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Hundreds of size matrices. Linear combination of hundreds of matrices to be called a matrix-stovpets, at the same time - acts of action or complex numbers, called by the coefficients of the line combination... As in the line combination of taking all the performance equal to zero, then the line combination of the pre-zero matrixes is zero.

Hundreds of matrices are called linearly independent If all the functions of the line combination are going to be zero, if all the functions of the line combination are going to be zero. Hundreds of matrices are called linear fallow , Yakshcho is a set of numbers, the middle one wants to be one from zero, and the line combination of hundreds of numbers with a number of coefficients goes to zero

Likewise, there can be data on the value of the linear fallowness and linear indentation of the rows in the matrix. All theorems are formulated for a hundred matrices.

Theorem 5

While the middle of the matrix is ​​zero, then the hundredth of the matrix is ​​linear.

Delivered. It is easy to understand the line combination, in all the performance it will return to zero with all non-zero bolts and one with zero hundred percent. There is zero, and the middle of the functions of the line combination is displayed as zero. Otzhe, a hundred matrices of linear fallow.

Theorem 6

yaksho 100% of matrices lіnіyno fallow, then all 100% of matrices of linear deposits.

Delivered. We will be for the value of respect, for the first hundred of the matrix linear fallow. According to the values ​​of the line of fallowness, there is a small set of numbers, the middle one would like to be one from zero, and the line combination of one hundred percent of the number of coefficients to zero

Easily a linear combination of all hundred matrices, including in it one hundred percent with zero coefficients

Ale. Otzhe, all hundreds of matrices are linearly fallow.

succession... In the midst of linear independent matrixes, be it linear independent. (This can be easily done by contradicting it.)

Theorem 7

In order for one hundred matrices to be baked on line, it is necessary and sufficient, if you want one hundred of matrices. line combinationіnshih.

Delivered.

Necessity. Do not have a hundred matrices of linear fallow, so that there is a simple set of numbers, the middle one wants to be one from zero, and a linear combination of hundred points from the number of parameters to zero

Acceptable for value, scho. Todi tobto is the first hundred percent of the line combination of the first.

abundance... Nekhai would like to have one hundred matrices in a linear combination of numbers, for example, de - decisive numbers.

That is, so that the line combination is zero, and the middle of the numbers of the line combination would be one (if) displayed as zero.

Find the rank of the matrix. Be aware from zero to the minor order to be called basis ... Rows і hundredptsі, on the overtakes of which stand the basic minor, are called baseline .

In the matrix A of sizes (m; n), select k rows and k 100% (k ≤ min (m; n)). Matrix elements, which stand on alternating vibrational rows and 100%, set up a square matrix of order k, the name of which is called the minor M kk order k y or the minor k-th order of the matrix A.

The rank of a matrix is ​​the maximal order r of the minors of the zero minors in the matrix A, and whether a minor is of the order r, the minor is the basic minor. Designation: rang A = r. If rang A = rang B and the size of the matrices A and B coincide, then the matrices A and B are called equivalent. Designation: A ~ B.

The main methods for calculating the rank of the matrix are the method to improve the minority and the method.

Method of healthy minors

The essence of the method is to make the minors of the field healthy in the offensive. Locate in the matrix the same values ​​of the minor order k, the value of zero. Toddlers look out of the order of k + 1 minors, as they are in their own (i.e. oblyamovuyut) minork-th order, as seen from zero. If all the stinks come up to zero, then the rank of the matrix is ​​k, in the middle of the middle the minor is (k + 1) -th order, there will be an indication of zero and the whole procedure is repeated.

Linearity of rows (100%) of the matrix

Understanding the rank of the matrix is ​​clearly linked to the understandings of the linear independence and the number of rows (100%).

Matrix rows:

they are called linearly fallow, if such numbers are known as λ 1, λ 2, λ k, but the equality is true:

Rows of matrices A are called linearly independent, as there are fewer equalities if all numbers are λ 1 = λ 2 = ... = λ k = 0

An analogous rank is the line of fallowness and indeterminacy of the hundredths of the matrix A.

