelli theorem:
In order for a system with p ranks of n unknowns (p can be one n ) to be strong, it is necessary and sufficient for the rank of the main matrix of the system to be equal to the rank of the extended matrix, so that Rank (A) = Rank (T).
Let's take a look at the application of the Kronecker-Capelli theorem on the strength of a system of linear levels.
butt.
Explain what the system of linear ranks is 
decision.
Decision.
. There is a quick way to frame minors. Minor of a different order 
External view of zero. Let's look at the third order minor: 
Since all minors are of the third order and equal to zero, then the rank of the main matrix is equal to two.
The drawing has its own rank of the extended matrix 
ancient threesome, fragments of minor third order 
External view of zero.
In such a manner Rang(A), then, according to the Kronecker–Capelli theorem, it is possible to arrive at a non-relevant equation such that the resulting system of linear ranks is inconsistent.
Subject:
The system has no solutions.
Then, we began to establish the absurdity of the system using the Kronecker–Capelli theorem.
But how do you know the decisions of the SLAU, where its capacity has been established?
For this we need to understand the basis minor of the matrix and the theorem about the rank of the matrix.
The minor of the highest order of matrix A, replaced by zero, is called basic.
The value of the basis minor means that its order is equal to the rank of the matrix. For a non-zero matrix of basic minors, or perhaps even more, one basic minor at a time.
For example, let's look at the matrix 
.
All minors of the third order of this matrix are equal to zero, since the elements of the third row of this matrix are the sum of similar elements of the first and other rows.
The basic ones are minors of a different order, the fragments of the stench are removed from zero 
Minori 
basic ones, the fragments are equal to zero.
Theorem about the rank of a matrix.
Since the rank of the matrix is of order p by n prior to r, then all the elements of the rows (and columns) of the matrix that do not equal the base minor are linearly expressed through the similar elements of the rows (and columns), How to establish the basic minor.
What does the theorem about the rank of a matrix give us?
Since we have established the complexity of the system using the Kronecker–Cappelli theorem, we select a basic minor of the main matrix of the system (its order is higher than r), and we exclude from the system all equalities that create the basis minor . The SLAE removed in this way will be equivalent to the output, the fragments of the thrown rows are all one statement (from the theorem about the rank of the matrix and the linear combination of rows that are lost).
The result after the release of all levels of the system is possible in two ways.
If the number of variables in a modern system is equal to the number of unknown variables, then a single solution can be found by the Cramer method, the matrix method or the Gauss method.
butt.
.
Decision.
Rank of the main matrix of the system 
high-grade two, minor minor fragments of a different order 
External view of zero. Rank of the extended matrix 
is also relative to two, leaving a single minor of the third order relative to zero 
and the above-mentioned minor is of a different order, different from zero. Using the Kronecker – Capelli theorem, it is possible to confirm the consistency of the output system of linear levels, the fragments Rank(A)=Rank(T)=2.
As a basic minor we can take it . This is confirmed by the coefficients of the first and other ranks: 
The third rank of the system does not take part in the established basic minor, which is excluded from the system on the basis of the theorem about the rank of the matrix: 
So we deduced an elementary system of linear algebra. Verified by Cramer's method: 
Subject:
x 1 = 1, x 2 = 2.
If the number of ranks r of the extracted SLAE is less than the number of unknown changeables n , then the left parts of the ranks are deprived of additions that establish the basic minor, other additions are transferred to the right side of the rank of the system with the protagonist sign.
The unknown changes (of them) that have been lost in the left parts of the country are called main.
Unseen changes (n - r pieces), which were found on the right parts, are called free.
Now it is important that significant unknown variables can accumulate significant values, such that the main unknown variables are determined through the independent variables in a single manner. Its expression can be found by deriving the SLAE using the Cramer method, the matrix method, or the Gauss method.
Let's take it out of the butt.
butt.
Unravel the system of linear algebraic equations 
.
Decision.
We know the rank of the main matrix of the system 
method of framing minors. As a non-zero minor of the first order, we take a 1 1 = 1 . Let’s now look for a non-zero minor of a different order, which oblyamov’s given minor: 
So we found a non-zero minor of a different order. Let’s look for a non-zero miner of the third order, which oblyamov: 
Thus, the rank of the main matrix is three. The rank of the extended matrix is also the same three, so the system is complete.
A non-zero minor of the third order is taken as a basic one.
