What is the image of the matrix | Formation of a matrix of a whole image with the addition of elements of a complex object. Elementary theory of linear operators
Value 1. The image of the linear operator A is the absence of all elements that appear.
The image of the linear operator A is a linear subspace. This size is called operator rank A.
Value 2. The kernel of the linear operator A is called the list of all vectors for which.
The core is a linear subspace of the X space. This dimension is called operator defect A.
Since the operator A is equal to the worldly space of X, then the relation + = is valid.
Operator A is called invirogenim yakshcho yogo core. The rank of a non-virgin operator is of the same size as X.
Let us go - the matrix of linear transformation And the space X has a basis, that is, the coordinates of the image and the prototype associated with the relationships
Therefore, the coordinates of any vector are satisfied with the systems of rulers
It turns out that the core of the linear operator is the linear shell of the fundamental system of the solution to this system.
Zavdannya
1. Make sure that the rank of the operator is equal to the rank of its matrix on a sufficient basis.
Compute the kernels of linear operators specified in the space X basis by the following matrices:
5. Bring it out.
Calculate the rank and defect of operators specified by the following matrices:
6. . 7. . 8. .
3. VALUE VECTORS AND VALUE OF LINEAR OPERATOR
Let's take a look at the linear operator A, what is in the peaceful space of X.
Viznachennya. The number l is called the powerful values of the operator A, yakscho, such, scho. In this case, the vector is called the power vector of operator A.
The most important power is the power vectors of the linear operator and those power vectors related to pairwise different power values linearly independent.
Since - the matrix of the linear operator A in the basis space X, the power values l and power vectors of the operator A are designated as follows:
1. Vlasni meanings are known as the root of the characteristic level (algebraic level of the th level):
2. The coordinates of all linearly independent power vectors that correspond to the skin's blood value are drawn, resulting in a system of uniform linear alignments:
the matrix of which has a rank. The fundamental solutions of the system are vector-based from the coordinates of power vectors.
The root of the characteristic value is also called the power values of the matrix, and the solutions of the system are called the power vectors of the matrix.
butt. Find the power vectors and power values of the operator A specified in the given basis by the matrix
1. To determine the power values, the following is added and most likely characteristically equal:
Stars are of greater importance, their multiplicity.
2. To determine the power vectors, a system of levels is formed and verified:
The equivalent system of basic levels looks like
Therefore, the skin power vector is a vector-stovpets, and it is a sufficient constant.
3.1. Simple structure operator.
Viznachennya. The linear operator A, which operates in n-dimensional space, is called the operator of simple structure, as it represents exactly n linearly independent power vectors. In this case, we can use the basis of the space of the vectors of the operator, in which the matrix of the operator has the simplest diagonal view
de – the authority of the operator. It is clear that this is the case: since the operator matrix of any basis space X has a diagonal appearance, then the basis consists of the power vectors of the operator.
Linear operator A is an operator of simple structure and method, if the skin power value of the multiplicity corresponds to linearly independent power vectors. The fragments of the vector are the solution of the rank system, then the cutaneous root of the characteristic rank multiplicity is responsible for the rank matrix.
Any matrix of size that corresponds to the operator of a simple structure, similar to a diagonal matrix
where the matrix of the transition T from the output basis to the basis of its vectors has its own vector-coordinates from the coordinates of its vectors of the matrix (operator A).
butt. Reduce the linear operator matrix to diagonal view
It is inherently characteristic that jealousy is known to its root.
Signs of power values of multiplicity and multiplicity.
Pershe vlasne znannya. This is indicated by power vectors, the coordinates of which are
system decisions
The rank of this system is higher than 3, so there is only one independent solution, for example, vector.
