The simplest consequences from the definition of a vector space. Vector space

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Vector (or linear) space - a mathematical structure, which is a set of elements called vectors, for which the operations of addition with each other and multiplication by a number are defined - a scalar. These operations are subject to eight axioms. Scalars can be elements of real, complex, or any other number field. A special case of such a space is the usual three-dimensional Euclidean space, the vectors of which are used, for example, to represent physical forces. It should be noted that the vector as an element of the vector space does not have to be specified in the form of a directed segment. Generalization of the concept "vector" to an element of a vector space of any nature not only does not cause confusion of terms, but also allows one to understand or even foresee a number of results that are valid for spaces of arbitrary nature.

Vector spaces are the subject of the study of linear algebra. One of the main characteristics of a vector space is its dimension. Dimension is the maximum number of linearly independent elements of space, that is, using a rough geometric description, the number of directions that cannot be expressed through each other by means of only addition and multiplication by a scalar. Vector space can be endowed with additional structures, for example, the norm or the dot product. Such spaces naturally appear in mathematical analysis, mainly in the form of infinite-dimensional function spaces ( english), where functions are used as vectors. Many analysis problems require finding out whether a sequence of vectors converges to a given vector. Consideration of such questions is possible in vector spaces with an additional structure, in most cases with a suitable topology, which allows one to define the concepts of proximity and continuity. Such topological vector spaces, in particular, Banach and Hilbert spaces, allow deeper study.

In addition to vectors, linear algebra also studies tensors of higher rank (a scalar is considered a rank 0 tensor, a vector is considered a rank 1 tensor).

The first works that anticipated the introduction of the concept of vector space date back to the 17th century. It was then that analytical geometry, the doctrine of matrices, systems of linear equations, and Euclidean vectors were developed.

Definition

Linear, or vector space $V\; \backslash \backslash \; left\; (F\; \backslash \backslash \; right)$ over the field $F$ is an ordered four $(V,\; F,\; +,\; \backslash \backslash \; cdot)$where

• $V$ - a non-empty set of elements of arbitrary nature, which are called vectors;
• $F$ - (algebraic) field, the elements of which are called scalars;
• Operation defined additions vectors $V\; \backslash \backslash \; times\; V\; \backslash \backslash \; to\; V$which assigns to each pair of elements $\backslash \backslash \; mathbf\; (x),\; \backslash \backslash \; mathbf\; (y)$ multitudes $V$ $V$called them sum and designated $\backslash \backslash \; mathbf\; (x)\; +\; \backslash \backslash \; mathbf\; (y)$;
• Operation defined multiplication of vectors by scalars $F\; \backslash \backslash \; times\; V\; \backslash \backslash \; to\; V$matching each element $\backslash \backslash \; lambda$ fields $F$ and each element $\backslash \backslash \; mathbf\; (x)$ multitudes $V$ single element of the set $V$designated $\backslash \backslash \; lambda\; \backslash \backslash \; cdot\; \backslash \backslash \; mathbf\; (x)$ or $\backslash \backslash \; lambda\; \backslash \backslash \; mathbf\; (x)$;

Vector spaces defined on the same set of elements, but over different fields, will be different vector spaces (for example, the set of pairs of real numbers $\backslash \backslash \; mathbb\; (R)\; ^\; 2$ can be a two-dimensional vector space over the field of real numbers or one-dimensional over the field of complex numbers).

