See pages where the term orthogonal system is mentioned. See pages where the term orthogonal system is mentioned French government system, suffrage and electoral system

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 linear 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

Definition. Vectorsa Andb are called orthogonal (perpendicular) to each other if their scalar product is equal to zero, i.e.a × b = 0.

For non-zero vectors a And b the equality of the scalar product to zero means that cos j= 0, i.e. . The zero vector is orthogonal to any vector, because a × 0 = 0.

Exercise. Let and be orthogonal vectors. Then it is natural to consider the diagonal of a rectangle with sides and . Prove that

those. the square of the length of the diagonal of a rectangle is equal to the sum of the squares of the lengths of its two non-parallel sides(Pythagorean theorem).

Definition. Vector systema 1 ,…, a m is called orthogonal if any two vectors of this system are orthogonal.

Thus, for an orthogonal system of vectors a 1 ,…,a m the equality is true: a i × a j= 0 at i¹ j, i= 1,…, m; j= 1,…,m.

Theorem 1.5. An orthogonal system consisting of nonzero vectors is linearly independent. .

□ We carry out the proof by contradiction. Suppose that the orthogonal system of nonzero vectors a 1 , …, a m linearly dependent. Then

l 1 a 1 + …+ l ma m= 0 , wherein . (1.15)

Let, for example, l 1 ¹ 0. Multiply by a 1 both sides of equality (1.15):

l 1 a 1× a 1 + …+ l m a m × a 1 = 0.

All terms except the first are equal to zero due to the orthogonality of the system a 1 , …, a m. Then l 1 a 1× a 1 =0, which follows a 1 = 0 , which contradicts the condition. Our assumption turned out to be wrong. This means that the orthogonal system of nonzero vectors is linearly independent. ■

The following theorem holds.

Theorem 1.6. In the space Rn there is always a basis consisting of orthogonal vectors (orthogonal basis)
(no proof).

Orthogonal bases are convenient primarily because the expansion coefficients of an arbitrary vector over such bases are simply determined.

Suppose we need to find the decomposition of an arbitrary vector b on an orthogonal basis e 1 ,…,e n. Let’s compose an expansion of this vector with still unknown expansion coefficients for this basis:

Let's multiply both sides of this equality scalarly by the vector e 1 . By virtue of axioms 2° and 3° of the scalar product of vectors, we obtain

Since the basis vectors e 1 ,…,e n are mutually orthogonal, then all scalar products of the basis vectors, with the exception of the first, are equal to zero, i.e. the coefficient is determined by the formula

Multiplying equality (1.16) one by one by other basis vectors, we obtain simple formulas for calculating the vector expansion coefficients b :

Formulas (1.17) make sense because .

Definition. Vectora is called normalized (or unit) if its length is equal to 1, i.e. (a , a )= 1.

Any nonzero vector can be normalized. Let a ¹ 0 . Then , and the vector is a normalized vector.

Definition. Vector system e 1 ,…,e n is called orthonormal if it is orthogonal and the length of each vector of the system is equal to 1, i.e.

Since there is always an orthogonal basis in the space Rn and the vectors of this basis can be normalized, then there is always an orthonormal basis in Rn.

An example of an orthonormal basis of the space R n is the system of vectors e 1 ,=(1,0,…,0),…, e n=(0,0,…,1) with the scalar product defined by equality (1.9). In an orthonormal basis e 1 ,=(1,0,…,0),…, e n=(0,0,…,1) formula (1.17) to determine the coordinates of the vector decomposition b have the simplest form:

Let a And b – two arbitrary vectors of the space R n with an orthonormal basis e 1 ,=(1,0,…,0),…, e n=(0,0,…,1). Let us denote the coordinates of the vectors a And b in the basis e 1 ,…,e n accordingly through a 1 ,…,a n And b 1 ,…, b n and find the expression for the scalar product of these vectors through their coordinates in on this basis, i.e. Let's pretend that

From the last equality, by virtue of the scalar product axioms and relations (1.18), we obtain

Finally we have

Thus, in an orthonormal basis, the scalar product of any two vectors is equal to the sum of the products of the corresponding coordinates of these vectors.