To be clear, let’s show the elements that create the basic minor: 
In the left part, the level of the supplementary system is removed, so that the participation of the base minor is taken from the base minor, and the others are transferred with the protagonist signs on the right part: 
There are some unknown changes x 2 and x 5 that are sufficient values, then it is acceptable 
de - more numbers. With this SLAU I can see 
An elementary system of linear algebraic levels has been derived using Cramer's method: 
Otje, .
In the video, do not forget to note any unknown changes.
Subject:
De – more numbers.
Let's bring the pouch.
In order to construct a system of linear algebras in the form of algebra, it is clear from the beginning that it is consistent with the Kronecker-Capelli theorem. Since the rank of the main matrix is not the same as the rank of the extended matrix, there is no doubt about the absurdity of the system.
Since the rank of the main matrix is equal to the rank of the extended matrix, we select a basic minor and select a system level that takes part in the illumination of the selected basic minor.
Since the order of the basic minor is equal to the number of unknown variables, then the SLAE has a single solution that can be found by any method known to us.
If the order of the basic minor is less than the number of unknown variables, then the left side of the system is deprived of the additions to the main unknown variables, other additions are transferred to the right side and is given to the other unknown variables more significant. From the derivation of a system of linear equations, the main unknown variables are found using the Cramer method, the matrix method and the Gauss method.
The Gaussian method for virtuous systems of linear algebras in a non-formal form.
Using the Gauss method, it is possible to construct systems of linear algebras of any kind without first examining them for complexity. The process of sequentially switching off unknown changes allows for a quiet conclusion about both the strength and insanity of the SLAU, and once a decision is made, it makes it possible to find out.
From a computational point of view, the Gaussian method is superior.
Watch his report on the application of the statistical method of Gauss for the development of linear systems of algebra in a formal form.
Recording of the formal solution of homogeneous and heterogeneous systems of algebraic lines for additional vectors of the fundamental solution system.
In this section we are talking about both homogeneous and heterogeneous systems of linear algebra, which can be solved without any doubt.
Let's take a look at the similar systems.
The fundamental system is decided A homogeneous system with p linear levels of algebra with n unknown variables is called the set of (n – r) linearly independent solutions of the system, where r is the order of the basic minor of the main matrix of the system.
How to determine linearly independent solutions of a homogeneous SLAE as X (1) , X (2) , …, X (n-r) (X (1) , X (2) , …, X (n-r) – based on a matrix of dimension n by 1 ) , then the native of the same one-rod systems is represented by the yak of the lilacum-shaped vector of fundamental systems of the systems of the river, the Keephiziyts z 1, s 2, ..., s (n-r), Tobto ,.
What does the term for the solution of a homogeneous system of linear algebra (oroslau) mean?
The sense is simple: the formula specifies all possible solutions of the output SLAE, in other words, taking any set of values of the constant C1, C2, …, C(n-r), we then remove one of the solutions of the output homogeneous C LAU.
In this way, once we know the fundamental system of solutions, we can define all solutions to the same SLAE as .
Let us show the process of creating a fundamental system for solving a homogeneous SLAE.
We select the basic minor of the output system of linear levels, turn on all other levels from the system and transfer the right side of the level system with the corresponding signs to all warehouses, so that no place can be found. houses of change. There are a lot of unknown variables in the values 1,0,0,...,0 and a lot of basic unknowns that have been removed from the elementary system of linear equations, in any way, such as Cramer’s method. Thus, X(1) will be eliminated – first of all the solution of the fundamental system. If you add the values 0,1,0,0,…,0 to the main unknowns and calculate the unknowns from your main ones, remove X (2) . And so on. If the values 0.0, ..., 0.1 and are computably unknown, then X (n-r) is removed. Thus, the fundamental system will be forced to solve a single SLAE and its secret solutions can be written down in sight.
For non-homogeneous systems of linear algebra, the secret solution is given in the form , de - the secret solution of the homogeneous system, and - the private solution of the output non-homogeneous SLAE, which we will remove So, by assigning the values 0,0, ..., 0 to the main unknowns and calculating the values of the main unknowns .
Let's look at the butts.
butt.
Find a fundamental system solution and a fundamental solution to a homogeneous system of linear levels of algebra 
.
Decision.
