Systems of linear lines. Elementary transformation of vector systems. Step-by-step system of vector systems

Appointment 5. Elementary transformations systems of linear alignments are called її advancing transformations:

1) permutation of whether or not two equal places;

2) multiplying both parts of the same equal number;

3) adding to both parts of one equal parts of the second equal, multiplied by the number k;

(at the same time, the rivers become permanent).

Zero equals called equal to the offensive mind:

Theorem 1. Be-like the last sequence of elementary transformations and the transformation of the Sunday of zero equalization to translate one system of linear equalities equally strong and another system of linear equalities.

Bringing. With a glance at the authority of the 4th paragraph, to bring the theorem to the skin for the transformation of the okremo.

1. In case of permutation of the ranks of the system, the ranks themselves do not change, so the system is equally strong for the appointments.

2. By virtue of the first part of the proof, it is enough to bring the firmness for the first equal. Multiplying the system (1) by the number , we take the system

(2)

Come on  system (1) . The same numbers satisfy the system's equalities (1). Since the oskіlki all the equals of the system (2) of the first one zbіgayutsya with the equals of the system (1), then the numbers satisfy all the equals. Shards of the number satisfy the first equality of the system (1), may be the first time the numerical equality:

Multiplying yogo by a number K, We take the correct numerical equality:

That. install, what  system (2).

Back, yakscho solution of the system (2), then the numbers satisfy the mustaches of the system (2). The oskіlki all the equals of the system (1) of the first one zbіgayutsya with the equals of the system (2), then the numbers satisfy all the equals. Shards of the number satisfy the first equality of the system (2), then the numerical equality (4) is valid. After dividing the insults into the number, we take away the numerical equality (3) and conclude that  decoupling of the system (1).

Zvіdsi for appointments 4 system (1) is equal to system (2).

3. By virtue of the first part of the proof, it is sufficient to bring firmness for the first and the other equal system. Dodamo to both parts of the first alignment of the system K, take the system

(5)

Come on system solution (1) . The same numbers satisfy the system's equalities (1). Since the numbers of all the equals of the system (5) of the first one are combined with the equals of the system (1), then the numbers satisfy all the equals. Shards of the number satisfy the first equivalence of the system (1)

Adding term by term to the first equality to a friend, multiplied by the number K we take the correct numerical equality.

§7. Line systems

Equal systems. Elementary transformation of the system of linear lines.

Come on W- field complex numbers. Equal to mind

de
, are called linear equals n nevidomimi
. Ordering set
,
called decisions equal (1), like .

system m linear rivnyan z n the system is called equal to the mind:

- Coefficients of the system of linear alignments, - Free members.

Rectangular table

,

called the matrix of the world
. Let's introduce the notation: - i-Ta row of the matrix,
- k-Ty stovpets matrix. Matrix BUT more signify
or
.

The coming transformation of the rows in the matrix BUT are called elementary:
) turning off the zero row; ) multiplication of all elements of any row by a number
; ) an addendum to any row of any other row, multiplied by
. Similar transformations of the matrix columns BUT are called elementary transformations of the matrix BUT.

The first non-zero element (more importantly to the right) of any row of the matrix BUT is called the conductive element of this row.

Appointment. matrix
it is called a step, as if they were consecrated like this:

1) zero rows of the matrix (like stink) are lower than non-zero ones;

2) yakscho
conduct elements of a row of a matrix, then

Be-like a non-zero matrix And in the case of ordinary elementary transformations, it can be reduced to a stepped matrix.

butt. Inducible matrix
to step matrix:
~
~
.

Matrix folded with system coefficients linear lines (2) are called the main matrix of the system. Matrix
, Otriman, with the admission of the free members, is called the expanded matrix of the system.

The orderings of the set are called the solutions of the system of linear alignments (2), as well as the decisions of the skin linear alignment of the system.

The system of linear alignments is called coherent, because it can only be one solution, and it is not crazy, because it can’t be solved.

The system of linear alignments is called singing, because there is only one solution, that one is not marked, because there is more than one solution.

The coming transformation of the system of linear alignments is called elementary:

) exclusion from the system equal to the mind;

) multiples of both parts, whether it be equal to
,
;

) adding to whether there is any other equal, multiplied by ,.

Two systems of linear lines n the unknown are called equally strong, because the stench is not coherent, but many of their decisions are taken.

Theorem. For example, one system of linear alignments was taken away from the other elementary transformations of the type ), ), ), it is equally strong as a visual one.

