Is the vector system orthogonal? Orthogonal vector system

44. The system of government bodies of France, suffrage and electoral system France is a mixed (semi-presidential) republic, the system of government bodies of which is based on the principle of separation of powers. France today is a republic with a strong

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The design of PLMs is a LSI, made in the form of a system of orthogonal buses, in the nodes of which basic semiconductor elements - transistors or diodes - are located. Setting up the PLM for the required logical transformation (PLM programming) consists in the appropriate organization of connections between the basic logical elements. Programming of the PLM is performed either during its manufacture, or by the user using a programmer device. Thanks to such properties of PLMs as simplicity of structural organization and high speed of logical transformations, as well as relatively low cost, determined by manufacturability and mass production, PLMs are widely used as an element base in the design of computer systems and production automation systems.

There are no good "mechanical systems" to follow even at this level. In my opinion, there has never been a successful “mechanical” system that could be described by a linear model. It does not exist now and, in all likelihood, never will exist, even with the use of artificial intelligence, analog processors, genetic algorithms, orthogonal regressions and neural networks.

Let us explain the meaning of the norm - G. In an (n+1)-dimensional space, an oblique coordinate system is introduced, one axis of which is the straight line Xe, and the second axis is the n-dimensional hyperplane G, orthogonal to g. Any vector x can be represented in the form

Parabolic regression and the system of orthogonal

For definiteness, let us limit ourselves to the case m = 2 (the transition to the general case m > 2 is carried out in an obvious way without any difficulties) and represent the regression function in the system of basis functions if>0 (n), (x), ip2 to) which are orthogonal (on totality of observed

The mutual orthogonality of the polynomials (7- (JK) (on the observation system xlt k..., xn) means that

With such planning, called orthogonal, the X X matrix will become diagonal, i.e. the system of normal equations splits into k+l independent equations

System of points with the fulfillment of the orthogonality condition (1st order plan)

It is obvious that the deformation tensor in rigid motion vanishes. It can be shown that the converse is also true: if at all points of the medium the deformation tensor is equal to zero, then the law of motion in some rectangular coordinate system of the observer has the form (3.31) with the orthogonal matrix a a. Thus, rigid motion can be defined as the motion of a continuous medium in which the distance between any two points of the medium does not change during the motion.

Two vectors are said to be orthogonal if their scalar product is zero. A system of vectors is called orthogonal if the vectors of this system are pairwise orthogonal.

O Example. System of vectors = (, O,..., 0), e% = = (O, 1,..., 0), . .., e = (0, 0,..., 1) is orthogonal.

The Fredholm operator with kernel k (to - TI, 4 - 12) has a complete orthogonal system of eigenvectors in the Hilbert space (according to Hilbert's theorem). This means that φ(t) form a complete basis in Lz(to, T). Therefore I am with I.

An orthogonal system of n-zero vectors is linearly independent.

The given method for constructing an orthogonal system of vectors t/i, yb,. ..> ym+t for a given linearly independent

For a biotechnical well drilling system, where the amount of physical work remains significant, studies of the biomechanical and motor-strength areas of activity are of particular interest. The composition and structure of labor movements, quantity, dynamic and static loads and developed forces were studied by us on Uralmash-ZD drilling rigs using stereoscopic filming (with two synchronously operating cameras using a special technique at a frequency of 24 frames per 1 s) and the ganiographic method using a three-channel medical oscilloscope. Rigid fixation of optical axes, parallel to each other and perpendicular to the base line (object of filming), made it possible to quantitatively study (based on perspective-orthogonal conjugate projections on film frames, as shown in Fig. 48) working poses, trajectories of movement of the centers of gravity of workers when performing individual operations, techniques, actions and determine efforts, energy costs, etc.

A promising approach to identifying independent alternatives must be the identification of independent synthetic factor indicators. The original system of factor indicators Xi is transformed into a system of new synthetic independent factor indicators FJ, which are orthogonal components of the system of indicators Xg. The transformation is carried out using methods of component analysis 1. Mathematical

One of the components of ADAD is a module for the three-dimensional design of complex piping systems. The module's graphical database contains volumetric elements of pipelines (connections, taps, flanges, pipes). The element selected from the library is automatically adjusted to the characteristics of the pipeline system of the model being designed. The module processes drawings and creates two- and three-dimensional images, including the construction of isometric models and orthogonal projections of objects. There is a choice of parts for pipelines, types of coatings and types of insulation according to a given specification.

From relations (2.49) it is clear how the solution to equations (2.47) should be constructed. First, the polar decomposition of the tensor of is constructed and the tensors p"b are determined. Since the tensors a"b and pI are equal, the matrix s has the form (2.44), (2.45) in the main coordinate system of the tensor p. We fix the matrix Su. Then aad = lp labsd. According to aad, au is calculated from the equation aad = = biljд x ad. The “orthogonal part” of the distortion is found from (2.49) id = nib sd.