The rank of the main matrix of homogeneous linear rank systems is always the same as the rank of the extended matrix. We know the rank of the main matrix using the method of framing minors. As a non-zero minor of the first order, we take the element a 1 1 = 9 of the main matrix of the system. We know a non-zero minor of a different order, which oblyamova: 
A minor of a different order, a different type of zero, has been found. Let's look at the third-order minori in non-zero searches: 
All frame minors of the third order to zero, therefore, the rank of the main and extended matrix is equal to two. Let's take the basic minor. It is significant for the accuracy of the elements of the system that are established: 
The third level of the output SLAE does not take part in the creation of the base minor, it can be turned off: 
The right-hand side is deprived of additional information to replace the main unknowns, and the right-hand side is transferring additional information with the main unknowns:
Let us create a fundamental system of decoupling of the output single-row system of linear levels. The fundamental system of solving a given SLAE consists of two solutions, the output of the SLAE is subject to several unknown changes, and the order of the basic minor is the same as two. To find X (1), the main unknowns are known from the equation system 
.
Matrix form
The system of linear rankings can be presented in matrix form as follows:
otherwise, using the matrix multiplication rule,
AX = B.If a matrix is supplemented with a hundred additional members, then A is called an extended matrix.
Methods
Direct (or more precise) methods allow you to find solutions in as many minutes as possible. Iterative methods are based on a random process that is repeated and allows solutions to be resolved as a result of successive approaches.
Direct methods
- Sweeping method (for three-diagonal matrices)
- Kholetsky's decomposition or the square root method (for positive-valued symmetric and Hermitian matrices)
Iterative methods
Version of the linear algebra system in VBA
Option Explicit Sub rewenie() Dim i As Integer Dim j As Integer Dim r() As Double Dim p As Double Dim x() As Double Dim k As Integer Dim n As Integer Dim b() As Double Dim file As Integer Dim y () As Double file = FreeFile Open "C:\data.txt" For Input As file r(0 To n - 1 ) As Double For i = 0 To n - 1 For j = 0 To n - 1 Input #file, x(i * n + j) Next j Input #file, y(i) Next i Close #file For i = 0 To n - 1 p = x (i * n + i) For j = 1 To n - 1 x (i * n + j) = x (i * n + j) / p Next j y (i) = y(i) / p For j = i + 1 To n - 1 p = x (j * n + i) For k = i To n - 1 x (j * n + k) = x (j * n + k) - x (i * n + k) * p Next k y (j) = y (j) - y (i ) * p Next j Next i Upper tricut matrix For i = n - 1 To 0 Step -1 p = y(i) For j = i + 1 To n - 1 p = p - x(i * n + j) * r(j) Next j r(i) = p / x (i * n + i) Next i " Return move For i = 0 To n - 1 MsgBox r (i) Next i "End Sub
Div. also
Posilannya
Notes
Wikimedia Foundation. 2010.
Wonder what “SLAU” is in other dictionaries:
SLAU- system of linear algebra levels... Vocabulary for short and abbreviations
This term has other meanings, div. Slough (meaning). Place and unitary unit of Slough English. Slough Country… Wikipedia
- (Slough) place near Great Britain, at the warehouse of the industrial belt that feeds Great London, on the London-Bristol railway. 101.8 thousand. meshkantsiv (1974). Mechanical, electrical, electronic, automotive and chemical. Great Radyanska Encyclopedia
Slough- (Slough) Slough, industrial and trading place in Berkshire County, south. England, west of London; 97,400 citizens (1981); light industry began to develop during the period between the World Wars. End of the world. Dictionary
Slough: Slough (eng. Slough) a place in England, near the county of Berkshire Slough System of linear levels of algebra ... Wikipedia
Municipality of Reslau Röslau Coat of arms ... Wikipedia
Town of Bad Vöslau Bad Vöslau Coat of arms ... Wikipedia
Projection methods for developing SLAEs are a class of iterative methods, which involve the task of projecting an unknown vector onto a certain space, optimally for another space. 1 Statement of the problem… Wikipedia
Town of Bad Vöslau Bad Vöslau Country AustriaAustria … Wikipedia
The fundamental solution system (FSR) is a set of linearly independent solutions to a homogeneous ranking system. Place 1 Single systems 1.1 Application 2 Non-homogeneous systems ... Wikipedia
Books
- Direct and reverse image updates, spectroscopy and tomography from MatLab (+CD), Valery Sergeyovich Sizikov. The book contains a description of the apparatus of integral equations (IU), systems of linear equations of algebra (SLAU) and systems of linear nonlinear equations (SLNU), as well as software features.
Linear systems. Lecture 6
Linear systems.
Basic understanding.
Mind system
called system - linear levels from unknown.
The numbers are called system coefficients.