The power vectors that they represent are indicated by the level system
The rank of any previous one is 1 and, therefore, there are three linearly independent solutions, for example,
Thus, the skin power value of the multiplicity corresponds to linearly independent power vectors and, therefore, the operator is an operator of a simple structure. Transition matrix T looks like
And the connections between similar matrices are assigned to the relationships
Zavdannya
Know the power vectors and meanings
linear operators assigned to each basis by matrices:
This means that from the previous linear operators we can lead to a diagonal-looking transition to a new basis. Find this basis and its corresponding matrix:
10. Make sure that the power vectors of the linear operator from different power values are linearly independent.
11. Show that any linear operator A that acts in has n different values, then any linear operator B commuting with A is a basis of power vectors, and any power vector A will be powerful for B.
INVARIANT PIDSPACES
Value 1.. The subspace L of the linear space X is called invariant with the operator A, which is equal to X, since for the skin vector its image also belongs.
The main powers of invariant subspaces are indicated by the following relationships:
1. If both subspaces are invariant to operator A, then their sum and span are also invariant to operator A.
2. Since the space X is decomposed into a direct sum of subspaces i () and invariantly to A, then the matrix of the operator in the basis, which is a combination of bases and a block matrix
de - Square matrices, 0 - zero matrix.
3. The skin invariant operator A subspace operator may have one power vector.
butt 1. Let's take a look at the kernel of the ordinary operator A, the decorous X. For the reasons. Let it go. Therefore, the fragments of the zero vector are located near the skin linear subspace. Well, the core is invariant and subspace.
butt 2. Let any basis space X operator A be given by the matrix assigned to the equals i
5. Bring that as a subspace, invariant to the ungenerated operator A, to be invariant to the reversal operator.
6. Let the linear transformation of the A-dimensional space in the basis form a diagonal matrix with different diagonal elements. Find all subspaces that are invariant to A, and then calculate its number.
IN vector space V over a pretty field P tasks linear operator .
Value9.8. Core linear operator is called an impersonal vector in space V, in the same way as a zero vector Accept meaning for this multiplier: Ker, then.
Ker = {x | (X) = o}.
Theorem 9.7. The core of the linear operator is a subspace V.
Vicenza 9.9. Size the kernel of a linear operator is called defect line operator. dim Ker = d.
December 9.10.In a manner linear operator is called impersonal vectors in space V. Designation for this multiplicity Im, then. Im = {(X) | X V}.
Theorem 9.8. Image line operator and subspace V.
December 9.11. Size The image of a linear operator is called rank line operator. dim Im = r.
Theorem 9.9. Space V is the direct sum of the kernel and the image of a given linear operator. The amount of rank and defect of the line operator is equal to the size of the space V.
Butt 9.3. 1) In space R[x] ( 3) know the rank and defect operator differentiation. We know those rich terms, similar to ancient zero. There are a lot of members of the zero stage, then, Ker = {f | f = c) that d= 1. Pokhіdnі rich members, the stage of which does not exceed three, create the absence of rich members, the stage of which does not exceed two, then, Im =R[x] ( 2) that r = 3.
2) Like linear matrix assignment operator M(), then to find the kernel you need to change Rivnyannya ( X) = about, which in matrix form looks like this: M()[x] = [about]. Z This shows that the basis of the kernel of a linear operator is a fundamental set of decouplings of a homogeneous system of linear equations from the main matrix M(). The system is created by images of a line operator add vectors ( e 1), (e 2), …, (e n). The basis of this system of vectors represents the basis of the image of the linear operator.
9.6. Reverse line operators
Viznachennya9.12. Linear operator is called werewolves how I sleep linear operator ψ such what does it mean? jealousy ψ = ψ = , where is the same operator.
Theorem 9.10. Like linear operator brutally, That operator ψ is designated by a single rank and is called gateway For operator .
I here is the operator, gate operator , designated –1.
Theorem 9.11. Line operator brutally, and only then, if the matrix is reversed M(), when M( –1) = (M()) –1 .
This theorem implies that the rank of the reciprocal linear operator is ancient dimensions space, and the defect is equal to zero.
Butt 9.4 1) Meaning, it's brutally linear operator , which is ( x) = (2X 1 – X 2 , –4X 1 + 2X 2).
