Algorithm for reducing the matrix to a stepwise look. Step Matrix. Matrix rank Gauss method for reducing a matrix to a stepped look

In 2020, NASA is launching an expedition to Mars. Deliver the spacecraft to Mars with an electronic carrier bearing the names of all registered participants in the expedition.

Registration of participants in the vote. Take away your ticket to Mars for the blessings.

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Chergovy before the New Rock... the weather is frosty and there are big discounts... Everything prompted me to write again about... fractals, and about those who know about Wolfram Alpha. From the first drive є tsіkava stattya, in yakіy є buttocks of two-dimensional fractal structures. Here we can look at foldable examples of trivial fractals.

A fractal can be visually manifested (described), as a geometric figure, but as a body (looming on the surface, that they are more impersonal, in a given direction, an impersonal point), details that may have the same shape, like the actual figure itself. That is, the structure is self-similar, looking at the details as if it were enlarged, we can still find the very form that is without enlargement. Similarly, in a visually striking geometric figure (not a fractal), with more minor details, as if the form can be simple, the figure itself is visible. For example, when you finish the great big part of the ellipse, it looks like a straight tree. This is not the case with fractals: in case of any improvement, we will renew that very folding form, as if repeating again and again with skin enhancements.

Benoit Mandelbrot, the founder of the science of fractals, wrote in his article "Fractals and Mystery" in the name of science: talu will be reduced to the extent of the whole, it will look like a whole, or exactly, or, perhaps, with a slight deformation.

Appointment. Step-frequent we will name the matrix, as we can advance the power:

1) if the i-th row is zero, then the (I + 1)-th row is also zero,

2) as the first non-zero elements of the i-th and (I + 1)-th rows in the rows in columns with numbers k and R, obviously, then k< R.

Umova 2) due to the effect of the language reduction zero in the transition from the i-th row to the (I + 1)-th row. For example, matrices

A 1 =, A 2 =
, A 3 =

є steps, and matrices

Y 1 = , B 2 = , B 3 =

steps not є.

Theorem 5.1. Whether a matrix can be brought up to a stepwise one for the help of elementary transformations of rows of a matrix.

We illustrate this theorem with an application.

A =

The matrix, which has become ─ stupіnchasta.

Appointment. Matrix rank naming the number of non-zero rows in the stepwise view of the matrix.

For example, the rank of the matrix A for the front butt is 3.

lecture 6.

Visionaries, authorities. The reverse matrix is that її calculation.

The clerks are of a different order.

Let's look at a square matrix of a different order

A =

Appointment. Vyznachnik of a different order, in the form of matrix A, the number is called, which is calculated after the formula

│А│= = .

Elements a ij are called the elements of the primate│A│, elements a 11, a 22 make main diagonal and elements a 12, a 21 ─ pobichnu.

butt. = -28 + 6 = -22

Visionaries of the third order.

Let's look at the square matrix of the third order

A =

Appointment. Third Order Vyznachnik, the dependent matrix A is the number that is calculated after the formula

│А│= =

To remember, how to do it in the right part of equanimity, take it with a “plus” sign, and if you ─ with a “minus” sign, remember the rule, call tricot rule.

= ─

Apply:

1) = - 4 + 0 + 4 – 0 + 2 +6 = 8

2) = 1, then. │Е 3 │= 1.

Let's take a look at one method of calculating the third-order debtor.

Appointment. Minor element a ij vyznazhnik is called the vyznazhnik, subtracting from the given chair of the i-th row and j-th column. Algebraic additions A ij of the element a ij of the sign is called yogo minor M ij taken with the sign (-1) i + j .

butt. Let's calculate the minor M 23 і the addition of the algebra A 23 of the element a 23 in the matrix

A =

Let's calculate the minor M 23:

M 23 = = = - 6 + 4 = -2

A 23 \u003d (-1) 2 + 3 M 23 \u003d 2

Theorem 1. The third-order leader is a richer sum of creations of elements, whether of a row (stovptsya) with their own algebraic additions.

Doc. For appointment

= (1)

Vibero, for example, another row i know the addition of the algebra A 21, A 22, A 23:

A 21 \u003d (-1) 2 + 1 = -() =

A 22 \u003d (-1) 2 + 2 =

A 23 \u003d (-1) 2 + 3 = - () =

Let's remake the formula (1)

│А│= ( ) + () + () \u003d A 21 + A 22 + A 23

│A│= A 21 + A 22 + A 23

called the appointment of the primate│A│ behind the elements of another row. Similarly, the layout can be taken away for the elements of other rows of that kind of column.

butt.

= (behind elements of another column) = 1× (-1) 1+2 + 2 × (-1) 2+2 +

+ (-1)(-1) 3+2 = - (0 + 15) + 2(-2 +20) + (-6 +0) = -15 +36 – 6 = 15.