The numbers are called free members of the system, – replacement systems. Matrix
called the main matrix of the system, and the matrix
– expanded matrix of the system. Matrices - stovpts
I am sure matrices of free members and unknown members of the system. Then, in matrix form, the ranking system can be written down visually. System decisions is called the value of the variables, when substituting them, all levels of the system are transformed into the correct numerical equality. Whether the system is resolved, you can submit taxes in the form of a matrix. Then matrix jealousy is fair.
The system of ranks is called sleeping room I guess I would like one solution crazy There is no urgent decision.
Developing a system of linear rankings means making sure that decisions are made at different times.
The system is called one-of-a-kind as their free terms are equal to zero. One system is always complete, which is why the decision can be made
Kronecker–Copelli theorem.
The answer to the nutritional solution of linear systems and their unity allows us to eliminate the current result that can be formulated from the view of the current firmness of the system of linear levels from the unknown
(1)
Theorem 2. The system of linear ranks (1) is either complete or not, if the rank of the main matrix is equal to the rank of the extended one (.
Theorem 3. Since the rank of the main matrix of the joint system of linear ranks is equal to the number of unknown ones, the system has a single solution.
Theorem 4. If the rank of the main matrix of the sleeping system is less than the number of unknown ones, then the system can make a neutral decision.
Rules for connecting systems.
3. Find out the main changes through the rules and identify the hidden solutions of the system.
4. As a result, additional values increase in all the values of the main changes.
Methods for untying systems of linear levels.
Return matrix method.
Moreover, the system is a single solution. Let's write the system in matrix form
de 
,
,
.
Multiply the offending parts of the matrix equal to the evil by the matrix
The fragments are, then, obviously, the stars are obviously jealous of finding the unknown.
Butt 27. Using the gate matrix method to unravel the system of linear levels 
Decision. Significantly through the main matrix of the system
.
Let us find the solution behind the formula.
Countable.
The fragments of this system can only be solved. We know all the additions to algebra
,
,
,
,
,
,
,
,
In this manner
.
We're looking forward to checking it out
.
The gate matrix is found correctly. Following the formula, we will know the matrix of variables.
.
The matrix of equal values is excerpted as follows: .
Kramer method.
Let us give you a system of linear rankings from the unknown
Moreover, the system is a single solution. Let's write down the solution of the system in matrix form or
Significantly
. . . . . . . . . . . . . . ,
Thus, we reject formulas for finding the meaning of unknowns, which are called Cramer formulas.
Butt 28 Using Cramer's method, figure out such a system of linear rankings 
.
Decision. We know the origin of the main matrix of the system
.
If there are fragments, then the system can be solved.
Let's know the results of Cramer's formulas
,
,
.
Behind Cramer's formulas we know the meaning of the changes
Gauss method.
The method involves sequential switching off of the changes.
Let us give you a system of linear rankings from the unknown.
The solution process using the Gauss method consists of two stages:
At the first stage, the matrix of the system was expanded to guide additional elementary changes to a similar appearance
,
what type of system does it represent?
After this change 
are respected by the free and in the skin equal to be transferred to the right part.
At the other end of the stage, the remaining level is shown as a change, and the value is substituted for the level. Whose jealousy
appears to be changeable. This process will continue until the first meeting. As a result, the main changes come out through the main ones. 
.
Butt 29. Virishity using the Gauss method will attack the system
Decision. We write the expanded matrix of the system and bring it to a step-by-step view
.
So yak 
There is more than the number of unknown people, then the system is weak and has a neutral solution. Let's write the system for the step-frequency matrix
The origin of the expanded matrix of the value of the system, the folds from the three first columns is not equal to zero, which is important to the basic one. Zminni
They will be basic, and change will be free. Transferable to all levels on the left side
The remaining jealousy is expressed
Having substituted this value for the transfer of the other equalities, it is rejected
stars 
. Having substituted the values of the changeable ones and first of all, we know 
. We'll write this down in front of you
Rivet systems have become widely used in economical practice with mathematical modeling of various processes. For example, at the moment there is a growing demand for the management and planning of production, logistics routes (transportation) and installation placement.
Rivalry systems are being developed in mathematics, physics, chemistry and biology, at the same time increasing the number of populations.
A system of linear ranks is called two or more ranks with many variables that need to know the secret solution. Such a sequence of numbers, if all the equations become true equalities and bring that there is no sequence.
Linear alignment
Similar to the form ax+by=c is called linear. The values of x, y are unknown values that need to be known, b, a are the coefficients of change, c is the variable term.
The solution is equal to the way of the daily graph, so it looks straight, all the points of which correspond to the solutions of the rich member.