Let us now consider a completely arbitrary (generally speaking, not orthonormal) basis in the n-dimensional Euclidean space R n and find an expression for the scalar product of two arbitrary vectors a And b through the coordinates of these vectors in the specified basis. f 1 ,…,f n Euclidean space R n the scalar product of any two vectors is equal to the sum of the products of the corresponding coordinates of these vectors, it is necessary and sufficient that the basis f 1 ,…,f n was orthonormal.

In fact, expression (1.20) goes into (1.19) if and only if the relations establishing the orthonormality of the basis are satisfied f 1 ,…,f n.

1) O. such that (x a , x ab)=0 at . If the norm of each vector is equal to one, then the system (x a) is called. orthonormal. Full O. s. (x a) called orthogonal (orthonormal) basis. M. I. Voitsekhovsky.

2) O. s. coordinates - a coordinate system in which coordinate lines (or surfaces) intersect at right angles. O. s. coordinates exist in any Euclidean space, but, generally speaking, do not exist in any space. In a two-dimensional smooth affine space O. s. can always be introduced at least in a sufficiently small neighborhood of each point. Sometimes it is possible to introduce O. s. coordinates in action. In O. s. metric tensor g ij diagonals; diagonal components gii accepted name Lamé coefficients. Lame coefficient O. s. in space are expressed by formulas

Where x, y And z- Cartesian rectangular coordinates. The element of length is expressed through the Lamé coefficients:

surface area element:

volume element:

vector differential operations:

The most frequently used O. s. coordinates: on the plane - Cartesian, polar, elliptical, parabolic; in space - spherical, cylindrical, paraboloidal, bicylindrical, bipolar. D. D. Sokolov.

3) O. s. functions - finite or countable system (j i(x)) functions belonging to the space

L 2(X, S, m) and satisfying the conditions

If l i=1 for all i, then the system is called orthonormal. It is assumed that the measure m(x), defined on the s-algebra S of subsets of the set X, is countably additive, complete, and has a countable base. This is the definition of O. s. includes all O. pages considered in modern analysis; they are obtained for various specific implementations of measure space ( X, S, m).

Of greatest interest are complete orthonormal systems (j n(x)), which have the property that for any function there is a unique series converging to f(x) in the metric of the space L 2(X, S, m) , while the coefficients s p are determined by the Fourier formulas

Such systems exist due to the separability of space L 2(X, S, m). A universal way to construct complete orthonormal systems is provided by the Schmidt orthogonalization method. To do this, it is enough to apply it to a certain swarm of complete L 2(S, X, m) a system of linearly independent functions.

In theory orthogonal series in mainly considered O. s. spaceLva L 2[a, b](that special case when X=[a, b], S- system of Lebesgue measurable sets, and m is the Lebesgue measure). Many theorems on the convergence or summability of series , , according to general mathematical systems. (j n(x)) spaces L 2[a, b] are also true for series in orthonormal systems of space L 2(X, S, m). At the same time, in this particular case, interesting specific O. systems have been constructed that have certain good properties. Such are, for example, the systems of Haar, Rademacher, Walsh-Paley, and Franklin.

1) Haar system

where m=2 n+k, , t=2, 3, ... . Haar series represent a typical example martingales and for them the general theorems from the theory of martingales are true. In addition, the system is the basis in Lp, , and the Fourier series in the Haar system of any integrable function converges almost everywhere.

2) Rademacher system

represents an important example of O. s. independent functions and has applications both in probability theory and in the theory of orthogonal and general functional series.

3) Walsh-Paley system is determined through the Rademacher functions:

where are the ti numbers q k are determined from the binary expansion of the number n:

4) The Franklin system is obtained by orthogonalizing the sequence of functions using the Schmidt method

It is an example of an orthogonal basis of the space C of continuous functions.

In the theory of multiple orthogonal series, systems of functions of the form are considered

where is the orthonormal system in L 2[a, b]. Such systems are orthonormal on the m-dimensional cube J m =[a, b]x . . .x[ a, b] and are complete if the system (j n(x))

Lit.:[l] Kaczmarz S., Shteingauz G., Theory of orthogonal series, trans. from German, M., 1958; Results of science. Mathematical analysis, 1970, M., 1971, p. 109-46; there, s. 147-202; Dub J., Probabilistic processes, trans. from English, M., 1956; Loeve M., Theory of Probability, trans. from English, M., 1962; Zygmund A., Trigonometric series, trans. from English, vol. 1-2, M., 1965. A. A. Talalyan.

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