Properties of vector length in Euclidean space. Euclidean spaces. Linear algebra. And quadratic forms

Corresponding to such a vector space. In this article, the first definition will be taken as the starting point.

$n$-dimensional Euclidean space is denoted by $\backslash mathbb\; E^n,$ the notation is also often used $\backslash mathbb\; R^n$(if it is clear from the context that the space has a Euclidean structure).

Formal definition

To define Euclidean space, the easiest way is to take as the main concept the scalar product. A Euclidean vector space is defined as a finite-dimensional vector space over the field of real numbers, on whose vectors a real-valued function is specified $(\backslash cdot,\; \backslash cdot),$ having the following three properties:

• Bilinearity: for any vectors $u,v,w$ and for any real numbers $a,\; b\backslash quad\; (au+bv,\; w)=a(u,w)+b(v,w)$ And $(u,\; av+bw)=a(u,v)+b(u,w);$
• Symmetry: for any vectors $u,v\backslash quad\; (u,v)=(v,u);$
• Positive certainty: for anyone $u\backslash quad\; (u,u)\backslash geqslant\; 0,$ and $(u,u)\; =\; 0\backslash Rightarrow\; u=0.$

Example of Euclidean space - coordinate space $\backslash mathbb\; R^n,$ consisting of all possible tuples of real numbers $(x\_1,\; x\_2,\; \backslash ldots,\; x\_n),$ scalar product in which is determined by the formula $(x,y)\; =\; \backslash sum\_(i=1)^n\; x\_iy\_i\; =\; x\_1y\_1+x\_2y\_2+\backslash cdots+x\_ny\_n.$

Lengths and angles

The scalar product defined on Euclidean space is sufficient to introduce the geometric concepts of length and angle. Vector length $u$ defined as $\backslash sqrt((u,u))$ and is designated $|u|.$ The positive definiteness of the scalar product guarantees that the length of the nonzero vector is nonzero, and from bilinearity it follows that $|au|=|a||u|,$ that is, the lengths of proportional vectors are proportional.

Angle between vectors $u$ And $v$ determined by the formula $\backslash varphi=\backslash arccos\; \backslash left(\backslash frac((x,y))(|x||y|)\backslash right).$ From the cosine theorem it follows that for a two-dimensional Euclidean space ( Euclidean plane) this definition angle coincides with the usual one. Orthogonal vectors, as in three-dimensional space, can be defined as vectors the angle between which is equal to $\backslash frac(\backslash pi)(2).$

The Cauchy-Bunyakovsky-Schwartz inequality and the triangle inequality

There is one gap left in the definition of angle given above: in order to $\backslash arccos\; \backslash left(\backslash frac((x,y))(|x||y|)\backslash right)$ has been defined, it is necessary that the inequality $\backslash left|\backslash frac((x,y))(|x||y|)\backslash right|\backslash leqslant\; 1.$ This inequality does hold in an arbitrary Euclidean space, and is called the Cauchy–Bunyakovsky–Schwartz inequality. From this inequality, in turn, follows the triangle inequality: $|u+v|\backslash leqslant\; |u|+|v|.$ The triangle inequality, together with the properties of length listed above, means that the length of a vector is a norm on the Euclidean vector space, and the function $d(x,y)=|x-y|$ defines the structure of a metric space on Euclidean space (this function is called the Euclidean metric). In particular, the distance between elements (points) $x$ And $y$ coordinate space $\backslash mathbb\; R^n$ is given by the formula $d(\backslash mathbf(x),\; \backslash mathbf(y))\; =\; \backslash |\backslash mathbf(x)\; -\; \backslash mathbf(y)\backslash |\; =\; \backslash sqrt(\backslash sum\_(i=1)^n\; (x\_i\; -\; y\_i)^2).$

Algebraic properties

Orthonormal bases

Conjugate spaces and operators

Any vector $x$ Euclidean space defines a linear functional $x^*$ on this space, defined as $x^*(y)=(x,y).$ This comparison is an isomorphism between Euclidean space and its dual space and allows them to be identified without compromising calculations. In particular, conjugate operators can be considered as acting on the original space, and not on its dual, and self-adjoint operators can be defined as operators that coincide with their conjugates. In an orthonormal basis, the matrix of the adjoint operator is transposed to the matrix of the original operator, and the matrix of the self-adjoint operator is symmetric.

Movements of Euclidean space

Examples

Illustrative examples of Euclidean spaces are the following spaces:

• $\backslash mathbb\; E^1$ dimensions $1$ (real line)
• $\backslash mathbb\; E^2$ dimensions $2$ (Euclidean plane)
• $\backslash mathbb\; E^3$ dimensions $3$ (Euclidean three-dimensional space)

More abstract example:

• space of real polynomials $p(x)$ degree not exceeding $n$, with scalar product, defined as the integral of the product over a finite segment (or over the entire line, but with a rapidly decreasing weight function, for example $e^(-x^2)$).