Like a row (a l) of a matrix A (de (a l) = (a l1, a l2, ..., a ln)) can be represented in a view

An analogous rank is given to the disciple of the line combination of the stovpts. The theorem about the basic minor is valid.

Basic rows and basic rows. Whether there is a row (or a stub) of a matrix A є a linear combination of basic rows (stoppts), i.e. E. Rows (stoppts), which cross the basic minor. Thus, the rank of the matrix A: rank A = k is suitable for the maximum number of linearly independent rows (one hundred percent) of the matrix A.

So the rank of the matrix is ​​the size of the largest square matrix in the middle of this matrix, for which it is necessary to value the rank, for which the designer is not zero. If the visible matrix is ​​not є square, because it’s square, but the size of the matrix is ​​zero, then for square matrices of a smaller order, the rows and 100% are selected.

By means of the matrix, the rank of the matrix can be determined by the number of linearly independent rows, but rather than the number of matrices. There are some of the most common rows of rows, but not too many of the fallowing ones from the one that is less. For example, if the matrix has 3 linear independent rows and 5 linear independent rows, then the second rank is three.

Apply the known rank of the matrix

Using the method of improving the minors to know the rank of the matrix

P і w e n і e. Minor of a different order

the bordering minor is M 2, also known as zero. However, insulting the minor of the fourth order, heal M 3.

return to zero. To that, the rank of the matrix A is previously 3, and the basic minor is, for example, the representations of the minor M 3.

The method of elementary reworking of the basis on the fact that the elementary reworking of the matrix does not change its rank. Vikoristovuchi and reimagining, you can bring the matrix to the view, if all the elements, crim a 11, a 22, ..., a rr (r ≤min (m, n)), add zero. Tse obviously means that rang A = r. Remarkably, if the matrix of the n-th order can be seen from the upper tricyte matrix, i.e. the Matrix, for all the elements from the head diagonal to zero, then you should make sure that there are more elements for the head. The price of power can be victorious when the rank of the matrix is ​​calculated by the method of elementary reconciliation: it is also necessary to bring the matrix to a triangular one, if you have seen a single source, you know zero, but the rank of the matrix of the other

Using the method of elementary recalibration to know the rank of the matrix

P і w e n і e. Significant i-th row of the matrix A by the symbol α i. At the first stage of the vicon's elementary re-adaptation

At another stage of the viconmo transformation

As a result, otrimaєmo

Line algebra

Slough theory

1. Matrix, with matrices, matrix wrapped. Matrix Rivnyannya and іkh Rіshennya.

matrix- a straight-line table of significant numbers, roztasvanih in singing order, size m * n (rows at 100). Matrix elements are identified, de i is the row number, and j is the hundredth number.

addition (vidnimannya) matrices are only for single-dimension matrices. The sum (difference) of the matrix is ​​a matrix, elements of which є according to the sum (difference) of elements of the outgoing matrices.

Multiplication (rospodil)by the number- multiple (distribution) of the skin element of the matrix on a whole number.

The multiplication of matrices is assigned only for matrices, the number of hundred of the first of the same number of rows in the other.

multiple matrices- matrix, elements of which are given by the formulas:

matrix transpose- such a matrix B, rows (one hundred percent) of which є one hundred percent (rows) at the main matrix A. signify

ring matrix

matrix rivnyannya- Rivnyannya vіda A * X = B є tvir matrices;

1. Linear fallowness and indeterminacy of hundreds (rows) of the matrix. Criterion of linear deposition, sufficient and linear deposition of 100% (rows) of the matrix.