Decision. Add the matrix of this linear operator: M() = . So yak 
= 0 then matrix M() non-negotiable, which means non-negotiable and linear
operator
.
2) Know linear operator, gateway operator , yakscho (x) = (2X 1 + X 2 , 3X 1 + 2X 2).
Decision. Matrix of this linear
operator, Rivna
M() = 
, werewolf, fragments | M()| ≠ 0.
(M()) –1 = 
that –1
= (2X 1 – X 2 ,
–3X 1 + 2X 2).
Explaining the principles of integration of discrete information with the close integration of elements of a foldable object is an urgent interdisciplinary problem. The article examines the process of creating an image of an object, which is a complex of blocks, from which a set of other elements is combined. As the investigation of the object resulted in a conflict situation, the field remained in constant respect for a consistent strategy for analyzing information. The surrounding situations were warehouse parts of the object and were clearly taken as prototypes of the conflict. The background of this work was based on a mathematically expressed matrix, which depicted the image of a problematic behavioral situation. The current task was based on visual analysis of the design of the graphic composition, the elements of which corresponded to the situational environment. The size and graphic features of the elements that are selected, as well as their division in the composition, served as a guide for identifying rows and columns in the image matrix. The investigation showed that the construction of the matrix is determined, firstly, by behavioral motivation, in another way, by the causal-hereditary inputs of situational elements and the sequence of information withdrawal, as well as in the third place, the fragments of information you see are assigned to your settings. It can be noted that the matrix vector principle of forming an image of a behavioral situation is characteristic of the formation of images and other objects on which respect is directly expressed.
visualization
spriinyattya
discreteness of information
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The results of tracking the perception of different images have expanded the prospect of developing principles that mean the integration of discrete information and the assembly of whole images. Analysis of the features of the recognition of fragmented images when presented with a number of fragments that change, allowing us to follow three strategies to induce a meaningful image in the minds of the information deficit. The strategies were expanded by assessing the importance of ready-made portions of information on the formation of a whole image. Otherwise, the skin strategy was characterized by the manipulation of bodily parameters of prepared portions of information. The first strategy conveyed the significance of the fragments of the image - whose recognition was achieved after the accumulation of information to a level sufficient for the full identification of the object that was being presented. Another strategy was based on a differentiated approach to the assessment of all fragments of ready-made information. The assessment was given in accordance with the hypothesis, which is based on the essence of the object. The third strategy was the motivation for maximizing the availability of ready-made information, which was endowed with high value and was considered a sign and prototype of a real object. An important point in the theory of robots was the consideration of brain mechanisms that ensure a change in strategy independently of the dominant emotion and behavioral motivation. The non-specific brain systems are affected by the heterogeneity of neural modules that operate under the control of the central control. The investigations carried out, as well as those from the literary sources, deprived of strict nutritional principles for the distribution of information in a whole image. To achieve this, it is necessary to take care of the formation of the image of that object, in which respect is lost at such a difficult time and the strategy of the image is lost. As such an object could serve as a conflict situation, the field consistently retained respect for the constant other strategy of analyzing the situation. The opposing sides abandoned the first strategy through increasing the triviality of the conflict and did not put up a third strategy, following a unique decision.
Purpose This work was based on the following principles: a matrix of images with the arrangement of elements of information, extracted from the separate integration of the components of a complex object, which was directly respected. The following problems prevailed: firstly, they chose an object at which time it would be important to respect; in another way, they used the method of visualization of the image in order to overcome the fragmentation of information captured during the interaction Ekta, and then thirdly, formulate principles of integral distribution of fragments in the matrix.