Revision of the system of linear alignments by the method of ignoring the unknown (by the Gauss method).

Let the system go m linear rivnyan z n unwidomimi:

Like a system (1) to avenge the mind

then the system is not coherent.

Let us assume that the system (1) is not equal to the form (2). Let the system (1) change coefficient x 1 at first equal
(as if it’s not so, then by rearranging equal places it’s not possible to reach what, so not all coefficients at x 1 equals zero). Zastosuyemo to the system of linear lines (1) advancing lancets of elementary transformations:

, Dodamo to another level;

First equal, multiplied by
, Dodamo to the third level and so on;

First equal, multiplied by
dodamo to the rest of the system.

As a result, we take away the system of linear alignments (we gave the shortest SLN for the system of linear alignments) equal to the strength of the system (1). You may find out that in the other system it is equal to the number i, i 2, do not take revenge on the unknown x 2. Come on k so least natural number, what is unknown x k I want to avenge myself in one equal number i, i 2. Todi otrimana system rivnyan maє vyglyad:

System (3) is equal to system (1). Zastosuєmo now to subsystem
systems of linear alignments (3) microscopy, which were stasted to SLN (1). And so far. As a result of this process, up to one of the two results comes.

1. We take away the SLU, which is equal to the mind (2). And here SLE (1) is inconsistent.

2. Elementary transformations, stasis to SLN (1), do not lead to a system that avenges the appearance (2). At tsomu vipadku SLP (1) by elementary transformations
point to the system equal to the mind:

(4)

de, 1< k < l < . . .< s,

The system of linear alignments in the form (4) is called stepwise. Here you can have two falls.

a) r= n then the system (4) may look

(5)

System (5) has only one solution. Again, system (1) can only be solved.

B) r< n. Whose mind does not have a home
in system (4) they are called head non-dominants, otherwise non-dominant in this system - free (six number one n- r). Nadamo quite a few numerical values ​​are not necessary, even SLU (4) matime looks the same as the system (5). From it, the headlines are unambiguous. In this rank, the system can be resolved, so it is a coherent one. Oskіlki vіlnim nevidomim gave quite a numerical value W, then system (4) is undefined. Again, system (1) is undefined. Viraziv in SLN (4) smut nevidomі through vіlnі nevidomі, otrimaemo system, which is called the wildest solutions of the system (1).

butt. Untie the system of linear alignments by the method G aussa

We write the expanded matrix of the system of linear alignments and, after the help of elementary row transformations, we bring it up to a stepped matrix:

~

~
~
~

~ . By omitting the matrix, we can find a system of linear alignments:
Tsya system is equal to the external system. Like a head of the unknown
vіlnі nevіdomі. By the way, the head of the unknown is only through the wild unknown:

We took away the full solution of SLN. Let me go

(5, 0, -5, 0, 1) is a private solution for SLP.

Task for independent vision

1. To know the global solution and one more solution of the equal system by the method of switching off the unknown:

1)
2)

4)
6)

2. Know for different values parameter a global solution of the system of rivers:

1)
2)

3)
4)

5)
6)

§eight. Vector spaces

Vector space concept. The simplest power.

Come on V ≠ Ø, ( F, +,∙) – field. The elements of the field are called scalars.

Fermentation φ : F× V –> V is called the operation of multiplying elements of multiplying V on scalars from the field F. Significantly φ (λ,a) through λа twir element a to a scalar λ .

Appointment. Bezlich V from a given algebraic operation by adding elements into a multiplier V that multiple elements V on scalars from the field F is called the vector space over the field F, which means the following axioms:

butt. Come on F field, F n = {(a 1 , a 2 , … , a n) | a i F (i=)). Leather element multiple F n called n-simple arithmetic vector. Let's introduce the operation of adding n-peace vectors and multiplication n-world vector per scalar z field F. Come on
. Let's do it = ( a 1 + b 1 , … , a n + b n), = (λ a 1 , λ a 2 , … , λ a n). Bezlich F n where the introduction of operations is vector space, and it is called n-simple arithmetic vector space over the field F.

Come on V- vector space over the field F, ,
. There are such characteristics:

1)
;

3)
;

4)
;

Proof of toughness 3.

Z of jealousy for the law of the fast group ( V,+) maybe
.

Linear fallow, independence of vector systems.

Come on V- Vector space over the field F,

. A vector is called a linear combination of a system of vectors
. Anonymity of all linear combinations of the vector system is called linear shell tsієyu system vektorіv i poznaєєєєєyu.