The system of rows (100%) is called linear square, As the line combination is trivial (the parity is shown only for a1 ... n = 0), where A1 ... n are a hundred (rows), aa1 ... n is a storage capacity.

criterion: In order for the system of vectors to be linearly dependent, it is necessary and sufficient, if you want one of the vectors of the system and linearly flowing through the vectors of the system.

umov's wealth:

1. Business cards for matrices and power

Matrix template (determinant)- the same number as for a square matrix A can be calculated for the elements of the matrix according to the formula:

, De - dodatkovy minor element

power:

1. Zvorotn_y matrix, algorithm for calculating the zvorotn_y matrix.

ring matrix- such a square matrix X, as in addition to the square matrix A of the same order, satisfies the definition:, where E is a single matrix, of the same order and іA. Whether it is a square matrix with a matrix, not equal to zero, but 1 rotary matrix. To be aware of the auxiliary method of elementary re-creation and the auxiliary formula:

Understand the rank of the matrix. Basic Minor Theorem. Criterion of equality to zero of matrix matrix. Elementary transformation of matrices. Calculated by the method of elementary conversions. Calculation of the wrapped matrix by the method of elementary transformation.

Matrix rank - basic minor order (rg A)

Basic minor - Minor order r not to zero, so all minors to order r + 1 and more to order zero or not.

The theorem about the basic minor - In the majority of the matrix A leather stovpez (row) є in the line combination of stovpts (rows), in some re-stitches, the basic minor.

Delivered: Find in the matrix of sizes m * n the basic minor of the seams in the first r rows and the first r ties. It is easy to understand the form, which is rendered ascribed to the base minor of the matrix A of the above elements s-th rowі of the k-th century.

Obviously, if there are any, the card holder will go to zero. If, then, the determinant is to replace two of the same rows or two of the same row. As well, then the nameplate D is leading to zero, so as is the minor (r + λ) -ro order. Raznkladayuchi viznachnik on the last row, we will recognize:, de-algebraic additional elements of the last row. Amazingly, well, it’s like a basic minor. Tom, de Write down the balance for, otrimuєmo , Tobto k-th hundred(In any case) є line combination of 100% of the basic minor, which needs to be brought.

criterion detA = 0- Business card holder for zero todi and only todi, if the rows (100) are lined with fallow.

elementary redevelopment:

1) multiple rows by a number, not from zero;

2) addition to the elements of one row of elements of the first row;

3) rearrangement of rows;

4) picking up one of the same rows (100%);

5) transposing;

Calculated to rank - With the theorems about the basic minority, the rank of the matrix A is suitable for the maximum number of linear independent rows (100% in the matrix), from the default of elementary conversions to know all the linear independent rows.

assessed wrapped matrix ­ - The re-implementation can be implemented on multiples on the matrix A of the same matrix T, as it is a complement of the following elementary matrices: TA = E.

Tse rіvnyannya means that the transformation matrix T is a vortex matrix for the matrix. Then, now,

A skin row of matrices А, meaningfully е i = (a i 1 a i 2 ..., a in) (for example,
e 1 = (a 11 a 12 ..., a 1 n), e 2 = (a 21 a 22 ..., a 2 n) etc.). The skin of them is a matrix-row, which can be multiplied by the number of a fold in a row behind the zagalny rules for matrices.

line combination rows e l, e 2, ... e k name the sum of the creatures of these rows on most of the dates:
e = l l e l + l 2 e 2 + ... + l k e k, de l l, l 2, ..., l k - most numbers (coefficients of linear combination).

Matrix rows e l, e 2, ... e m are called linear fallow, Whenever there are such numbers l l, l 2, ..., l m, unevenly zero for an hour, but the line combination of rows in the matrix goes to zero in a row:
l l e l + l 2 e 2 + ... + l m e m = 0, de 0 = (0 0 ... 0).

Linear accumulation of rows in the matrix means that one row of the matrix is ​​needed in a linear combination of those. Anyway, do not know for the value of the residual efficiency l m ¹ 0.Todi, having divided the offense parts of the equality by l m, can be used for the last row, like a linear combination of the other rows:
e m = (l l / l m) e l + (l 2 / l m) e 2 + ... + (l m-1 / l m) e m-1.

As soon as there is a line combination of rows on the way to zero, then the rows are called linearly independent.