6.3. Vyznachnik n-th order (n N).

Appointment. n-th order vyznachnik, matching matrices of the n-th order

A =

The number is called, which is more expensive than the sum of the creations of the elements, whether there is a row (stupptsya) from the їхної algebraic additions, tobto.

│A│= A i1 + A i2 + ... + A in = A 1j + A 2j + ... + A nj

It is not important to remember that for n = 2 there is a formula for the calculation of a variable in a different order.

butt. = (behind the elements of the 4th row) = 3×(-1) 4+2 +

2×(-1) 4+4 = 3 (-6 + 20 - 2 - 32) +2 (-6 +16 +60 +2) = 3 (-20) +2 × 72 = -60 +144 = 84.

Respectfully, if the chief has all the elements of this row (stovptsya), except for one, add to zero, then when calculating the chief of yogo, manually lay out the elements of that row (stowptsya).

butt.

│Е n │= = 1 × │E n -1 │ = ... = │E 3 │= 1

The power of the appointees.

Appointment. mind matrix

we will name tricot matrix.

Power 1. The signifier of the tricot matrix is more advanced in the process of supplementing the elements of the head diagonal, tobto.

= =

Power 2. Significant matrix with a zero row or a zero row leading to zero.

3. . When the matrix is transposed, the signifier changes, so.

│А│= │А t │.

Power 4. If the matrix goes from matrix A to the multiplication of the skin element of the third row by the number k, then

│В│= k│А│

Power 5.

= =

Power 6. If the matrix comes out of matrix A by permuting two rows, then │В│= −│А│.

Power 7. The index of the matrix with proportional rows is equal to zero, the zero is the index of the matrix with two equal rows.

Power 8. The index of the matrix is not changed, so to the elements of one row add the elements of the next row of the matrix, multiplied by the kilka.

Respect. So, as for the power of the matrix 3, the matrix does not change when transposed, all the powers of the matrix rows are correct for the columns.

Power 9. If A and B are square matrices of order n, then │AB│=│A││B│.

The back matrix.

Appointment. A square matrix A of order n is called vicious, yakscho іsnuє matrix is such that AB = BA = E n . Which way the matrix is called back to matrix A i is denoted by A-1.

Theorem 2. Fairly so assertion:

1) as the matrix A is reverse, there is exactly one reverse matrix;

2) the reversal matrix may be a signifier, a reference to zero;

3) if A and B are reverse matrices of order n, then the matrix AB is reverse, and (AB) -1 =

B -1 ×A -1 .

Proof.

1) Let B and C - matrices, return to matrix A, tobto. AB \u003d BA \u003d E n and AC \u003d CA \u003d E n. Todi B \u003d BE n \u003d B (AC) \u003d (VA) C \u003d E n C \u003d C.

2) Let matrix A be inverted. Then there is the matrix A-1, which is reversed, moreover

For power 9 vyznachnik │АА -1 │=│А││А -1 │. Todі │А││А -1 │=│Е n │, stars

│А││А -1 │= 1.

Also, │А│¹ 0.

3) True,

(AB) (B -1 A -1) \u003d (A (BB -1)) A -1 \u003d (AE n) A -1 \u003d AA -1 \u003d E n.

(B -1 A -1) (AB) \u003d (B -1 (A -1 A)) B \u003d (B -1 E n) B \u003d B -1 B \u003d E n.

Later, AB is a reverse matrix, and (AB) -1 \u003d B -1 A -1.

The theorem is coming, giving the criterion for the basis of the pivot matrix and the її calculation.

Theorem 3. The square matrix A is reversible and then, if it is a sign of zero. Yakscho │А│¹ 0, then

A -1 ==

butt. Know the matrix wrapped for the matrix A =

Solution.│А│= = 6 + 1 = 7.

Oskіlki │А│¹ 0, іsnuє revvotna matrix

A -1 ==

Calculating A11 = 3, A12 = 1, A21 = -1, A22 = 2.

A -1 = .

lecture 7.

Systems of linear lines. Criteria for the sleepiness of the system of linear alignments. Gauss method of solving systems of linear alignments. Cramer's rule and the matrix method for solving systems of linear alignments.

Systems of linear lines.

Sukupnіst is equal to the mind

(1)

called a system of m linear lines from n nevidomimi x 1, x 2, ..., x n. Numbers a ij are called system coefficients, and numbers b i ─ independent members.

System solutions (1) the collection of numbers is called z 1, z 2, ..., n, when adding them to the system (1) instead of x 1, x 2, ..., x n, it is necessary to correct the number of equalities.

Check the system─ it means to know all the solutions, or to bring what you can’t. The system is called sleepy I can’t even want to make one decision, and crazy there is no solution.

Matrix stacked with system coefficients

A =

It is called the matrix of the system (1). If we add a hundred of free members to the matrix of the system, then we take away the matrix

B =
,

yaku is called extended matrix of system (1).