Examples of geometric shapes in multidimensional Euclidean space

• Regular multidimensional polyhedra (specifically N-dimensional cube, N-dimensional octahedron, N-dimensional tetrahedron)

Related definitions

• Under Euclidean metric can be understood as the metric described above as well as the corresponding Riemannian metric.

• By local Euclideanity we usually mean that each tangent space of a Riemannian manifold is a Euclidean space with all the ensuing properties, for example, the ability (due to the smoothness of the metric) to introduce coordinates in a small neighborhood of a point in which the distance is expressed (up to some order of magnitude) ) as described above.

• A metric space is also called locally Euclidean if it is possible to introduce coordinates on it in which the metric will be Euclidean (in the sense of the second definition) everywhere (or at least on a finite domain) - which, for example, is a Riemannian manifold of zero curvature.

Variations and generalizations

• Replacing the basic field from the field of real numbers to the field of complex numbers gives the definition of a unitary (or Hermitian) space.
• Refusal of the finite-dimensionality requirement gives the definition of a pre-Hilbert space.
• Refusal of the requirement of positive definiteness of the scalar product leads to the definition of pseudo-Euclidean space.

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Notes

Literature

• Gelfand I. M. Lectures on linear algebra. - 5th. - M.: Dobrosvet, MTsNMO, 1998. - 319 p. - ISBN 5-7913-0015-8.
• Kostrikin A. I., Manin Yu. I. Linear algebra and geometry. - M.: Nauka, 1986. - 304 p.

Vectors and matrices Vectors Basic Concepts Types of vectors Operations on vectors Types of spaces Matrices Basic Concepts Other

An excerpt characterizing Euclidean space

Sonya walked across the hall to the buffet with a glass. Natasha looked at her, at the crack in the pantry door, and it seemed to her that she remembered that light was falling through the crack from the pantry door and that Sonya walked through with a glass. “Yes, and it was exactly the same,” thought Natasha. - Sonya, what is this? – Natasha shouted, fingering the thick string.
- Oh, you’re here! - Sonya said, shuddering, and came up and listened. - Don't know. Storm? – she said timidly, afraid of making a mistake.
“Well, in exactly the same way she shuddered, in the same way she came up and smiled timidly then, when it was already happening,” Natasha thought, “and in the same way... I thought that something was missing in her.”
- No, this is the choir from the Water-bearer, do you hear! – And Natasha finished singing the choir’s tune to make it clear to Sonya.
-Where did you go? – Natasha asked.
- Change the water in the glass. I'll finish the pattern now.
“You’re always busy, but I can’t do it,” said Natasha. -Where is Nikolai?
- He seems to be sleeping.
“Sonya, go wake him up,” said Natasha. - Tell him that I call him to sing. “She sat and thought about what it meant, that it all happened, and, without resolving this question and not at all regretting it, again in her imagination she was transported to the time when she was with him, and he looked with loving eyes looked at her.
“Oh, I wish he would come soon. I'm so afraid that this won't happen! And most importantly: I'm getting old, that's what! What is now in me will no longer exist. Or maybe he’ll come today, he’ll come now. Maybe he came and is sitting there in the living room. Maybe he arrived yesterday and I forgot.” She stood up, put down the guitar and went into the living room. All the household, teachers, governesses and guests were already sitting at the tea table. People stood around the table, but Prince Andrei was not there, and life was still the same.
“Oh, here she is,” said Ilya Andreich, seeing Natasha enter. - Well, sit down with me. “But Natasha stopped next to her mother, looking around, as if she was looking for something.
- Mother! - she said. “Give it to me, give it to me, mom, quickly, quickly,” and again she could hardly hold back her sobs.
She sat down at the table and listened to the conversations of the elders and Nikolai, who also came to the table. “My God, my God, the same faces, the same conversations, dad holding the cup in the same way and blowing in the same way!” thought Natasha, feeling with horror the disgust rising in her against everyone at home because they were still the same.
After tea, Nikolai, Sonya and Natasha went to the sofa, to their favorite corner, where their most intimate conversations always began.