Types of linear systems
The simplest applications are the systems of linear alignments with two changes: X and Y.
F1(x, y) = 0 and F2(x, y) = 0, where F1,2 are functions, and (x, y) are variable functions.
Unlock the system of ranks - This means knowing the values (x, y) for which the system is converted to the correct value and establishing that there are no similar values of x and y.
A pair of values (x, y), written as the coordinates of a point, is called a solution to the linear alignment system.
As systems are subject to the same secret decisions and decisions, they cannot be called equally strong.
Some systems of linear rulers and systems of law are partly as old as zero. Since the rights after the sign “jealousy” are partly significant and expressed by the function, such a system is heterogeneous.
The number of changes may be more than two, then we are talking about the application of a system of linear rankings with three changes or more.
Having exited the systems, schoolchildren admit that many equals can easily escape from many unknowns, but this is not the case. The system has a lot of reserves, which can be quite rich.
Simple and easy methods for improving level systems
There is no formal analytical method for developing such systems; all methods are based on numerical solutions. In the school mathematics course, methods such as permutation, folding algebra, substitution, as well as the graphical and matrix method, solved by the Gaussian method, are described.
The main task when learning the best methods is to learn to correctly analyze the system and find the optimal solution algorithm for the skin problem. It’s important not to memorize the system of rules for the skin method, but to understand the principles of establishing one or another method
The solutions for applications of linear level systems for 7th grade programs of the backlight school are simple and very clear. Any teacher of mathematics should have enough respect for this department. Solutions for the applications of linear equation systems using the Gauss and Cramer method are reported in the first courses of the most basic mortgages.
Solving systems using the substitution method
This method of direct substitution changes the value of one variable through another. Viraz is introduced into the line that was lost, then it is brought to the surface with one change. The action is repeated indefinitely due to the number of people unknown to the system
Let's solve the problem of the 7th grade linear ranking system using the substitution method:
As can be seen from the butt, the variable x was expressed through F(X) = 7 + Y. Subtracting the expression from the 2nd level of the system in place of X, helped to subtract one variable Y from the 2nd level. This solution does not require any difficulties and allows you to cancel the Y values. The remaining procedure involves checking the canceled values.
It will never be possible to improve the butt of the system of linear rankings by substitution. The line may be foldable and changeable through an unknown source and appear cumbersome for subsequent calculations. If the system has more than three unknown solutions, substitution is also ineffective.
Connection to the butt system of linear heterogeneous levels:
Solution for additional algebraic folding
When searching for solutions to systems, it is possible to add term-by-term addition and multiplication of different numbers. The end method of mathematical processes is the comparison with one change.
To perfect this method, practice and caution are required. It is difficult to create a system of linear rankings by adding routes when the number of changes is 3 or more. Algebraic additions are manually added when present fractions and tens numbers are added.
Algorithm for action:
- Multiply the offending parts equal to the number. As a result of the arithmetic operation, one of the coefficients when changing is guilty of being equal to 1.
- Item-by-member the tendency to reject the expression and to know one of the unknown.
- Substitute the same values for the 2nd level of the system to search for the change that was lost.
A method for promoting the development of a new change
A new change can be entered if the system needs to know a solution for no more than two levels, since the number of unknowns can also be no more than two.
The method is being used in order to eliminate one of the reasons for the introduction of a new change. The new grain is likely to be introduced into the unknown, and the value of the cob is removed.
From the butt you can see that, having introduced a new change t, it was possible to reduce the 1st level of the system to the standard quadratic trinomial. You can determine the polynomial by knowing the discriminant.
It is necessary to know the values of the discriminant behind the given formula: D = b2 - 4*a*c, where D is the discriminant, which is considered, b, a, c are the multipliers of the polynomial. For a given application a=1, b=16, c=39, then, D=100. If the discriminant is greater than zero, then there are two solutions: t = -b±√D / 2*a, if the discriminant is less than zero, then there is one solution: x = -b / 2*a.
The solution for taking away from the results of the system is to know the way of folding.
Scientific method of system development
Suitable for systems with three levels. The method is located on the coordinate axis of the skin level graphs to enter the system. The coordinates of the points and the span of the curves will be the underlying solutions of the system.
The graphical method has low aspects. Let's take a look at a number of applications for uncoupling linear alignment systems in a clear way.
As can be seen from the butt, two points were created for the skin line, the values of the variable x were chosen quite: 0 and 3. Coming from the value of x, the values for y were found: 3 and 0. Points with coordinates (0, 3) and (3, 0) were assigned to the graph and connected by a line.