How meaningful

Х = , З = , then system (1) can be written as a matrix equalization АХ=С.

In order to bring the matrix to a stepped look (Fig. 1.4), follow the vikonate so.

1. At the first column, select the element, the input of zero ( passing element ). A row with a conductive element ( provіdny row ), as if it’s not persha, rearrange it on the plate of the first row (reformation of type I). If there is no leading element in the first column (all elements add to zero), then this column is turned off, and we continue to search for the leading element in the part of the matrix that is missing. The transformation will end, as all the elements are excluded, or in the part of the matrix that is left out, all the elements are zero.

2. Divide all elements of the wire row into a wire element (type II conversion). If the last row is left, then on which one the transformation should be completed.

3. To the skin row, stitched below the leading one, add a leading row, multiplications are similar to such a number, so that the elements that stand below the leading one are added to zero (type III transformation).

4. Turning off the row and stovpets, on the back of which there is a conductive element, go to point 1, in which all the descriptions will stop to the part of the matrix that is left out.

Theorem for the distribution of the signifier for the elements of a row.

The theorem about the distribution of the arbitrator for the elements of the row, or else it allows you to calculate the calculus of the arbiter - first order () to the calculation of the chiefs in order .

If the leader can equal zero elements, then it is easier to lay out the leader according to the elements of that row or else, which will avenge the largest number of zeros.

Vykoristuyuchi power vyznachnikiv, you can remake the vyznachnik - th order so that all the elements of the deyago row of abostovptsya, krіm one, became equal to zero. In this rank, the counting of the vyznachnik - in the first order, as a rule, vin vіdminniy vіd zero, zvedetsya up to the calculation of one vznachnik - in order.

Task 3.1. Calculate the winner

Solution. Having added the first to the next row, to the third - the first, multiplied by 2, to the fourth - the first, multiplied by -5, we take

Laying out the chief for the elements of the first column, perhaps

In the otrimanomu vyznachnik of the 3rd order, all the elements of the first column, crimson of the first, are set to zero. For this to the next row, do the first, multiplications by (-1), to the third, multiplications by 5, dodamo the first, multiplications by 8. Oskilki multiplied the third row by 5, then (in order not to change the signifier) we multiply yogo by. Maymo

The otrimaniy vyznachnik is laid out behind the elements of the first column:

Laplace's theorem(1). Alien extension theorem(2)

1) The leader of the richest sum of creative elements of any order on their additions to algebra.

2) The sum of the creations of the elements of any row of the leader on the additions of the algebra of the second row of elements to the second row is equal to zero (the theorem of multiplication on foreign algebraic additions).

Be it a point on the plane, when the coordinate system is chosen, it is represented by a pair (α, β) of its coordinates; the numbers α and β can also be understood as the coordinates of the radius-vector with the end of the point. Similarly, in space, the triple (α, β, γ) assigns a point or a vector іz with coordinates α, β, γ. Itself on the basis of the goodness of the reading house, the geometric interpretation of the systems of linear alignments from two or three is nevidomimi. So, at the time of the system of two linear lines from two nevidomimi

a 1 x + b 1 y \u003d c 1

a 2 x + b 2 y \u003d c 2

the skin of the lines tumbles like a straight line on the plane (div. Fig. 26), and the tie (α, β) is like a crossing point of the lines or a vector with the coordinates аїр (the figure turns into a drop, if the system can be a single solution).

Rice. 26

Similarly, it is possible to find out from the system of linear lines from the trioma nevidomimi, interpreting the skin level as a leveling of the area near the open space.

Mathematics and other additions (zokrema, in the theory of coding) are brought to the right by the systems of linear equalities, so that more than three unknowns can revenge themselves. The system of linear alignments with n indistinguishable x 1 , x 2 , ..., x n is called the consistency of alignment of mind

a 11 x 1 + a 12 x 2 + ... + a 1n x n = b 1

a 21 x 1 + a 22 x 2 + ... + a 2n x n = b 2

. . . . . . . . . . . . . . . . . . . . . . (1)

a m1 x 1 + a m2 x 2 + ... + a mn x n = b m

Whether the solution of the system is understood as typing (real) values of unknown (α 1 , α 2 , ..., α n ), what to wrap the skin with the correct evenness.

If you want a bezperednє geometric clouding of the system (1) with n > 3 it is no longer possible, it is possible to easily expand the geometrical space of two or three of the worlds by a certain amount of n. Tsіy metі і є far away vyznachennya.

The skin of orderings is a set of n real numbers (α 1 , α 2 , ..., α n ) is called an n-world arithmetic vector, and the numbers themselves α 1 , α 2 , ..., α n - Coordination of that vector.