“It happens to you,” Natasha said to her brother when they sat down in the sofa, “it happens to you that it seems to you that nothing will happen - nothing; what was all that was good? And not just boring, but sad?
- And how! - he said. “It happened to me that everything was fine, everyone was cheerful, but it would come to my mind that I was already tired of all this and that everyone needed to die.” Once I didn’t go to the regiment for a walk, but there was music playing there... and so I suddenly became bored...
- Oh, I know that. I know, I know,” Natasha picked up. – I was still little, this happened to me. Do you remember, once I was punished for plums and you all danced, and I sat in the classroom and sobbed, I will never forget: I was sad and I felt sorry for everyone, and myself, and I felt sorry for everyone. And, most importantly, it wasn’t my fault,” Natasha said, “do you remember?
“I remember,” said Nikolai. “I remember that I came to you later and I wanted to console you and, you know, I was ashamed. We were terribly funny. I had a bobblehead toy then and I wanted to give it to you. Do you remember?
“Do you remember,” Natasha said with a thoughtful smile, how long, long ago, we were still very little, an uncle called us into the office, back in the old house, and it was dark - we came and suddenly there was standing there...
“Arap,” Nikolai finished with a joyful smile, “how can I not remember?” Even now I don’t know that it was a blackamoor, or we saw it in a dream, or we were told.
- He was gray, remember, and had white teeth - he stood and looked at us...
– Do you remember, Sonya? - Nikolai asked...
“Yes, yes, I remember something too,” Sonya answered timidly...
“I asked my father and mother about this blackamoor,” said Natasha. - They say that there was no blackamoor. But you remember!
- Oh, how I remember his teeth now.
- How strange it is, it was like a dream. I like it.
- Do you remember how we were rolling eggs in the hall and suddenly two old women began to spin around on the carpet? Was it or not? Do you remember how good it was?
- Yes. Do you remember how dad in a blue fur coat fired a gun on the porch? “They turned over, smiling with pleasure, memories, not sad old ones, but poetic youthful memories, those impressions from the most distant past, where dreams merge with reality, and laughed quietly, rejoicing at something.
Sonya, as always, lagged behind them, although their memories were common.
Sonya did not remember much of what they remembered, and what she did remember did not arouse in her the poetic feeling that they experienced. She only enjoyed their joy, trying to imitate it.
She took part only when they remembered Sonya's first visit. Sonya told how she was afraid of Nikolai, because he had strings on his jacket, and the nanny told her that they would sew her into strings too.
“And I remember: they told me that you were born under cabbage,” said Natasha, “and I remember that I didn’t dare not believe it then, but I knew that it wasn’t true, and I was so embarrassed.”
During this conversation, the maid's head poked out of the back door of the sofa room. “Miss, they brought the rooster,” the girl said in a whisper.
“No need, Polya, tell me to carry it,” said Natasha.
In the middle of the conversations going on in the sofa, Dimmler entered the room and approached the harp that stood in the corner. He took off the cloth and the harp made a false sound.
“Eduard Karlych, please play my beloved Nocturiene by Monsieur Field,” said the voice of the old countess from the living room.
Dimmler struck a chord and, turning to Natasha, Nikolai and Sonya, said: “Young people, how quietly they sit!”
“Yes, we are philosophizing,” Natasha said, looking around for a minute and continuing the conversation. The conversation was now about dreams.
Dimmer started to play. Natasha silently, on tiptoe, walked up to the table, took the candle, took it out and, returning, quietly sat down in her place. It was dark in the room, especially on the sofa on which they were sitting, but through the large windows the silver light of the full moon fell onto the floor.
“You know, I think,” Natasha said in a whisper, moving closer to Nikolai and Sonya, when Dimmler had already finished and was still sitting, weakly plucking the strings, apparently indecisive to leave or start something new, “that when you remember like that, you remember, you remember everything.” , you remember so much that you remember what happened before I was in the world...
“This is Metampsic,” said Sonya, who always studied well and remembered everything. – The Egyptians believed that our souls were in animals and would go back to animals.
“No, you know, I don’t believe it, that we were animals,” Natasha said in the same whisper, although the music had ended, “but I know for sure that we were angels here and there somewhere, and that’s why we remember everything.” ...
-Can I join you? - said Dimmler, who approached quietly and sat down next to them.
- If we were angels, then why did we fall lower? - said Nikolai. - No, this cannot be!
“Not lower, who told you that lower?... Why do I know what I was before,” Natasha objected with conviction. - After all, the soul is immortal... therefore, if I live forever, that’s how I lived before, lived for all eternity.
“Yes, but it’s hard for us to imagine eternity,” said Dimmler, who approached the young people with a meek, contemptuous smile, but now spoke as quietly and seriously as they did.
– Why is it difficult to imagine eternity? - Natasha said. - Today it will be, tomorrow it will be, it will always be and yesterday it was and yesterday it was...
- Natasha! now it's your turn. “Sing me something,” the countess’s voice was heard. - That you sat down like conspirators.
- Mother! “I don’t want to do that,” Natasha said, but at the same time she stood up.
All of them, even the middle-aged Dimmler, did not want to interrupt the conversation and leave the corner of the sofa, but Natasha stood up, and Nikolai sat down at the clavichord. As always, standing in the middle of the hall and choosing the most advantageous place for resonance, Natasha began to sing her mother’s favorite piece.
She said that she did not want to sing, but she had not sung for a long time before, and for a long time since, the way she sang that evening. Count Ilya Andreich, from the office where he was talking with Mitinka, heard her singing, and like a student, in a hurry to go play, finishing the lesson, he got confused in his words, giving orders to the manager and finally fell silent, and Mitinka, also listening, silently with a smile, stood in front of count. Nikolai did not take his eyes off his sister, and took a breath with her. Sonya, listening, thought about what a huge difference there was between her and her friend and how impossible it was for her to be even remotely as charming as her cousin. The old countess sat with a happily sad smile and tears in her eyes, occasionally shaking her head. She thought about Natasha, and about her youth, and about how there was something unnatural and terrible in this upcoming marriage of Natasha with Prince Andrei.
Dimmler sat down next to the countess and closed his eyes, listening.
“No, Countess,” he said finally, “this is a European talent, she has nothing to learn, this softness, tenderness, strength...”
- Ah! “how I’m afraid for her, how afraid I am,” said the countess, not remembering who she was talking to. Her maternal instinct told her that there was too much of something in Natasha, and that this would not make her happy. Natasha had not yet finished singing when an enthusiastic fourteen-year-old Petya ran into the room with the news that the mummers had arrived.
Natasha suddenly stopped.
- Fool! - she screamed at her brother, ran up to the chair, fell on it and sobbed so much that she could not stop for a long time.
“Nothing, Mama, really nothing, just like this: Petya scared me,” she said, trying to smile, but the tears kept flowing and sobs were choking her throat.
Dressed up servants, bears, Turks, innkeepers, ladies, scary and funny, bringing with them coldness and fun, at first timidly huddled in the hallway; then, hiding one behind the other, they were forced into the hall; and at first shyly, and then more and more cheerfully and amicably, songs, dances, choral and Christmas games began. The Countess, recognizing the faces and laughing at those dressed up, went into the living room. Count Ilya Andreich sat in the hall with a radiant smile, approving of the players. The youth disappeared somewhere.