Matrix rank theorem... The rank of the matrix is ​​appropriate for the maximum number of line-independent rows, but not all, through which all the other rows can be displayed.

Let us prove the theorem. Find the matrix A size m х n maє rank r (r (A) £ min (m; n)). Also, it is shown in the form of zero to the minor of the r-th order. Any such minor will be nazivat basis... Come on for the value of the tse minor

The rows of the minor will also be nazivat baseline.

Brought to you, that the same rows of the matrix e l, e 2, ... e r linearly square. Suppose there is a guide, that is one of a series of rows, for example, the r-th, in a linear combination of those: e r = l l e l + l 2 e 2 + ... + l r-1 e r-1 = 0. element r-th rows of elements of the 1st row, multiplied by l l, elements of the 2nd row, multiplied by l 2, etc., nareshti, elements of the (r-1) -th row, multiplied by l r-1, then r-th row become zero. In the presence of the authorities of the visitor of the vishchenavia, the visitor is not guilty of change, and in the case of a large amount of money, it is zero. Otrimano rubbing, line indentation of rows has been brought.

Now it is clear that if there are (r + 1) rows in the matrix, there are linear fallows, so that whether a row can be violated through the baseline.

Advantageously, one row (i-th) and one hundredth (j-th) were previously minor. As a result, the minor (r + 1) -th order is recognized, which for the given rank is zero.

Understanding the rank of the matrix is ​​clearly linked to the understanding of the lineage (independent) and the number of rows, for example. Nadal will be a material for rows, for hundreds of viclades it is similar.

Matrix A meaningfully її rows of the offensive rank:

Two rows of the matrix are called equal, Yaksho rіvnі іх types of items :, yakshо ,.

Arithmetic operations on the rows of the matrix (multiple rows by a number, additional rows) are entered as operations, which are carried out element by element:

row e be called a line combination of rows..., matrixes, if you have a large number of creations in the same series on the majority of the numbers:

Matrix rows are called linear fallow If there are such numbers, not equal to zero at once, but the line combination of rows in the matrix goes to zero in a row:

, =(0,0,...,0). (3.3)

Theorem 3.3Rows of matrices are lined with fallow, if you want one row of matrices in a linear combination of the first.

□ Dіysno, do not know for the value of the formula (3.3), todі

In such a rank, a row is a line of combinates of other rows. ■

If there is a line combination of rows (3.3), then the rows are called line-by-line.

Theorem 3.4.(About the rank of the matrix) The rank of the matrix is ​​suitable for the maximum number of line-independent rows, but all the rows (hundred) are rotated through them.

□ Come on matrix A size m n maє rank r(r min). Tse means, that isnu vidminny from zero minor r th order. Any non-null minor r The th order will be called the basic minor.

Nokhay for the value of the basic minor є provinny abo kutoviy minor. The same rows of the matrix are linearly independent. Admittedly, a guide, tobto one of these rows, for example, is a linear combination of those. See the items r- the first row of elements of the 1st row, multiplied by, instead of the elements of the 2nd row, multiplied by, ... and elements ( r - 1) - the first row, multiplied by. At the head of the government 8, with such re-implementation of the matrix, the designer D does not change, albeit so yak r- I row will now fold from one zeros, then D = 0 - rubbing. Otzhe, our pripuschennya is about those rows of matrices that are linearly fallow, not really.

rows named baseline... It will be shown that there are (r + 1) rows in the matrix of linear fallow, so that whether a row rotates through the baseline.

The minor (r +1) - the first order, which is visible when the minor is added with more than one row of elements iі stovptsі j... Tsei minor is door-to-door to zero, so as the rank of the matrix r To that, be-like a minor in order to order zero.

Laying down yogo for the elements of the rest (dodany) stovpchik, obsessed

De module of the remainder algebraic supplement get out of basic minor D and it seems to be from zero, to that is 0.

3. Voevodin V.V., Kuznetsov Yu.A.

4. Ilyin V.A., Poznyak E.G. Linin algebra.- M.: "Science", 1978.- 304s.