These steps must be repeated for another level. The point of the crossbar of the straight lines is the decoupling of the system.
For the next application, you need to know the graphical solution of the linear alignment system: 0.5x-y+2=0 and 0.5x-y-1=0.
As can be seen from the example, the system has no solution, since the graphs are parallel and do not move along their entire length.
The systems from applications 2 and 3 are similar, but upon examination it becomes obvious that their solutions are different. Keep in mind that you can never tell whether the system is being solved or not, but you will need to create a schedule.
The matrix is of the same variety
The matrices are used to briefly record the system of linear rows. A matrix is a special type of table filled with numbers. n*m may n - rows and m - stacks.
The matrix is square if the number of columns and rows are similar to each other. A matrix-vector is a matrix of one row with an infinite number of rows. A matrix with one of the diagonals and other zero elements is called single.
Return matrix - this matrix, when multiplied by the output, is transformed into a single one, such a matrix is only used for the output square.
Rules for converting the ranking system onto the matrix
In one hundred systems of rows, as matrix numbers, we write down the coefficients and members of the rows, one row - one row of the matrix.
A matrix row is called non-zero if one element of the row is not equal to zero. Therefore, if in any case the number of variables varies, then it is necessary to enter a zero in place of the unknown.
The elements of the matrix are very likely to change. This means that the coefficients of the variable x can only be recorded in one column, for example the first, and the coefficients of the unknown y - only in the other.
When a matrix is multiplied, all elements of the matrix are successively multiplied in number.
Options for finding the gate matrix
The formula for finding the return matrix is simple: K -1 = 1 / | K |, de K -1 - Return matrix, and | K | - The leader of the matrix. |K| It is not necessary to add zero, then the system may be solved.
The variable is easily calculated for a two-by-two matrix; you just need to multiply one by one elements along the diagonal. For the “three by three” option, the following formula is | K | b 2 c 1 . You can quickly use the formula, or you can remember that it is necessary to take one element from each row and row of skin elements so that the creation does not repeat the numbers of columns and rows of elements.
Connection of applications of linear alignment systems using the matrix method
The matrix method of finding a solution allows you to speed up cumbersome recordings during the development of systems with a large number of variables and levels.
In the application, a nm is the coefficient of equalities, the matrix is a vector x n are variables, and b n are free terms.
Solving systems using the Gauss method
In general mathematics, the Gaussian method is related to the Cramer method, and the process of searching for solutions to systems is called the Gauss-Cramer solution method. These methods work well with existing replacement systems with a large number of linear levels.
The Gaus method is very similar to the solution using additional substitutions and algebraic folding, but is more systematic. In the school course, the solution using the Gaussian method stagnates for systems with 3 and 4 levels. The method described above resembles an inverted trapezoid. The way to transform algebra and substitutions is to find the values of one variable in one of the equal systems. Another level is related to two unknown ones, and 3 and 4 are apparently related to three and several changes.
After the system has been brought to the described form, it is decided to proceed to the subsequent installation of the main variable levels of the system.
In school textbooks for 7th grade, the descriptions are solved using the Gaussian method in the following order:
As can be seen from the butt, two lines 3x3 -2x4 = 11 and 3x3 +2x4 =7 were cut out on the cut (3). It is a decision to allow one of these changes to be recognized x n.
Theorem 5, as mentioned in the text, shows that if one of the system's equals is replaced with an equivalent one, then the system will be equally strong.
The Gaussian method is important for the study of secondary school students, and is one of the best ways to develop the intelligence of children who begin the advanced learning program in math and physics classes.
For simplicity, I will write the calculations as follows:
The coefficients of the rows and their members are recorded in the form of a matrix, where each row of the matrix corresponds to one of the rows of the system. The left side is strengthened by the alignment to the right. Roman numerals indicate the number of levels in the system.
First write down the matrix to be worked on, then all the steps that need to be carried out in one row. Once the matrix is obtained, write it down after the “arrow” sign and continue to complete the necessary algebra until the result is reached.
The result should be a matrix in which one of the diagonals is 1, and all other coefficients are equal to zero, so that the matrix is reduced to the same appearance. It is impossible to forget the efficiency of calculation with the figures of both parts of the equation.
This recording method is less cumbersome and allows you to avoid having to deal with numerous unknowns.
In any case, it will require respect and honesty. Not all methods are of an applied nature. Some methods of searching for solutions are more important in this other area of activity of people who otherwise study with the method of learning.