Euclidean space

T.A. Volkova, T.P. Knysh.

AND SQUARE SHAPES

EUCLIDEAN SPACE

Saint Petersburg

Reviewer: candidate technical sciences, Associate Professor Shkadova A.R.

Euclidean space and quadratic forms: lecture notes. – St. Petersburg: SPGUVK, 2012 – p.

Lecture notes are intended for second-year students of the bachelor's degree 010400.62 "Applied Mathematics and Computer Science" and first-year students of the bachelor's degree 090900.62 "Information Security".

The manual contains a complete lecture notes on one of the sections of the discipline “Geometry and Algebra” for direction 010400.62 and the discipline “Algebra and Geometry” for direction 090900.62 Tutorial corresponds to the work programs of the disciplines, the standards of the specified specialties and can be used in preparing for the exam by students and teachers.

©St. Petersburg State

University of Water Communications, 2012

Many properties of objects found in geometry are closely related to the ability to measure the lengths of segments and the angle between straight lines. In linear space we cannot yet make such measurements, as a result of which the scope general theory linear spaces to geometry and a number of others mathematical disciplines narrows quite a bit. This difficulty, however, can be eliminated by introducing the concept of the scalar product of two vectors. Namely, let be a linear -dimensional real space. Let us associate each pair of vectors with a real number and call this number scalar product vectors and if the following requirements are met:

1. (commutative law).

3. for any real.

4. for any non-zero vector.

The scalar product is a special case of the concept numeric function of two vector arguments, i.e. functions whose values ​​are numbers. We can therefore call the scalar product such numerical function vector arguments , , whose values ​​are valid for any values ​​of the arguments from and for which requirements 1 − 4 are satisfied.

A real linear space in which the scalar product is defined will be called Euclidean and will be denoted by .

Note that in Euclidean space the scalar product of a zero vector and any vector is equal to zero: . Indeed, and due to requirement 3. Assuming , we get that . Hence, in particular, .

1. Let be an ordinary three-dimensional space geometric vectors with a common origin at the point . In analytical geometry, the scalar product of two such vectors is a real number equal to , where and are the lengths of the vectors and , and is the angle between the vectors , , and it is proved that for this number all requirements 1 − 4 are satisfied.

Thus, the concept of a scalar product introduced by us is a generalization of the concept of a scalar product of geometric vectors.

2. Consider the space of dimensional rows with real coordinates and assign a real number to each pair of such row vectors

It is easy to check that all requirements 1 − 4 are satisfied for this number:

and similarly. Finally,

since at least one of the numbers at is different from zero.

We see from here that this number is the scalar product of the string vectors and , and the space, after we introduced such a scalar product, becomes Euclidean.

3. Let be a linear real -dimensional space and be some of its basis. Let us associate each pair of vectors with a real number. Then the space will turn into Euclidean, i.e. the number will be the scalar product of the vectors and . Indeed:

We can even turn our space into a Euclidean space in other ways, for example, we could associate a pair of vectors with a real number

and it is easy to check that for such a number all requirements 1 − 4, characterizing the scalar product, are satisfied. But since here (with the same basis) we have defined a different numerical function, then we get a different Euclidean space with a different “measure definition”.

4. Finally, turning to the same space, consider the numerical function, which, for , is defined by the equality . This function is no longer a scalar product, since requirement 4 is violated: when , the vector is equal to , a . Thus, Euclidean space cannot be obtained here.

