Euclidean understanding. Definition and examples of Euclidean spaces. See what “Euclidean space” is in other dictionaries

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 whose angle 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 length properties listed above, means that the length of a vector is a norm on 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 the scalar product defined as the integral of the product over a finite segment (or over the entire line, but with a rapidly decaying 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 ago, 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. The room, especially the sofa on which they were sitting, was dark, but big windows The silver light of the full month fell on 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.

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.

Euclidean space with practical point vision 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 a 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. In addition, in addition to the scalar sum of 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 the relative position of 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.

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

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

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✪ 04 - Linear algebra. Euclidean space

✪ Non-Euclidean geometry. Part one.

✪ Non-Euclidean geometry. Part two

✪ 01 - Linear algebra. Linear (vector) space

✪ 8. Euclidean spaces

Subtitles

Formal definition

To define Euclidean space, the easiest way is to take as the main concept the scalar product. 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 (⋅ , ⋅) , (\displaystyle (\cdot ,\cdot),) having the following three properties:

Example of Euclidean space - coordinate space R n , (\displaystyle \mathbb (R) ^(n),) consisting of all possible tuples of real numbers (x 1 , x 2 , … , x n) , (\displaystyle (x_(1),x_(2),\ldots ,x_(n)),) scalar product in which is determined by the formula (x , y) = ∑ i = 1 n x i y i = x 1 y 1 + x 2 y 2 + ⋯ + x n y n . (\displaystyle (x,y)=\sum _(i=1)^(n)x_(i)y_(i)=x_(1)y_(1)+x_(2)y_(2)+\cdots +x_(n)y_(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 (\displaystyle u) defined as (u , u) (\displaystyle (\sqrt ((u,u)))) and is designated | u | . (\displaystyle |u|.) The positive definiteness of the scalar product guarantees that the length of the nonzero vector is nonzero, and from bilinearity it follows that | a u | = | a | | u | , (\displaystyle |au|=|a||u|,) that is, the lengths of proportional vectors are proportional.

Angle between vectors u (\displaystyle u) And v (\displaystyle v) determined by the formula φ = arccos ⁡ ((x , y) | x | | y |) . (\displaystyle \varphi =\arccos \left((\frac ((x,y))(|x||y|))\right).) From the cosine theorem it follows that for a two-dimensional Euclidean space ( Euclidean plane) this definition of 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 π 2. (\displaystyle (\frac (\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 arccos ⁡ ((x , y) | x | | y |) (\displaystyle \arccos \left((\frac ((x,y))(|x||y|))\right)) has been defined, it is necessary that the inequality | (x, y) | x | | y | | ⩽ 1. (\displaystyle \left|(\frac ((x,y))(|x||y|))\right|\leqslant 1.) This inequality actually holds in an arbitrary Euclidean space; it is called the Cauchy-Bunyakovsky-Schwartz inequality. From this inequality, in turn, follows the triangle inequality: | u + v | ⩽ | u | + | v | . (\displaystyle |u+v|\leqslant |u|+|v|.) The triangle inequality, together with the length properties listed above, means that the length of a vector is a norm on Euclidean vector space, and the function d(x, y) = | x − y | (\displaystyle 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 (\displaystyle x) And y (\displaystyle y) coordinate space R n (\displaystyle \mathbb (R) ^(n)) is given by the formula d (x , y) = ‖ x − y ‖ = ∑ i = 1 n (x i − y i) 2 . (\displaystyle d(\mathbf (x) ,\mathbf (y))=\|\mathbf (x) -\mathbf (y) \|=(\sqrt (\sum _(i=1)^(n) (x_(i)-y_(i))^(2))).)

Algebraic properties

Orthonormal bases

Conjugate spaces and operators

Any vector x (\displaystyle x) Euclidean space defines a linear functional x ∗ (\displaystyle x^(*)) on this space, defined as x ∗ (y) = (x , y) . (\displaystyle x^(*)(y)=(x,y).) This mapping is an isomorphism between Euclidean space and

§3. Dimension and basis of vector space

Linear combination of vectors

Trivial and non-trivial linear combination

Linearly dependent and linearly independent vectors

Properties of vector space associated with linear dependence of vectors

P-dimensional vector space

Dimension of vector space

Decomposition of a vector into a basis

§4. Transition to a new basis

Transition matrix from the old basis to the new one

Vector coordinates in the new basis

§5. Euclidean space

Scalar product

Euclidean space

Length (norm) of the vector

Properties of vector length

Angle between vectors

Orthogonal vectors

Orthonormal basis

§ 3. Dimension and basis of vector space

Consider some vector space (V, Å, ∘) over the field R. Let be some elements of the set V, i.e. vectors.