For the recognition of vectors, as a rule, bold font i is used for a vector with coordinates α 1, α 2, ..., α n, the primary form is taken:

a = (α 1, α 2, ..., α n).

By analogy with the sizable plane of the impersonal plane of all n-world vectors, which satisfies the linear alignment with n nevidomim, is called the hyperplane in the n-world space. With such a designation of the impersonality of all solutions of the system (1), there is nothing else, like a retinal decal of hyperplanes.

The folding and multiplication of n-world vectors are governed by the same rules as for the largest vectors. And yourself, like

a = (α 1 , α 2 , ..., α n), b = (β 1 , β 2 , ..., β n) (2)

Two n-world vectors, then their sum is called a vector

α + β = (α 1 + β 1 , α 2 + β 2 ..., α n + β n). (3)

The subset of a vector by the number λ is called the vector

λа = (λα 1 , λα 2 , ..., λα n). (4)

A lot of n-world arithmetic vectors with operations of adding vectors and multiplying a vector by a number is called an arithmetic n-world vector space L n .

With the introduction of operations, you can see quite a few linear combinations of several vectors, so you can see

λ 1 a 1 + λ 2 a 2 + ... + λ k a k ,

de λ i - actual numbers. For example, a linear combination of vectors (2) with coefficients λ and μ - ce vector

λа + μb = (λα 1 + μβ 1 , λα 2 + μβ 2 , ..., λα n + μβ n).

In the trivial space of vectors, a special role is played by the trinity of vectors i, j, k (coordinate vectors), which sort of vector a:

a = xi + yj + zk,

de x, y, z are real numbers (coordinates of the vector a).

The n-world type has the same role played by the system of vectors:

e 1 = (1, 0, 0, ..., 0),

e 2 = (0, 1, 0, ..., 0),

e 3 = (0, 0, 1, ..., 0),

. . . . . . . . . . . . (5)

n = (0, 0, 0, ..., 1).

Any vector є, obviously, is a linear combination of vectors e 1 , e 2 , ..., e n:

a = a 1 e 1 + a 2 e 2 + ... + a n e n (6)

moreover, the coefficients 1 , 2 , ..., n are taken from the coordinates of the vector a.

Denoting a vector through 0, all coordinates of which are equal to zero (in short, a zero vector), we introduce the following important designation:

The system of vectors a 1 , a 2 , ..., and k is called linearly fallow, as it is equal to the zero vector linear combination

λ 1 a 1 + λ 2 a 2 + ... + λ k a k = 0,

in yakіy want b one of the coefficients h 1 , 2 , ..., λ k vіdminny vіd zero. Otherwise, the system is called linearly independent.

Yes, vector

and 1 = (1, 0, 1, 1) and 2 = (1, 2, 1, 1) and 3 = (2, 2, 2, 2)

linear deposits, shards

a 1 + a 2 - a 3 = 0.

Linear fallow, as can be seen from the designation, is equal (at k ≥ 2) to the fact that one of the vectors in the system wants a linear combination of others.

If the system is composed of two vectors a 1, and 2 then the linearity of the system means that one of the vectors is proportional to the other, say, and 1 = λа 2; in the trivi- merous type, it is equal to the collinearity of the vectors a 1 and a 2 . So the very linearity of the system I of the three vectors in the greatest space means the complanarity of these vectors. The concept of linear fallow is such a rank of natural zagalnennyam to understand the colinearity and coplanarity.

It doesn't matter if the vectors e 1 , e 2 , ..., e n іz system (5) are linearly independent. Also, in the n-world space, systems are developed from n linearly independent vectors. It is possible to show that a system with a larger number of vectors can be linearly deposited.

Whether the system is a 1 , a 2 , ..., and n iz n linearly independent vectors in the n-world space L n is called its basis.

Whether a vector to the space L n is expanded, and, moreover, in the same order, according to the vectors of a sufficient basis a 1, a 2, ..., and n:

a = λ 1 a 1 + λ 2 a 2 + ... + λ n a n.

This fact can be easily established from a specific basis.

Continuing the analogy with the trivum space, it is possible to designate the scalar vertex a b vector in the n-virtual scale, depending on

a b = α 1 β 1 + α 2 β 2 + ... + α n β n .

For such a designation, all the main powers of the scalar creation of trivi- mer vectors are taken care of. The vectors a and b are called orthogonal, since their scalar addition is equal to zero:

α 1 β 1 + α 2 β 2 + ... + α n β n = 0.

Theoretically, there is one more important understanding of the line codes - the understanding of the space. The subdivision of the V expanse L n is called the subspace of this expanse, as

1) for any vectors a, b, which lie within V, if the sum a + b also lie within V;

2) for any vector that lies within V, and for any real number λ, the vector λand also lies with V.

For example, the absence of all linear combinations of vectors e 1 e 2 from system (5) will be a subspace to the space L n .