The remaining branches do not satisfy condition (2.5 1). Let's prove this statement. The matrix x = A 5, f = X Mfs is orthogonal. Let us denote by X j the matrix corresponding to the first matrix s" (2.44), and by X j the matrix corresponding to any other choice of matrix sa (2.44). The sum "a + Aza by construction s" is equal to either the double value of one of the diagonal

Equal to zero:

An orthogonal system, if complete, can be used as a basis for space. In this case, the decomposition of any element can be calculated using the formulas: , where .

The case when the norm of all elements is called an orthonormal system.

Orthogonalization

Any complete linearly independent system in a finite-dimensional space is a basis. From a simple basis, therefore, one can go to an orthonormal basis.

Orthogonal decomposition

When decomposing the vectors of a vector space according to an orthonormal basis, the calculation of the scalar product is simplified: , where and .

see also

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- (Greek orthogonios rectangular) a finite or countable system of functions belonging to the (separable) Hilbert space L2(a,b) (quadratically integrable functions) and satisfying the conditions F tion g(x) called. weighing O. s. f.,* means... ... Physical encyclopedia

System of functions??n(x)?, n=1, 2,..., specified on the segment ORTHOGONAL TRANSFORMATION linear transformation of Euclidean vector space, preserving unchanged lengths or (which is equivalent to this) scalar products of vectors ... Big Encyclopedic Dictionary

A system of functions (φn(x)), n = 1, 2, ..., defined on the interval [a, b] and satisfying the following orthogonality condition: for k≠l, where ρ(x) is some function called weight. For example, the trigonometric system is 1, sin x, cos x, sin 2x,... ... encyclopedic Dictionary

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Such a subset of vectors $\backslash left\backslash (\; \backslash varphi\_i\; \backslash right\backslash )\backslash subset\; H$ that any distinct two of them are orthogonal, that is, their scalar product is equal to zero:

$(\backslash varphi\_i,\; \backslash varphi\_j)\; =\; 0$.

An orthogonal system, if complete, can be used as a basis for space. Moreover, the decomposition of any element $\backslash vec\; a$ can be calculated using the formulas: $\backslash vec\; a\; =\; \backslash sum\_(k)\; \backslash alpha\_i\; \backslash varphi\_i$, Where $\backslash alpha\_i\; =\; \backslash frac((\backslash vec\; a,\; \backslash varphi\_i))((\backslash varphi\_i,\; \backslash varphi\_i))$.

The case when the norm of all elements $||\backslash varphi\_i||=1$, is called an orthonormal system.

Orthogonalization

Any complete linearly independent system in a finite-dimensional space is a basis. From a simple basis, therefore, one can go to an orthonormal basis.

Orthogonal decomposition

When decomposing the vectors of a vector space according to an orthonormal basis, the calculation of the scalar product is simplified: $(\backslash vec\; a,\; \backslash vec\; b)\; =\; \backslash sum\_(k)\; \backslash alpha\_k\backslash beta\_k$, Where $\backslash vec\; a\; =\; \backslash sum\_(k)\; \backslash alpha\_k\; \backslash varphi\_k$ And $\backslash vec\; b\; =\; \backslash sum\_(k)\; \backslash beta\_k\; \backslash varphi\_k$.

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An excerpt characterizing the Orthogonal system

If we choose any two mutually perpendicular vectors of unit length on a plane (Fig. 7), then an arbitrary vector in the same plane can be expanded in the directions of these two vectors, i.e., represented in the form

where are numbers equal to the projections of the vector onto the directions of the axes. Since the projection onto the axis is equal to the product of the length and the cosine of the angle with the axis, then, recalling the definition of the scalar product, we can write

Similarly, if in three-dimensional space we choose any three mutually perpendicular vectors of unit length, then an arbitrary vector in this space can be represented as

In a Hilbert space, one can also consider systems of pairwise orthogonal vectors of this space, i.e., functions

Such systems of functions are called orthogonal systems of functions and play an important role in analysis. They are found in a wide variety of questions of mathematical physics, integral equations, approximate calculations, theory of functions of a real variable, etc. The ordering and unification of concepts related to such systems was one of the incentives that led at the beginning of the 20th century. towards the creation of a general concept of Hilbert space.

Let us give precise definitions. Function system

is called orthogonal if any two functions of this system are orthogonal to each other, i.e. if

In three-dimensional space, we required that the lengths of the system vectors be equal to one. Recalling the definition of vector length, we see that in the case of a Hilbert space this requirement is written as follows:

A system of functions that satisfies requirements (13) and (14) is called orthogonal and normalized.