Appointment. The system of vectors is called linear fallow, as such scalars are used
not all are equal to zero, so

How equivalence (1) is victorious either or less than that, if λ 1 = λ 2 = … = =λ m=0, the system of vectors is called linearly independent.

butt. Chi z'yasuvati chi є system of vectors = (1,-2,2), =(2,0, 1), = (-1, 3, 4) space R 3 linear fallow or independent.

Solution. Let λ 1 , λ 2 , λ 3
і

 |=> (0,0,0) – system solution. Otzhe, the vector system is linearly independent.

The dominance of the linear fallacy and the independence of the vector system.

1. The system of vectors, which wants to avenge one zero vector, is linearly fallow.

2. A system of vectors to avenge a linear fallow subsystem, a linear fallow one.

3. System of vectors, de
є linearly fallow even and only once, if you want one vector of the system, a single vector, є a linear combination of forward vectors.

4. As a system of vectors is linearly independent, but a system of vectors
linearly fallow, then the vector you can look at a linear combination of vectors and up to the same rank.

Bringing. If the vector system is linearly fallow, then
not all are equal to zero, so

In vector equivalence (2) λ m+1 ≠ 0 λ m+1 \u003d 0, then s (2) \u003d\u003e We see that the system of vectors is linearly fallow, shards λ 1 , λ 2 , … , λ m not all equal to zero. They came to wipe their minds. Z (1) => de
.

Let the vector be shown in the same way as you see it: Todo with vector equality
through the linear independence of the vector system, we can see that
1 = β 1 , …, m = β m .

5. Give data to two systems of vectors and
, m>k. If the vector of the vector system can be combined as a linear combination of the vector system, then the vector system is linearly fallow.

Basis, rank of the system of vectors.

Kіntseva vector system in space V over the field F meaningfully through S.

Appointment. Be-yaka linearly independent subsystem of the vector system S is called the basis of the system of vectors S yakscho be-yaky vector system S you can look at the linear combination of the vector system.

butt. Find the basis of the system of vectors = (1, 0, 0), = (0, 1, 0),

= (-2, 3, 0) R3. The system of vectors, linearly independent, oskіlki, vіdpovіdno to dominion 5 the system of vectors was removed from the system of vectors additional help basics electromechanotronics: initialadditional help foundation electrical engineering"; ...

Before the elementary transformations one can see:

1) An addition to both parts of one equal parts of the other, multiplied by the same number that is not equal to zero.

2) Permutation of the equals of the missions.

3).

THEOREM OF THE KRONECKER - CAPELLI

(Umova system integrity)

(Leopold Kronecker (1823–1891) German mathematician)

Theorem: The system is split (may want one solution) either or less if the rank of the matrix of the system is equal to the rank of the extended matrix.

Obviously, system (1) can be written as:

x 1 + x 2 + … + x n

Bringing.

1) If the decision is made, then the column of the free members is a linear combination of the columns of the matrix A, which is also added to the matrix, that is. transition А®А* does not change the rank.

2) Yakshcho RgA = RgA * , tse means that the stench can be in the same basic minor. Stovpets vіlnyh termіnі - linear combination of stovptsіv base minor, tі correct notation, pointed higher.

butt. Calculate the consistency of the system of linear alignments:

~ . Rga = 2.

A* = Rga * = 3.

The system is insane.

butt. Determine the sum of the system of linear alignments.

A =; = 2 + 12 = 14 ¹ 0; RgA = 2;

A* =

RgA* = 2.

Sleep system. Solution: x1 = 1; x2 = 1/2.

2.6 GAUSS METHOD

(Karl Friedrich Gaus (1777-1855) German mathematician)

On the basis of the matrix method and the Cramer method, the Gauss method can be converted to systems of linear alignments from a large number of alignments and unknowns. The essence of the method is based on the subsequent inclusion of non-domestic patients.

Let's take a look at the system of linear alignments:

Let's divide the insulting parts of the 1st equal on a 11 ¹ 0, then:

1) multiply by a 21 i see from another equal

2) multiply by a 31 i see from the third equal

, de d 1 j = a 1 j /a 11 j = 2, 3, …, n+1.

d ij = a ij - a i1 d 1j i = 2, 3, …, n; j = 2, 3, …, n+1.

butt. Reveal the system of linear lines using the Gaussian method.

, Stars are acceptable: x 3 \u003d 2; x 2 \u003d 5; x1=1.

butt. Check the system by the Gauss method.

Let's expand the system matrix.