In linear algebra it can be explained that the skin subspace V has such a linearly independent system of vectors a 1 , a 2 , ..., a k

a = λ 1 a 1 + λ 2 a 2 + ... + λ k a k.

The system of vectors is given and is called the basis of subspace V.

Z is assigned to the expanse of that subspace without a middle, that the expanse of L n є is a commutative group for the operation of the folding of vectors, and be it yoga subspace of the V є subgroup of the group. In this sense, for example, you can look at the sum of the classes of space L n subspace V.

At the end, it’s necessary to look at the elements of the sufficient field F, even though theoretically n-world arithmetic space, the replacement of real numbers (that is, the elements of the field of real numbers), all designations and facts, pointing to more, saved b force.

The theory of coding plays an important role in vipadok, if the field F is a field in Z p , like, as we know, it is great. In this way, in the n-peace, the expanse is also sig- nificantly and vengeance, no matter how bad the bachiti, p n elements.

Understanding the space, like understanding the group and the circle, admitting the same axiomatic designation. For the details, let us look at the Zhiver to the course of linear algebra.

Linear combination. Linear fallow and independent vector systems.

other combination of vectors

Linear combination of vectors name the vector

de - Linear combination coefficients. Yakscho the combination is called trivial, yakscho-nontrivial.

Linear fallow and independence of vectors

System linear fallow

System linearly independent

Criterion of linear occurrence of vectors

In order to vector (r > 1) are linearly fallow, necessary and sufficient, if only one of these vectors is used, a linear combination of others.

Openness of linear space

Linear space V called n-peaceful (maє razmirnіst n), just like in newcomer:

1) іsnuє n linearly independent vectors;

2) be it a system n + 1 vectorіv linearly fallow.

Designation: n= dim V;.

The vector system is called linear fallow, as is true non-zero set of numbers such that a linear combination

The vector system is called linearly independent, yakscho z equal to zero linear combination

next equal to zero all coefficients

The feeding of the linear occurrence of vectors in the fall is built up to the feeding of a non-zero solution in a homogeneous system of linear alignments with coefficients equal to the respective coordinates of these vectors.

To better understand the concept of "linear staleness", "linear independence" of the vector system, to articulate the tasks of the offensive type:

Linear fallow. I and II criteria of linear fallow.

Vector system linearly fallow even if one of the vectors in the system is a linear combination of other vectors in the system.

proof. Let the vector system be linearly fallow. Then use such a set of coefficients , What, moreover, I want one coefficient of vіdminny vіd zero. Let's say what. Todi

that is a linear combination of other vectors in the system.

Let one of the vectors in the system be a linear combination of other vectors. It is acceptable that the vector is . Obviously what. It was taken into account that the linear combination of vectors in the system is equal to zero, and one of the coefficients of input is equal to zero (equal).

Rechennya10 . 7 As a vector system to replace a linearly fallow subsystem, the entire system is linearly fallow.

proof.

Come to the system vector_v subsystem , є linearly fallow, tobto , and wanting one coefficient of input zero Then we will store a linear combination. It is obvious that this linear combination is closer to zero, and the middle coefficients are non-zero.

The base of the system of vectors, її the main power.

The base of a non-zero system of vectors is called an equivalent and linearly independent subsystem. There is no zero basis system.

Power 1: The base of the linear independent system is generated from itself.

Butt: The system of linearly independent vectors, but the shards of the vector from the vectors cannot but linearly degenerate through others.

Dominion 2: (Bazi Criterion) The subsystem of the given system is linearly independent as the base and the same, if it is maximally linearly independent.

Proof: Given system Necessity Come on base. Even for the appointments of i, as well as, de, the system is linearly fallow, to the fact that it linearly evolves through, it is also as linearly independent as possible. Prosperity Let the subsystem be as linear as possible, de . linearly fallow linearly evolves through the same base of the system.

Dominion 3: (Basic Dominion) The skin vector of the system is virojuєtsya through the base in a single rank.

proof Let the vector rotate through the base in two ways, then: , then

The rank of the system of vectors.

Appointment: The rank of a non-zero vector system in a linear space is the number of vectors in a base. The rank of the zero system after the appointment is closer to zero.

Power rank: 1) The rank of a linearly independent system is based on the number of vectors. 2) The rank of the linear fallow system is smaller for the number of vectors. 3) The ranks of equivalent systems are changed -rankrank. 4) The rank of the subsystem is less or more equal to the rank of the system. 5) As a rankrank, you can build a high base. 6) Do not change the rank of the system, just add a vector to it, which is a linear combination of other vectors in the system. 7) The rank of the system does not change, just as a vector is removed from it, but it is a linear combination of other vectors.

For znahodzhennya rank of the vector system, it is necessary to use the Gaussian method to bring the system to a tricot or trapezoidal form.