Using requirements 2 and 3 included in the definition of the scalar product, it is easy to obtain the following formula:

where , are two arbitrary systems of vectors. From here, in particular, it turns out for an arbitrary basis and for any pair of vectors , , that

Where . The expression on the right side of equality (1) is a polynomial in and and is called bilinear form from and (each of its terms is linear, i.e. of the first degree, both with respect to and with respect to ). The bilinear form is called symmetrical, if for each of its coefficients the symmetry condition is satisfied. Thus, scalar product in an arbitrary basis expressed as a bilinear symmetric form of the vector coordinates , with real odds. But this is still not enough. Namely, setting , we obtain from equality (1) that

Even at school, all students are introduced to the concept of “Euclidean geometry,” the main provisions of which are focused around several axioms based on such geometric elements as a point, a plane, a straight line, and motion. All of them together form what has long been known as “Euclidean space”.

Euclidean, which is based on the principle of scalar multiplication of vectors, is a special case of a linear (affine) space that satisfies a number of requirements. Firstly, the scalar product of vectors is absolutely symmetrical, that is, a vector with coordinates (x;y) is quantitatively identical to a vector with coordinates (y;x), but opposite in direction.

Secondly, if the scalar product of a vector with itself is performed, then the result of this action will be positive character. The only exception will be the case when the initial and final coordinates of this vector are equal to zero: in this case, its product with itself will also be equal to zero.

Thirdly, the scalar product is distributive, that is, the possibility of decomposing one of its coordinates into the sum of two values, which will not entail any changes in the final result of the scalar multiplication of vectors. Finally, fourthly, when multiplying vectors by the same thing, their scalar product will also increase by the same amount.

If all these four conditions are met, we can confidently say that this is Euclidean space.

From a practical point of view, Euclidean space can be characterized by the following specific examples:

1. The simplest case is the presence of a set of vectors with a scalar product defined according to the basic laws of geometry.
2. Euclidean space will also be obtained if by vectors we understand a certain finite set real numbers With given formula, describing their scalar sum or product.
3. A special case of Euclidean space should be recognized as the so-called null space, which is obtained if the scalar length of both vectors is equal to zero.

Euclidean space has a number of specific properties. Firstly, the scalar factor can be taken out of brackets from both the first and second factors of the scalar product, the result will not undergo any changes. Secondly, along with the distributivity of the first element of the scalar product, the distributivity of the second element also operates. Moreover, in addition to scalar sum vectors, distributivity also occurs in the case of subtraction of vectors. Finally, thirdly, when scalar multiplying a vector by zero, the result will also be equal to zero.

Thus, Euclidean space is the most important geometric concept, used in solving problems with relative position vectors relative to each other, to characterize which a concept such as a scalar product is used.

Definition of Euclidean space

Definition 1. A real linear space is called Euclidean, If it defines an operation that associates any two vectors x And y from this space number called the scalar product of vectors x And y and designated(x,y), for which the following conditions are met:

1. (x,y) = (y,x);

2. (x + y,z) = (x,z) + (y,z) , where z- any vector belonging to a given linear space;

3. (?x,y) = ? (x,y) , where ? - any number;

4. (x,x) ? 0 , and (x,x) = 0 x = 0.

For example, in a linear space of single-column matrices, the scalar product of vectors

can be determined by the formula

Euclidean dimension space n denote En. notice, that There are both finite-dimensional and infinite-dimensional Euclidean spaces.

Definition 2. Length (modulus) of vector x in Euclidean space En called (x,x) and denote it like this: |x| = (x,x). For any vector of Euclidean spacethere is a length, and the zero vector has it equal to zero.

Multiplying a non-zero vector x per number , we get a vector, length which is equal to one. This operation is called rationing vector x.

For example, in the space of single-column matrices the length of the vector can be determined by the formula:

Cauchy-Bunyakovsky inequality

Let x? En and y? En – any two vectors. Let us prove that the inequality holds for them:

(Cauchy-Bunyakovsky inequality)

Proof. Let be? - any real number. It's obvious that (?x ? y,?x ? y) ? 0. On the other hand, due to the properties of the scalar product we can write

Got that

The discriminant of this quadratic trinomial cannot be positive, i.e. , from which it follows:

The inequality has been proven.

Triangle inequality

Let x And y- arbitrary vectors of the Euclidean space En, i.e. x? En and y? En.

Let's prove that . (Triangle inequality).

Proof. It's obvious that On the other side,. Taking into account the Cauchy-Bunyakovsky inequality, we obtain

The triangle inequality has been proven.

Norm of Euclidean space

Definition 1 . Linear space ?called metric, if any two elements of this space x And y matched non-negativenumber? (x,y), called the distance between x And y , (? (x,y)? 0), and are executedconditions (axioms):

1) ? (x,y) = 0 x = y

2) ? (x,y) = ? (y,x)(symmetry);

3) for any three vectors x, y And z this space? (x,y) ? ? (x,z) + ? (z,y).

Comment. Elements of a metric space are usually called points.

The Euclidean space En is metric, and as the distance between vectors x? En and y? En can be taken x ? y.