Let us give examples of such systems of functions.

1. On the interval, consider the sequence of functions

Every two functions from this sequence are orthogonal to each other. This can be verified by simply calculating the corresponding integrals. The square of the length of a vector in a Hilbert space is the integral of the square of the function. Thus, the squared lengths of the sequence vectors

the essence of integrals

i.e. the sequence of our vectors is orthogonal, but not normalized. The length of the first vector of the sequence is equal to

the rest have length . By dividing each vector by its length, we obtain an orthogonal and normalized system of trigonometric functions

This system is historically one of the first and most important examples of orthogonal systems. It arose in the works of Euler, D. Bernoulli, and d'Alembert in connection with the problem of string vibrations. Her study played a significant role in the development of the entire analysis.

The appearance of an orthogonal system of trigonometric functions in connection with the problem of string vibrations is not accidental. Each problem about small oscillations of a medium leads to a certain system of orthogonal functions that describe the so-called natural oscillations of a given system (see § 4). For example, in connection with the problem of oscillations of a sphere, so-called spherical functions appear, in connection with the problem of oscillations of a round membrane or cylinder, so-called cylindrical functions appear, etc.

2. You can give an example of an orthogonal system of functions, each function of which is a polynomial. Such an example is the sequence of Legendre polynomials

i.e. there is (up to a constant factor) the order derivative of . Let's write down the first few polynomials of this sequence:

It is obvious that in general there is a polynomial of degree. We leave it to the reader to see for himself that these polynomials represent an orthogonal sequence on the interval

The general theory of orthogonal polynomials (the so-called orthogonal polynomials with weight) was developed by the remarkable Russian mathematician P. L. Chebyshev in the second half of the 19th century.

Expansion in orthogonal systems of functions. Just as in three-dimensional space each vector can be represented

as a linear combination of three pairwise orthogonal vectors of unit length

in the function space, the problem arises of expanding an arbitrary function into a series in an orthogonal and normalized system of functions, i.e., representing the function in the form

In this case, the convergence of series (15) to a function is understood in the sense of the distance between elements in the Hilbert space. This means that the root mean square deviation of the partial sum of the series from the function tends to zero as , i.e.

This convergence is usually called “convergence on average.”

Expansions in terms of certain systems of orthogonal functions are often found in analysis and are an important method for solving problems of mathematical physics. So, for example, if an orthogonal system is a system of trigonometric functions on the interval

then such an expansion is the classical expansion of a function in a trigonometric series

Let us assume that expansion (15) is possible for any function from the Hilbert space and find the coefficients of such expansion. To do this, let's multiply both sides of the equality scalarly by the same function of our system. We will get equality

from which, due to the fact that when the value of the coefficient is determined

We see that, as in ordinary three-dimensional space (see the beginning of this section), the coefficients are equal to the projections of the vector onto the directions of the vectors.

Recalling the definition of the scalar product, we find that the coefficients of the expansion of a function in an orthogonal and normalized system of functions

determined by formulas

As an example, consider the orthogonal normalized trigonometric system of functions given above:

We have obtained a formula for calculating the coefficients of the expansion of a function into a trigonometric series, assuming, of course, that this expansion is possible.

We have established the form of coefficients of expansion (18) of a function in an orthogonal system of functions under the assumption that such an expansion takes place. However, an infinite orthogonal system of functions may not be sufficient for it to be possible to expand any function from a Hilbert space. For such an expansion to be possible, the system of orthogonal functions must satisfy an additional condition - the so-called completeness condition.

An orthogonal system of functions is called complete if it is impossible to add to it a single non-identically zero function orthogonal to all functions of the system.

It is easy to give an example of an incomplete orthogonal system. To do this, let’s take some orthogonal system, for example the same

system of trigonometric functions, and eliminate one of the functions of this system, for example, Remaining infinite system of functions

will still be orthogonal, of course, it will not be complete, since the function we excluded is orthogonal to all functions of the system.

If a system of functions is not complete, then not every function from a Hilbert space can be expanded over it. Indeed, if we try to expand in such a system a zero function orthogonal to all functions of the system, then, by virtue of formulas (18), all coefficients will be equal to zero, while the function is not equal to zero.

The following theorem holds: if a complete orthogonal and normalized system of functions in a Hilbert space is given, then any function can be expanded into a series in terms of the functions of this system

In this case, the expansion coefficients are equal to the projections of the vectors onto the elements of the orthogonal normalized system

The Pythagorean theorem in § 2 in Hilbert space allows us to find an interesting relationship between the coefficients and the function. Let us denote by the difference between and the sum of the first terms of its series, i.e.