Equivalent systems of vectors.

Butt:

Let's convert the vector data into a matrix for the base value. We take:

Now, in addition to the Gauss method, we will transform the matrix to a trapezoidal look:

1) In our main matrix, we will annul the entire first column of the crim of the first row, we will see the first row multiplied by the second, we will see the third row multiplied by the third, and we will not see anything like the first element of the fourth row, then bto retin the first column and the fourth row, dorivnyuє zero. Take away the matrix: 2) Now in the matrix, we remember the rows 2, 3 and 4 for simplicity of the solution, so there was one on the space of the element. The fourth row is remembered as a substitute for another, another for a third and a third on the fourth place. Take away the matrix: 3) In the matrix, all elements under the element are annulled. Oskіlki renew the element of our matrix to zero, we can’t see anything in the fourth row, and up to the third we add another multiplication by . Take away the matrix: 4) I remember again in the matrix of rows 3 and 4 months. Take away the matrix: 5) At the matrix, add to the worm'ya row of the third, multiplied by 5. We take away the matrix, as a matima tricutny look:

Systems , їх ranks zbіgayutsya from power rank and їх rank more rank rank

Respect: 1) On the basis of the traditional Gauss method, as in the row of the matrix all the elements are subdivided into one number, we do not have the right to shorten the row of the matrix through the power of the matrix. If you want to speed up a row on a single number, then you can speed up the entire matrix by a whole number. 2) In times, as we take into account a linear fallow row, we can remove it from our matrix and replace it with a zero row. Butt: You can clearly see that the other row turns over the first one, so multiply the first one by 2. You can replace the entire other row by zero. We take: In the result, having grafted the matrix, either to a tricot, or to a trapezoidal look, where it does not have many linearly fallow vectors, all non-zero vectors of the matrix i will be the base of the matrix, and their number will be the rank.

So the very application of the system vector_v yak graphics: Given the system de , , i . The basis of this system will obviously be the vector i, the shards through them are vectors. The system is given to the graphical viewer:

Elementary transformation. Step vision systems.

Elementary transformation of the matrix- tse such a transformation of the matrix, in which the equivalence of matrices is taken. In this way, elementary transformations do not change the impersonal solution of the system of linear alignments of algebra, as the matrix is represented.

Elementary transformations are achieved by the Gauss method for reducing the matrix to a tricot or stepped look.

Elementary transformations of rows name:

In some courses of linear algebra, the permutation of the rows of the matrix is not seen in the context of elementary transformation through those that the permutation of the places of any two rows in the matrix can be taken away, victoriously multiplying any row of the matrix by a constant, and adding to any row matrices of the next row, multiplied by a constant.

Similarly, they are appointed elementary transformation of the stovptsiv.

Elementary transformation werewolves.

The designation points to those that the matrix can be cut off by a path of elementary transformations (or navpaki).

In order to bring the matrix to a stepped look (Fig. 1.4), follow the vikonate so.

1. At the first column, select the element, the input of zero ( passing element ). A row with a conductive element ( provіdny row ), as if it’s not persha, rearrange it on the plate of the first row (reformation of type I). If there is no leading element in the first column (all elements add to zero), then this column is turned off, and we continue to search for the leading element in the part of the matrix that is missing. The transformation will end, as all the elements are excluded, or in the part of the matrix that is left out, all the elements are zero.

2. Divide all elements of the wire row into a wire element (type II conversion). If the last row is left, then on which one the transformation should be completed.

4. Turning off the row and stovpets, on the back of which there is a conductive element, go to point 1, in which all the descriptions will stop to the part of the matrix that is left out.

7. Theorem for the distribution of the signifier behind the elements of a row.

If the leader can equal zero elements, then it is easier to lay out the leader according to the elements of that row or else, which will avenge the largest number of zeros.

Task 3.1. Calculate the winner

Solution. Having added the first to the next row, to the third - the first, multiplied by 2, to the fourth - the first, multiplied by -5, we take

Laying out the chief for the elements of the first column, perhaps

The otrimaniy vyznachnik is laid out behind the elements of the first column:

8. Laplace's theorem (1). Alien extension theorem(2)

1) The leader of the richest sum of creative elements of any order on their algebraic additions.

2) The sum of creations of the elements of any row of the algebraic addition on the algebraic complement of the second element of the next row is equal to zero (the theorem of multiplication on foreign algebraic complements).