So, for example, in the space of single-column matrices, where

hence

Definition 2 . Linear space?called normalized, If each vector x from this space is associated with a non-negative number called it the norm x. In this case, the axioms are satisfied:

It is easy to see that a normed space is a metric space stvom. In fact, as the distance between x And y can be taken . In Euclideanspace En as the norm of any vector x? En is its length, those. .

So, the Euclidean space En is a metric space and, moreover, The Euclidean space En is a normed space.

Angle between vectors

Definition 1 . Angle between non-zero vectors a And b Euclidean spacequality E n name the number for which

Definition 2 . Vectors x And y Euclidean space En are called orthogonlinen, if equality holds for them (x,y) = 0.

If x And y- are non-zero, then from the definition it follows that the angle between them is equal

Note that the zero vector is, by definition, considered orthogonal to any vector.

Example . In geometric (coordinate) space?3, which is a special case of Euclidean space, unit vectors i, j And k mutually orthogonal.

Orthonormal basis

Definition 1 . Basis e1,e2 ,...,en the Euclidean space En is called orthogonlinen, if the vectors of this basis are pairwise orthogonal, i.e. If

Definition 2 . If all vectors of the orthogonal basis e1, e2 ,...,en are unitary, i.e. e i = 1 (i = 1,2,...,n) , then the basis is called orthonormal, i.e. Fororthonormal basis

Theorem. (on the construction of an orthonormal basis)

In any Euclidean space E n there exist orthonormal bases.

Proof . Let us prove the theorem for the case n = 3.

Let E1 ,E2 ,E3 be some arbitrary basis of the Euclidean space E3 Let's construct some orthonormal basisin this space.Let's put where ? - some real number that we chooseso that (e1 ,e2 ) = 0, then we get

and what is obvious? = 0 if E1 and E2 are orthogonal, i.e. in this case e2 = E2, and , because this is the basis vector.

Considering that (e1 ,e2 ) = 0, we get

It is obvious that if e1 and e2 are orthogonal to the vector E3, i.e. in this case we should take e3 = E3. Vector E3? 0 because E1, E2 and E3 are linearly independent,therefore e3 ? 0.

In addition, from the above reasoning it follows that e3 cannot be represented in the form linear combination of vectors e1 and e2, therefore vectors e1, e2, e3 are linearly independentsims and are pairwise orthogonal, therefore, they can be taken as a basis for the Euclideanspace E3. All that remains is to normalize the constructed basis, for which it is sufficientdivide each of the constructed vectors by its length. Then we get

So we have built a basis - orthonormal basis. The theorem has been proven.

The applied method for constructing an orthonormal basis from an arbitrary basis is called orthogonalization process . Note that in the process of prooftheorem, we established that pairwise orthogonal vectors are linearly independent. Except if is an orthonormal basis in En, then for any vector x? Enthere is only one decomposition

where x1, x2,..., xn are the coordinates of the vector x in this orthonormal basis.

Because

then scalarly multiplying equality (*) by, we get .

In what follows we will consider only orthonormal bases, and therefore for ease of writing, zeroes are on top of the basis vectorswe will omit.

Euclidean spaces
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Chapter 4
EUCLIDAN SPACES

From the course of analytical geometry, the reader is familiar with the concept of the scalar product of two free vectors and with the four main properties of the specified scalar product. In this chapter, linear spaces of any nature are studied, for the elements of which a rule is defined in some way (and it doesn’t matter what) that associates any two elements with a number called the scalar product of these elements. In this case, it is only important that this rule has the same four properties as the rule for composing the scalar product of two free vectors. Linear spaces in which the specified rule is defined are called Euclidean spaces. This chapter explains the basic properties of arbitrary Euclidean spaces.

§ 1. Real Euclidean space and its simplest properties

1. Definition of real Euclidean space. A real linear space R is called real Euclidean space(or simply Euclidean space) if the following two requirements are met.
I. There is a rule by which any two elements of this space x and y are associated with a real number called scalar product of these elements and denoted by the symbol (x, y).
P. This rule is subject to the following four axioms:
1°. (x, y) = (y, x) (commutative property or symmetry);
2°. (x 1 + x 2, y) = (x 1, y) + (x 2, y) (distribution property);
3°. (λ x, y) = λ (x, y) for any real λ;
4°. (x, x) > 0 if x is a non-zero element; (x, x) = 0 if x is the zero element.
We emphasize that when introducing the concept of Euclidean space, we abstract not only from the nature of the objects under study, but also from the specific type of rules for the formation of the sum of elements, the product of an element by a number and the scalar product of elements (it is only important that these rules satisfy the eight axioms of linear space and the four axioms scalar product).
If the nature of the objects being studied and the type of the listed rules are indicated, then the Euclidean space is called specific.
Let us give examples of specific Euclidean spaces.
Example 1. Consider the linear space B 3 of all free vectors. We define the scalar product of any two vectors as it was done in analytical geometry (i.e., as the product of the lengths of these vectors and the cosine of the angle between them). In the course of analytical geometry, the validity of the so-defined scalar product of axioms 1°-4° was proven (see issue “Analytical Geometry”, Chapter 2, §2, item 3). Therefore, the space B 3 with the scalar product so defined is a Euclidean space.
Example 2. Consider the infinite-dimensional linear space C [a, b] of all functions x(t), defined and continuous on the segment a ≤ t ≤ b. We define the scalar product of two such functions x(t) and y(t) as the integral (in the range from a to b) of the product of these functions