Materials and research methods
As a richly-component object that was consistently in the field of respect with a constant strategy for analyzing prepared information, a problematic behavioral situation served. The problem caused conflict among various family members, as well as among the workers of lighting and lighting installations. Experiments in which the analysis of the image of the situation was carried out were carried out using mediation, direct regulation of friction between the opposing sides. Before the start of mediation negotiations, representatives of the disputing parties decided to take part as they tried experiments with different methods that would support the analysis of the situation. The visualization technique conveyed a visual graphic composition, which depicted the design of the image that emerged from the close integration of the components of a complex object. The technique served as a tool for investigating the processes of forming a complete set of elements, the basic parts of an object. The last group consisted of 19 women and 8 men, ranging from 28 to 65 people. To create a complete visual image of the situation, the following actions were recommended: 1) to bring to mind the situation of the conflict situation - ideas, stories about people, motives of power behavior Inki and estranged people; 2) evaluate the situation based on the significance of the overall essence of the situation; 3) divide the situations into friendly and hostile to increase the conflict and try to overcome their mutual connections; 4) select the appropriate graphic element (column, square, triangle, line or point) for the skin that characterizes the situation; 5) form a composition from graphic elements, highlighting the significance and interconnections of the furnishings that are conveyed by these elements, and paint the resulting composition on a paper arch. Graphic compositions were subject to analysis - the ordering and size relationships of the image elements were assessed. The randomly unordered compositions were thrown out, and through testing it was necessary to look again at the interconnections of situational causes. The results of the formalized analysis of the composition served as a guideline for the formulation of the mathematical derivation of the image matrix.
Results of investigation and discussion
The skin graphic composition, through some testing, representing the design as an image of a behavioral situation, was original. Apply the composition to illustrate the baby.
Graphic compositions that evoke images of problematic behavioral situations that have been tested (the skin element of the composition corresponds to situational settings)
The uniqueness of the composition indicated a consistent approach to the analysis of the situation with the resolution of their important figures. The number of elements in the composition and the size of the elements, as well as the design of the composition reflected the assessment of the set of furnishings.
After the originality of the composition was determined, the investigation continued to identify important features of the design of the image. Coming up with a complete composition that reflects the image of the situation, the tested elements were divided according to their individual similarities, as well as to the arrangements of the causal and hereditary elements of the situation. that change of furnishings within an hour. These last ones decided to mount the composition around the appearance of a baby, which was supposed to be a folded figurative plan from afar. In Fig. 1 (a, b, d) the butts of such compositions are used. The last two, before putting together the composition, chose the idea that formed the basis of the plan, knowingly, and five intuitively, without giving a logical explanation of why they decided on the chosen option. The last twenty created a schematic composition, focusing mainly on the causal-hereditary connections of the furnishings and the subsequent furnishings hour after hour (Fig. 1, c, e, f). Knitted and run after an hour, the furnishings were joined by the composition. In the pre-investigations, there was no interpretation of the essence of the conflict from the historical data of the graphic composition. This interpretation has been used for years within the framework of mediation, as long as the parties were ready before negotiations.
Analysis of the composition allows us to understand both the sublimity and the universality of the principles of shaping the image of the situation. First of all, the compositions were made up of graphic elements, skins and various furnishings, with little complexity. The complexity of the furnishings was explained by causal and temporal changes. In another way, we present the unequal significance of the underlying essence of the problem situation. Then the furnishings were tested according to your parameters. Highly significant furnishings were represented by graphic elements of greater size, versus less significant ones. The designated features of the image were sealed when the matrix was folded into the image. It is important to note that the size and graphic features of the elements that are selected, as well as their spaciousness in the graphic composition, served as a guideline for creating an information matrix that represented the image situation and the mathematical model. The matrix is rectangular, presented in a visual table, divided into rows and columns. In a well-formed image of the problem situation in the matrix, rows were seen in which there were important elements of the images, united by cause-hereditary and temporal lines, and a series of elements that corresponded to the They are searching for your parameters.
(1)
The skin is surrounded by a row of molded parts of the image or, otherwise seemingly, the prototype of the object. The more rows and the larger the m, the more completely the object is perceived, and the more strongly the structural and functional authorities that served as its prototypes were emphasized. The number of verses was indicated by the number of details that are identified in the hour of the prototype. It can be taken into account that the more information fragments of high and low value that have been accumulated, this will increase the prototype of reality. The matrix (1) was characterized by dynamism, the fragments of its dimensionality changed, apparently to the full image of the captured object.