9. Arithmetic vector spaces.

a 1 x + b 1 y \u003d c 1

a 2 x + b 2 y \u003d c 2

Rice. 26

Similarly, it is possible to find out from the system of linear lines from the trioma nevidomimi, interpreting the skin level as a leveling of the area near the open space.

a 11 x 1 + a 12 x 2 + ... + a 1n x n = b 1

a 21 x 1 + a 22 x 2 + ... + a 2n x n = b 2

. . . . . . . . . . . . . . . . . . . . . . (1)

a m1 x 1 + a m2 x 2 + ... + a mn x n = b m

de a ij і b i – good old numbers. The number of equals in the system can be like that and in no way connected with the number of unknowns. Coefficients for unknowns and ij can be assigned numbering: the first index i indicates the number of the equal, the other index j is the number of the unknown, for which the coefficient is worth. Whether the solution of the system is understood as typing (real) meanings of unknown (α 1 , α 2 , ..., α n), which wrap the skin's alignment with the correct alignment.

Any ordering of a set of n real numbers (α 1 , α 2 , ..., α n) is called an n-dimensional arithmetic vector, and the numbers themselves α 1 , α 2 , ..., α n are the coordinates of the th vector.

For the recognition of vectors, as a rule, bold font i is used for a vector with coordinates α 1, α 2, ..., α n, the primary form is taken:

a = (α 1, α 2, ..., α n).

The folding and multiplication of n-world vectors are governed by the same rules as for the largest vectors. And yourself, like

a = (α 1 , α 2 , ..., α n), b = (β 1 , β 2 , ..., β n) (2)

Two n-world vectors, then their sum is called a vector

α + β = (α 1 + β 1 , α 2 + β 2 ..., α n + β n). (3)

The subset of a vector by the number λ is called the vector

λа = (λα 1 , λα 2 , ..., λα n). (4)

A lot of n-world arithmetic vectors with operations of adding vectors and multiplying a vector by a number is called an arithmetic n-world vector space L n .

With the introduction of operations, you can see quite a few linear combinations of several vectors, so you can see

λ 1 a 1 + λ 2 a 2 + ... + λ k a k ,

de λ i - actual numbers. For example, a linear combination of vectors (2) with coefficients λ and μ - ce vector

λа + μb = (λα 1 + μβ 1 , λα 2 + μβ 2 , ..., λα n + μβ n).

In the trivial space of vectors, a special role is played by the trinity of vectors i, j, k (coordinate vectors), which sort of vector a:

a = xi + yj + zk,

de x, y, z are real numbers (coordinates of the vector a).

The n-world type has the same role played by the system of vectors:

e 1 = (1, 0, 0, ..., 0),

e 2 = (0, 1, 0, ..., 0),

e 3 = (0, 0, 1, ..., 0),

. . . . . . . . . . . . (5)

n = (0, 0, 0, ..., 1).

Any vector є, obviously, is a linear combination of vectors e 1 , e 2 , ..., e n:

a = a 1 e 1 + a 2 e 2 + ... + a n e n (6)

moreover, the coefficients 1 , 2 , ..., n are taken from the coordinates of the vector a.

Denoting a vector through 0, all coordinates of which are equal to zero (in short, a zero vector), we introduce the following important designation:

The system of vectors a 1 , a 2 , ..., and k is called linearly fallow, as it is equal to the zero vector linear combination

λ 1 a 1 + λ 2 a 2 + ... + λ k a k = 0,

in yakіy want b one of the coefficients h 1 , 2 , ..., λ k vіdminny vіd zero. Otherwise, the system is called linearly independent.

Yes, vector

and 1 = (1, 0, 1, 1) and 2 = (1, 2, 1, 1) and 3 = (2, 2, 2, 2)

linear deposits, shards

a 1 + a 2 - a 3 = 0.

Linear fallow, as can be seen from the designation, is equal (at k ≥ 2) to the fact that one of the vectors in the system wants a linear combination of others.

Whether the system is a 1 , a 2 , ..., and n iz n linearly independent vectors in the n-world space L n is called its basis.

Whether a vector to the space L n is expanded, and, moreover, in the same order, according to the vectors of a sufficient basis a 1, a 2, ..., and n:

a = λ 1 a 1 + λ 2 a 2 + ... + λ n a n.

This fact can be easily established from a specific basis.

Continuing the analogy with the trivum space, it is possible to designate the scalar vertex a b vector in the n-virtual scale, depending on

a b = α 1 β 1 + α 2 β 2 + ... + α n β n .

For such a designation, all the main powers of the scalar creation of trivi- mer vectors are taken care of. The vectors a and b are called orthogonal, since their scalar addition is equal to zero:

α 1 β 1 + α 2 β 2 + ... + α n β n = 0.

Theoretically, there is one more important understanding of the line codes - the understanding of the space. The subdivision of the V expanse L n is called the subspace of this expanse, as

1) for any vectors a, b, which lie within V, if the sum a + b also lie within V;

2) for any vector that lies within V, and for any real number λ, the vector λand also lies with V.

For example, the absence of all linear combinations of vectors e 1 e 2 from system (5) will be a subspace to the space L n .

In linear algebra it can be explained that the skin subspace V has such a linearly independent system of vectors a 1 , a 2 , ..., a k

a = λ 1 a 1 + λ 2 a 2 + ... + λ k a k.