The validity of the so-defined scalar product of axioms 1°-4° is checked in an elementary way. Indeed, the validity of axiom 1° is obvious; the validity of axioms 2° and 3° follows from the linear properties of the definite integral; the validity of axiom 4° follows from the fact that the integral of a continuous non-negative function x 2 (t) is non-negative and vanishes only when this function is identically equal to zero on the segment a ≤ t ≤ b (see the issue “Fundamentals of Mathematical Analysis”, part I, properties 1° and 2° from paragraph 1 §6 chapter 10) (i.e. it is the zero element of the space under consideration).
Thus, the space C[a, b] with the scalar product so defined is infinite-dimensional Euclidean space.
Example 3. The following example of Euclidean space gives an n-dimensional linear space A n of ordered collections of n real numbers, the scalar product of any two elements x = (x 1, x 2,..., x n) and y = (y 1, y 2 ,...,y n) which is defined by the equality

(x, y) = x 1 y 1 + x 2 y 2 + ... + x n y n. (4.2)

The validity of axiom 1° for such a defined scalar product is obvious; The validity of axioms 2° and 3° can be easily verified; just remember the definition of the operations of adding elements and multiplying them by numbers:

(x 1 , x 2 ,...,x n) + (y 1 , y 2 ,...,y n) = (x 1 + y 1 , x 2 + y 2 ,...,x n + y n) ,

λ (x 1, x 2,..., x n) = (λ x 1, λ x 2,..., λ x n);

finally, the validity of axiom 4° follows from the fact that (x, x) = x 1 2 + x 2 2 + ...+ x n 2 is always a non-negative number and vanishes only under the condition x 1 = x 2 = . .. = x n = 0.
The Euclidean space considered in this example is often denoted by the symbol E n.
Example 4. In the same linear space A n, we introduce the scalar product of any two elements x = (x 1, x 2,..., x n) and y = (y 1, y 2,..., y n) not relation (4.2), but in another, more general way.
To do this, consider a square matrix of order n

Using matrix (4.3), let us compose a homogeneous polynomial of the second order with respect to n variables x 1, x 2,..., x n

Looking ahead, we note that such a polynomial is called quadratic form(generated by matrix (4.3)) (quadratic forms are systematically studied in Chapter 7 of this book).
The quadratic form (4.4) is called positive definite if she takes it strictly positive values for all values ​​of the variables x 1, x 2,..., x n, which are not equal to zero at the same time (in Chapter 7 of this book the necessary and sufficient condition for the positive definiteness of the quadratic form will be indicated).
Since for x 1 = x 2 = ... = x n = 0 the quadratic form (4.4) is obviously equal to zero, we can say that positive definite
the quadratic form vanishes only under the condition x 1 = x 2 = ... = x n = 0.
We require that matrix (4.3) satisfy two conditions.
1°. Generated a positive definite quadratic form (4.4).
2°. It was symmetrical (relative to the main diagonal), i.e. satisfied the condition a ik = a ki for all i = 1, 2,..., n and k = I, 2,..., n.
Using matrix (4.3), satisfying conditions 1° and 2°, we define the scalar product of any two elements x = (x 1, x 2,..., x n) and y = (y 1, y 2,... ,y n) of space A n by the relation

It is easy to check the validity of the so-defined scalar product of all axioms 1°-4°. Indeed, axioms 2° and 3° are obviously valid for a completely arbitrary matrix (4.3); the validity of axiom 1° follows from the symmetry condition of the matrix (4.3), and the validity of axiom 4° follows from the fact that the quadratic form (4.4), which is the scalar product (x, x), is positive definite.
Thus, the space A n with the scalar product defined by equality (4.5), provided that the matrix (4.3) is symmetric and the quadratic form generated by it is positive definite, is a Euclidean space.
If we take the identity matrix as matrix (4.3), then relation (4.4) turns into (4.2), and we obtain the Euclidean space E n , considered in Example 3.
2. The simplest properties of an arbitrary Euclidean space. The properties established in this paragraph are valid for a completely arbitrary Euclidean space of both finite and infinite dimensions.
Theorem 4.1.For any two elements x and y of an arbitrary Euclidean space, the following inequality holds:

(x, y ) 2 ≤ (x, x )(y, y ), (4.6)

called the Cauchy-Bunyakovsky inequality.
Proof. For any real number λ, by virtue of axiom 4° of the scalar product, the inequality (λ x - y, λ x - y) > 0 is true. By virtue of axioms 1°-3°, the last inequality can be rewritten as