Here it is correct to mean that repetition is not the only indicator of the sweetness of the image. The images presented on the canvases of artists are most often reproduced in photographs in detail and for the appearance of reality, but in this case they can reverse the association with other images, destroying the appearance and provoking emotions ій. Careful attention helps to understand the meaning of amn parameters, which indicate the value of information fragments. The increase in water meant that there was no shortage of cooking. As shown by the study of the strategy of insignificance, the recognition of the high significance of ready-made fragments of information hastened the adoption of a solution to a problem situation.
Furthermore, the process of forming a whole image is subject to interpretation, as it relates to the manipulation of information within the matrix. The manipulation is expressed as a fairly or fleeting (obviously purposeful or intuitively unknown) change in the parameters of information fragments, such as a change in the amn value. In this case, the value bm increases or changes, which characterizes the significance of the prototype, and the resulting image br changes simultaneously. As soon as one turns to a matrix model of forming an image that embraces the totality of data of an object, the organization of the image is described in the next order. There is a significant vector of inverse images to move the m component through
where T is a transposition sign, and the skin element of the prototype vector looks like this:
The choice of the resulting image can be determined using Laplace’s rule:
where br is the final result of the formation of a whole image, which has its own components, the values bm, amn - a complex value, which indicates the position and parameters of the variable in the row, which represents the prototype. In the minds of the exchanged information, the final result may increase due to the additional shift in the meaning of the available data.
Upon completion of the discussion of the principles of forming an image presented in the material, respect is raised for the need to specify the term “image”, since there is a widespread lack of confusion in literature today. The term, first of all, means the formation of an entire system of information fragments that represent the details of the object that is in the field of respect. Moreover, the great details of the object are embodied by subsystems of information fragments, which become prototypes. An object can be an object, a phenomenon, a process, or a behavioral situation. The formation of the image is ensured by the association of the information contained and that which resides in the memory and is associated with the object being received. The consolidation of information fragments and association with a created image is realized within the framework of a matrix, the design and vector of which are chosen both intuitively and intuitively. The choice is to lie under the influence that determines the motivations of behavior. Here, attention is especially paid to the main point - the discreteness of information, which is used to assemble a whole image matrix. Integrity, as shown, is ensured by nonspecific brain systems that control the processes of analysis of captured information and integration in memory. The consistency may be lost at minimum values of n and m equal to one. The image acquires high value by increasing the parameters of the cooking information, and the completeness of the image increases by increasing the values of n and m (1).
Visnovok
Visualization of the elements of the image made it possible to follow the principles of its design for the separate identification of the causes of the problematic behavioral situation. As a result of the work, it was shown that in a holistic manner it is possible to see how the distribution of information fragments in the structure of the matrix. This structure and vector are determined, firstly, by behavioral motivation, and secondly, by the causal and hereditary elements of the environment and the time-to-hour sequence of information, and also, thirdly, by the visible fragments The information is consistent with your settings. The integrity of the image matrix is ensured by the integration of discrete information that represents the object being compressed. Non-specific systems can form a mechanism responsible for integrating information into a whole image. The application of matrix principles to the formation of the image of a foldable object expands the perspective of understanding nature as wholeness, and other powers of the image. The integrity and preservation of the image system, as well as the value and subjectivity, are due to the lack of new information about the object.
Bibliographic mailing
Lavrov V.V., Rudinsky A.V. FORMATION OF THE MATRIX IN AN INTEGRATED IMAGE WHEN SPRING ELEMENTS OF A COMPLEX OBJECT // International Journal of Applied and Fundamental Research. - 2016. - No. 7-1. - pp. 91-95;URL: https://applied-research.ru/ru/article/view?id=9764 (date of publication: 01/15/2020). We would like to present to you the magazines that are available at the Academy of Natural Sciences