In this rank, the external system can be presented in the following way:

, Stars are acceptable: z = 3; y=2; x = 1.

Otriman v_dpovіd zbіgaєtsya vіdpovіddu, otrimana for this system by the Cramer method and the matrix method.

For an independent vision:

Suggestion: (1, 2, 3, 4).

TOPIC 3. ELEMENTS OF VECTOR ALGEBRI

BASIC DESIGNATION

Appointment. Vector called straight lines (a couple of points are ordered). Before vector_v_vіdnosti also zero vector, the cob of that kind of zbіgayutsya.

Appointment. Dovzhina (module) the vector is called between the cob and the end of the vector.

Appointment. The vectors are called collinear like stench spread on one or the parallel lines. The null vector is collinear to any vector.

Appointment. The vectors are called coplanar like a real flat, like a parallel stink.

Colinear vectors are always coplanar, but not all coplanar vectors are collinear.

Appointment. The vectors are called equal as if they are collinear, however, they are straightened and can be the same modules.

Be-yaki vectors and can bring to the hearty cob, tobto. to induce vectors and vidpovidno equal data and make a hot cob. From the designation of the vector equality, it is obvious that whether a vector can be an impersonal vector, equal to you.

Appointment. Line operations over vectors is called addition and multiplication by a number.

Sumoyu vector_v є vector -

Tvir - , at which kolіnearen .

Direction vector іz vector ( ), so a > 0.

The vector of protivolezhnoy directives with the vector (?), so that a< 0.

POWER OF VECTORIV

1) + = + - commutativity.

2) + ( + ) = ( + )+

5) (a×b) = a(b) – associativity

6) (a + b) = a + b - distributivity

7) a(+) = a + a

Appointment.

1) Basis the space is called as if 3 non-coplanar vectors, taken in the same order.

2) Basis on the flat are called 2 non-collinear vectors, taken in the same order.

3)Basis on a straight line is called a non-zero vector.

Two systems of linear alignments in one set x 1 ..., x n

They are called equivalent, because their impersonal decisions are avoided (therefore, multiplications and K n are avoided,). Tse means, sho: or stench at once є empty submultiples (so offending systems (I) and (II) unsettled), or stench at once not empty, i (so skin solution of system I є solutions of system II і skin solution of System II є solutions of system I ).

Stock 3.2.1.

Gaus method

The plan for the algorithm proposed by Gaus is rather simple:

1. zastosovuvat to the system of linear alignments sequentially, so as not to change the impersonal solution (in this way, we take the impersonal solution of the visual system), and go to the equivalent system, which can be "simple looking" (this is the name of the step form);
2. for the "simple mind" of the system (with a stepwise matrix) describe the impersonal solution that is used for the impersonal solution of the visual system.

It is significant that the close method "fan-chen" used to be used already in ancient Chinese mathematics.

Elementary transformation of systems of linear alignments (row of matrices)

Designation 3.4.1 (elementary transformation of the 1st type). When up to the i-th level of the system, the k-th level is added, multiplied by the number (signed: (i) "=(i) + c(k); then only one i-th level (i) is replaced by a new level (i) "=(i)+c(k)). New i-e equal may look (a i1 + ca k1) x 1 + ... + (a in + ca kn) x n = b i + cb k, or, briefly,

That is, in the new i-th district a ij " = a ij + ca kj , b i " = bi + cb k.

Designation 3.4.2 (elementary conversion type 2). For i -е і k -е the equals are changed by the ranks, the other equals are not changed (signs: (i)"=(k) , (k)"=(i) ; .,n

Respect 3.4.3. For clarity, for specific calculations, you can add elementary transformations of the 3rd type: the i-th calculation is multiplied by a non-zero number , (i)" = c (i) .

Proposition 3.4.4. Just as the type of system I passed to system II for the help of the final number of elementary transformations of the 1st and 2nd type, then in the form of system II you can turn to system I as well as elementary transformations of the 1st and 2nd type.

Bringing.

Respect 3.4.5. The firmness is true and s included to the elementary transformations of the elementary transformation of the 3rd type. Yakscho i (i)"=c(i) , then ta (i)=c -1 (i)" .

Theorem 3.4.6.After the last stop of the last number of elementary transformations of the 1st or 2nd type, the system of linear alignments, equivalent to the cob, comes up to the system of linear alignments.

Bringing. It is important to take a look at the transition from system I to system II with the help of one elementary transformation and to bring the solution of inclusion to riches (shards through the brought proposition of system II can be turned to system I and to that, inclusion, to be brought equanimity).