λ 2 (x, x) - 2 λ (x, y) + (y, y) ≤ 0

A necessary and sufficient condition for the non-negativity of the last quadratic trinomial is the non-positivity of its discriminant, i.e. the inequality (in the case (x, x) = 0 quadratic trinomial degenerates into a linear function, but in this case the element x is zero, so (x, y) = 0 and inequality (4.7) is also true)

(x, y ) 2 - (x, x )(y, y ) ≤ 0. (4.7)

Inequality (4.6) immediately follows from (4.7). The theorem has been proven.
Our next task is to introduce the concept norms(or length) of each element. To do this, we introduce the concept of a linear normed space.
Definition. The linear space R is called normalized, if the following two requirements are met.
I. There is a rule by which each element x of the space R is associated with a real number called the norm(or length) of the specified element and denoted by the symbol ||x||.
P. This rule is subject to the following three axioms:
1°. ||x|| > 0 if x is a non-zero element; ||x|| = 0 if x is a zero element;
2°. ||λ x|| = |λ | ||x|| for any element x and any real number λ;
3°. for any two elements x and y the following inequality is true

||x + y || ≤ ||x|| + ||y ||, (4.8)

called the triangle inequality (or Minkowski inequality).
Theorem 4.2. Any Euclidean space is normed if the norm of any element x in it is defined by the equality

Proof. It is enough to prove that for the norm defined by relation (4.9), axioms 1°-3° from the definition of a normed space are valid.
The validity of the norm of axiom 1° immediately follows from the axiom 4° of the scalar product. The validity of the norm of axiom 2° follows almost directly from axioms 1° and 3° of the scalar product.
It remains to verify the validity of Axiom 3° for the norm, i.e., inequality (4.8). We will rely on the Cauchy-Bunyakovsky inequality (4.6), which we will rewrite in the form

Using the last inequality, axioms 1°-4° of the scalar product and the definition of the norm, we obtain

The theorem has been proven.
Consequence. In any Euclidean space with the norm of elements determined by relation (4.9), for any two elements x and y the triangle inequality (4.8) holds.

We further note that in any real Euclidean space we can introduce the concept of an angle between two arbitrary elements x and y of this space. In complete analogy with vector algebra, we call angleφ between elements X And at that (varying from 0 to π) angle whose cosine is determined by the relation

Our definition of the angle is correct, because due to the Cauchy-Bunyakovsky inequality (4.7"), the fraction on the right side of the last equality does not exceed one in modulus.
Next, we will agree to call two arbitrary elements x and y of the Euclidean space E orthogonal if the scalar product of these elements (x, y) is equal to zero (in this case, the cosine of the angle (φ between the elements x and y will be equal to zero).
Again appealing to vector algebra, let's call the sum x + y of two orthogonal elements x and y the hypotenuse right triangle, built on the elements x and y.
Note that in any Euclidean space the Pythagorean theorem is valid: the square of the hypotenuse is equal to the sum of the squares of the legs. In fact, since x and y are orthogonal and (x, y) = 0, then by virtue of the axioms and definition of the norm

||x + y || 2 = ( x+y, x+y ) = (x, x ) + 2(x, y ) + (y, y) = (x,x) + (y, y) =||x|| 2 + ||y || 2.

This result is generalized to n pairwise orthogonal elements x 1, x 2,..., x n: if z = x 1 + x 2 + ...+ x n, then

||x|| 2 = (x 1 + x 2 + ...+ x n, x 1 + x 2 + ...+ x n) = (x 1, x 1) + (x 2, x 2) + .... + (x n,x n) = ||x 1 || 2 + ||x 1 || 2 +... +||x 1 || 2.

In conclusion, we write down the norm, the Cauchy-Bunyakovsky inequality and the triangle inequality in each of the specific Euclidean spaces considered in the previous paragraph.
In the Euclidean space of all free vectors with the usual definition of the scalar product, the norm of a vector a coincides with its length |a|, the Cauchy-Bunyakovsky inequality is reduced to the form ((a,b) 2 ≤ |a| 2 |b | 2, and the triangle inequality - to the form |a + b| ≤ |a| + |b | (If we add the vectors a and b according to the triangle rule, then this inequality trivially reduces to the fact that one side of a triangle does not exceed the sum of its two other sides).
In the Euclidean space C [a, b] of all functions x = x(t) continuous on the segment a ≤ t ≤ b with scalar product (4.1), the norm of the element x = x(t) is equal to , and the Cauchy-Bunyakovsky and triangle inequalities have the form

Both of these inequalities play important role in various sections of mathematical analysis.
In the Euclidean space E n of ordered collections of n real numbers with scalar product (4.2), the norm of any element x = (x 1 , x 2 ,..., x n) is equal

Finally, in the Euclidean space of ordered collections of n real numbers with scalar product (4.5), the norm of any element x = (x 1, x 2,..., x n) is equal to 0 (we remind you that in this case matrix (4.3) is symmetric and generates positive definite quadratic form (4.4)).

and the Cauchy-Bunyakovsky and triangle inequalities have the form