Linear combination vectors is any vector equal to the sum of the products of these vectors by arbitrary elements of the field R(i.e. on scalars):

If all scalars are equal to zero, then such a linear combination is called trivial(the simplest), and .

If at least one scalar is nonzero, the linear combination is called non-trivial.

The vectors are called linearly independent, if only the trivial linear combination of these vectors is equal to:

The vectors are called linearly dependent, if there is at least one non-trivial linear combination of these vectors equal to .

Example. Consider the set of ordered sets of quadruples of real numbers - this is a vector space over the field of real numbers. Task: find out whether the vectors are , And linearly dependent.

Solution.

Let's make a linear combination of these vectors: , where are unknown numbers. We require that this linear combination be equal to the zero vector: .

In this equality we write the vectors as columns of numbers:

If there are numbers for which this equality holds, and at least one of the numbers is not equal to zero, then this is a non-trivial linear combination and the vectors are linearly dependent.

Let's do the following:

Thus, the problem is reduced to solving a system of linear equations:

Solving it, we get:

The ranks of the extended and main matrices of the system are equal and less than the number of unknowns, therefore, the system has an infinite number of solutions.

Let , then and .

So, for these vectors there is a non-trivial linear combination, for example at , which is equal to the zero vector, which means that these vectors are linearly dependent.

Let's note some properties of vector space associated with linear dependence of vectors:

1. If the vectors are linearly dependent, then at least one of them is a linear combination of the others.

2. If among the vectors there is a zero vector, then these vectors are linearly dependent.

3. If some of the vectors are linearly dependent, then all of these vectors are linearly dependent.

The vector space V is called P-dimensional vector space, if it contains P linearly independent vectors, and any set of ( P+ 1) vectors is linearly dependent.

Number P called dimension of the vector space, and is denoted dim(V) from the English “dimension” - dimension (measurement, size, dimension, size, length, etc.).

Totality P linearly independent vectors P-dimensional vector space is called basis.

(*)

Theorem(about the decomposition of a vector by basis): Each vector of a vector space can be represented (and in a unique way) as a linear combination of basis vectors:

The formula (*) is called vector decomposition by basis, and the numbers – vector coordinates in this basis .

A vector space can have more than one or even infinitely many bases. In each new basis, the same vector will have different coordinates.

§ 4. Transition to a new basis

In linear algebra, the problem often arises of finding the coordinates of a vector in a new basis if its coordinates in the old basis are known.

Let's look at some P-dimensional vector space (V, +, ·) over the field R. Let there be two bases in this space: old and new .

Task: find the coordinates of the vector in the new basis.

Let the vectors of the new basis in the old basis have the expansion:

,

Let's write the coordinates of the vectors into the matrix not in rows, as they are written in the system, but in columns:

The resulting matrix is ​​called transition matrix from the old basis to the new.

The transition matrix connects the coordinates of any vector in the old and new basis by the following relation:

,

where are the desired coordinates of the vector in the new basis.

Thus, the task of finding the vector coordinates in a new basis is reduced to solving the matrix equation: , where X– matrix-column of vector coordinates in the old basis, A– transition matrix from the old basis to the new one, X* – the required matrix-column of vector coordinates in the new basis. From the matrix equation we get:

So, vector coordinates in a new basis are found from the equality:

.

Example. In a certain basis, the vector decompositions are given:

Find the coordinates of the vector in the basis.

Solution.

1. Let’s write out the transition matrix to a new basis, i.e. We will write the coordinates of the vectors in the old basis in columns:

2. Find the matrix A –1:

3. Perform multiplication , where are the coordinates of the vector:

Answer: .

§ 5. Euclidean space

Let's look at some P-dimensional vector space (V, +, ·) over the field of real numbers R. Let be some basis of this space.

Let us introduce in this vector space metric, i.e. Let's determine a method for measuring lengths and angles. To do this, we define the concept of a scalar product.