Appointment 1. The system of linear alignments mind (1) , de , field, is called a system of m linear lines from n nevidomimi over the field, - Coefficients for non-domic, , , - free members of the system (1).

Appointment 2. Ordered n-ka (), de, called to the top of the system of linear lines(1), even when replacing the change on the skin, the system (1) is changed to the correct numerical alignment.

Appointment 3. sleepy yakscho vain may want to make one decision. Otherwise, the system (1) is called crazy.

Appointment 4. The system of linear alignments (1) is called singing there can be only one solution. Otherwise, the system (1) is called unappointed.

System of linear lines

(є decision) (no decision)

sleepy crazy

(one decision) (not one decision)

pevna is unknown

Appointment 5. The system of linear lines over the field R called homogeneous yakscho all її vіlnі terms equal to zero. Otherwise the system is called heterogeneous.

Let's look at the system of linear lines (1). That same homogeneous system in mind is called a homogeneous system, associated from system (1). Homogeneous SLN for the first time, oskolki may be decided.

For cutaneous SLN, two matrices can be introduced at a glance - the main one is extended.

Appointment 6. The main matrix of the system of linear alignments(1) the matrix is ​​called, it is composed of coefficients with no offensive type: .

Appointment 7. Expanded matrix of the system of linear alignments(1) the matrix is ​​called, truncated from the matrix by a path adjoining to it a set of free members: .

Appointment 8.Elementary transformations of the system of linear alignments are called as follows: 1) multiplying both parts of the same equal system by a scalar; 2) adding to both parts of one level of the system of second parts of the other level, multiplied by an element; 3) supplementing or proving equal to the mind.

Appointment 9. Two systems of linear lines over the field R what the change is called equally strong, as their impersonal decisions are avoided.

Theorem 1 . Just as one system of linear equalities was taken away from another for the help of elementary transformations, such systems are equally strong.

Manually elementary transformations are not brought up to a system of linear alignments, but to an expanded matrix.

Appointment 10. Let's give a matrix with elements from the field R. Elementary transformations matrices are called like this:

1) multiplication of all elements of any row on the matrix by aО Р # ;

2) multiplying all the elements of any row on the matrix by aО Р # and adding the other elements of the next row;

3) permutation of the places by two rows of the matrix;

4) adding or releasing the zero row.

8. SLU solution: m method of subsequent exclusion of unknowns (Gauss method).

Let's take a look at one of the main methods of decoupling systems of linear alignments, which is called by the method of subsequent inclusion of unknown, what else, Gauss method. Take a look at the system(1) m linear rivnyan z n nevidomimi over the field R:(1) .

The system (1) wants one of the coefficients if not good 0 . Іnakshe (1) - the system of equals from () nevіdomimi - tse superechit minds. We remember the equalities by the months so that the coefficient at the first equalization is not good 0 . In this rank, you can vvazhati, sho. Multiply the offending parts of the first equal and add to the second parts of the other, third, ..., m th equal. We take the system mind: , de s- the smallest number, so I want one of the coefficients if not healthy 0 . We remember the equalities by the months so that the other row has a coefficient when changing the cost 0 , then. we can guess what. Let's multiply the insulting parts of the other equal and add to the equal parts of the third, ..., m th equal. Continuing this process, we take the system into account:

The system of linear equalities, yak, according to Theorem 1, is equal to the system (1) . The system is called a stepped system of linear alignments. There are two possibilities: 1) Wanting one of the elements is not good 0 . Come on, for example. Same with the system of linear alignments, it’s similar to the mind that it’s impossible. Tse means that the system does not have a solution, and therefore the system (1) cannot have a solution (at times (1) is an inconsistent system).

2) Come on, ...,. Todi for the help of elementary transformation Z) we take away the system - the system r linear rivnyan z n unknown. At any change, for the coefficients they are called head change(tse), їх total r. Інші ( n-r) change names free.

There are two possibilities: 1) Yakshcho r=n, then - the system of tricot look. For this one, from the last equal, we know change, from the last one - change, from the first equal - change. Also, there is only one solution for the system of linear alignments, and also for the system of linear alignments (1) (at times the system (1) is assigned).

2) Come on r . And here the main changes turn through the viles and win the decisive solution of the system of linear lines (1). Nadayuyuschie vіlnym zmіnnym sovіlnі znachenya, nabuvayut different private solutions of the system of linear lines (1) (system (1) is not visible in this case).