The system of vectors is given and is called the basis of subspace V.

Understanding the space, like understanding the group and the circle, admitting the same axiomatic designation. For the details, let us look at the Zhiver to the course of linear algebra.

10. Linear combination. Linear fallow and independent vector systems.

Matrix, see matrices, go over matrices.

See matrix:

1. Rectangular: mі n- pretty positive numbers

2. Square: m=n

3. matrix row: m=1. For example, (1 3 5 7) - for many practical tasks, such a matrix is called a vector

4. matrix: n=1. For example

5. Diagonal matrix: m=nі a ij =0, like i≠j. For example

6. single matrix: m=nі

7. Zero matrix: a ij =0, i=1,2,...,m

j=1,2,...,n

8. Tricut Matrix: all elements below the headline diagonal add 0.

9. Symmetric matrix:m=nі a ij = a ji(to stand equal elements on symmetrical head diagonals), and also A"=A

For example,

10. skew matrix: m=nі a ij =-a ji(That's why on the symmetrical main diagonals there are protilene elements). Also, there are zeros on the head diagonal (because i=j maybe a ii =-a ii)

Move over matrices:

1. Addition

2. Vidnimannya matrix - elementwise operation

3. tvir, dobutok matrix to number – elementwise operation

4. Reproduction A*B matrix by rule row on top(the number of columns in matrix A can be equal to the number of rows in matrix B)

Amk * Bkn = Cmn why the skin element h ij matrices Cmn add the sum of the elements of the i-th row of the matrix A and the other elements of the j-th column of the matrix B, tobto.

Demonstrate the operation of multiplying matrices on the application

5. Matrix transposition A. A transposed matrix is denoted by A T or A

for example

Rows and columns were commemorated by missions

Dominance of operations on matrices:

(A+B)+C=A+(B+C)

λ(A+B)=λA+λB

A(B+C)=AB+AC

(A+B)C=AC+BC

λ(AB)=(λA)B=A(λB)

A(BC)=(AB)C

(λA)"=λ(A)"

(A+B)"=A"+B"

(AB)"=B"A"

2. Signifiers of another and third order (basic understanding, sv-va, calculation)

Power 1. The signifier does not change at the hour of the transposition, that is.

Proof.

Respect. The advances in the power of the chiefs are formulated only for the ranks. With all the powers that be, 1 screams, that by the very powers of power, the stovpts are driven.

Power 2. With the multiplication of the elements of the row of the primate, the number of the entire primate is multiplied by that number, that is.

Proof.

Power 3. Vyznachnik, what can be the zero row, dorivnyu 0.

The proof of the authority's authority is evident from the authority 2 at k = 0.

Power 4. Vyznachnik, who can have two equal rows, finish 0.

Proof.

Power 5. Vyznachnik, two rows of some sort of proportional, good 0.

The proof is strong from the powers 2 and 4.

Power 6. When rearranging two rows of the arbitrator, the vin is increased by -1.

Proof.

Power 7.

The proof of power can be carried out independently, having equalized the meaning of the left and right parts of equanimity, known for the help of appointment 1.5.

Power 8. The value of the variable does not change, but to the elements of one row, add the second row elements of the next row, multiplied by the same number.

Minor. Algebraic addition. Laplace's theorem.

The method of bringing to a tricot look polagaє in such a transformation of the given primate, if all the elements of it, which lie on one side of one of its diagonals, become equal to zero.

Example 8. Calculate the winner

Brought to trikutny look.

Solution. We can see the first row of the vyznachnik of the other rows. Todi otrimaєmo

Tsey vyznachnik is more expensive to add the elements of the head diagonal. In this manner, please

Respect. Everything you look at can be mentioned for the n-th order.

Bringing the matrix up to a step-by-step look. Elementary transformation of rows and columns.

Elementary transformations of the matrix are called like this її transformation:

I. Permutation of two columns (rows) of a matrix.

ІІ. The multiplication of all elements of one column (row) of the matrix by the same number, not equal to zero.

ІІІ. Addition to the elements of one column (row) of the same elements of the next column (row), multiplied by the same number.

The matrix, otrimana z vihіdnoї matrixі kіntsemu number of elementary transformations, is called equivalent . Tse is indicated.

Elementary transformations are to be made for the forgiveness of matrices, which will be victorious for the cherry blossoms.

In order to bring the matrix to a stepped look (Fig. 1.4), follow the vikonate so.

2. Divide all elements of the wire row into a wire element (type II conversion). If the last row is left, then on which one the transformation should be completed.

4. Turning off the row and stovpets, on the back of which there is a conductive element, go to point 1, in which all the descriptions will stop to the part of the matrix that is left out.

Stock 1.29. Bring to the step-like look of the matrix

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