The simplest properties

1. Vector space is an abelian addition group.
2. Neutral element $\backslash \backslash \; mathbf\; (0)\; \backslash \backslash \; in\; V$
3. $0\; \backslash \backslash \; cdot\; \backslash \backslash \; mathbf\; (x)\; \backslash u003d\; \backslash \backslash \; mathbf\; (0)$ for anyone $\backslash \backslash \; mathbf\; (x)\; \backslash \backslash \; in\; V$.
4. For anyone $\backslash \backslash \; mathbf\; (x)\; \backslash \backslash \; in\; V$ opposite element $-\; \backslash \backslash \; mathbf\; (x)\; \backslash \backslash \; in\; V$ is the only thing that follows from the group properties.
5. $1\; \backslash \backslash \; cdot\; \backslash \backslash \; mathbf\; (x)\; \backslash u003d\; \backslash \backslash \; mathbf\; (x)$ for anyone $\backslash \backslash \; mathbf\; (x)\; \backslash \backslash \; in\; V$.
6. $(-\; \backslash \backslash \; alpha)\; \backslash \backslash \; cdot\; \backslash \backslash \; mathbf\; (x)\; \backslash u003d\; \backslash \backslash \; alpha\; \backslash \backslash \; cdot\; (-\; \backslash \backslash \; mathbf\; (x))\; \backslash u003d\; -\; (\backslash \backslash \; alpha\; \backslash \backslash \; mathbf\; (x))$ for any $\backslash \backslash \; alpha\; \backslash \backslash \; in\; F$ and $\backslash \backslash \; mathbf\; (x)\; \backslash \backslash \; in\; V$.
7. $\backslash \backslash \; alpha\; \backslash \backslash \; cdot\; \backslash \backslash \; mathbf\; (0)\; \backslash u003d\; \backslash \backslash \; mathbf\; (0)$ for anyone $\backslash \backslash \; alpha\; \backslash \backslash \; in\; F$.

Related definitions and properties

Subspace

Algebraic definition: Linear subspace or vector subspace - non-empty subset $K$ linear space $V$ such that $K$ itself is a linear space in relation to the $V$ operations of addition and multiplication by a scalar. The set of all subspaces is usually denoted as $\backslash \backslash \; mathrm\; (Lat)\; (V)$... For a subset to be a subspace, it is necessary and sufficient that

1. for any vector $\backslash \backslash \; mathbf\; (x)\; \backslash \backslash \; in\; K$, vector $\backslash \backslash \; alpha\; \backslash \backslash \; mathbf\; (x)$ also belonged $K$, for any $\backslash \backslash \; alpha\; \backslash \backslash \; in\; F$;
2. for any vectors $\backslash \backslash \; mathbf\; (x),\; \backslash \backslash \; mathbf\; (y)\; \backslash \backslash \; in\; K$, vector $\backslash \backslash \; mathbf\; (x)\; +\; \backslash \backslash \; mathbf\; (y)$ also belonged $K$.

The last two statements are equivalent to the following:

For any vectors $\backslash \backslash \; mathbf\; (x),\; \backslash \backslash \; mathbf\; (y)\; \backslash \backslash \; in\; K$, vector $\backslash \backslash \; alpha\; \backslash \backslash \; mathbf\; (x)\; +\; \backslash \backslash \; beta\; \backslash \backslash \; mathbf\; (y)$ also belonged $K$ for any $\backslash \backslash \; alpha,\; \backslash \backslash \; beta\; \backslash \backslash \; in\; F$.

In particular, a vector space consisting of only one zero vector is a subspace of any space; any space is a subspace of itself. Subspaces that do not coincide with these two are called own or non-trivial.

Properties of subspaces

• The intersection of any family of subspaces is again a subspace;
• Sum of subspaces $\backslash \backslash \; (K\_i\; \backslash \backslash \; quad\; |\; \backslash \backslash \; quad\; i\; \backslash \backslash \; in\; 1\; \backslash \backslash \; ldots\; N\; \backslash \backslash )$ defined as a set containing all possible sums of elements $K\_i$: $\backslash \backslash \; sum\_\; (i\; \backslash u003d\; 1)\; ^\; N\; (K\_i):\; \backslash u003d\; \backslash \backslash \; (\backslash \backslash \; mathbf\; (x)\; \_1\; +\; \backslash \backslash \; mathbf\; (x)\; \_2\; +\; \backslash \backslash \; ldots\; +\; \backslash \backslash \; mathbf\; (x)\; \_N\; \backslash \backslash \; quad\; |\; \backslash \backslash \; quad\; \backslash \backslash \; mathbf\; (x)\; \_i\; \backslash \backslash \; in\; K\_i\; \backslash \backslash \; quad\; (i\; \backslash \backslash \; in\; 1\; \backslash \backslash \; ldots\; N)\; \backslash \backslash )$.
  • The sum of a finite family of subspaces is again a subspace.

Linear combinations

The final sum of the form

$\backslash \backslash \; alpha\_1\; \backslash \backslash \; mathbf\; (x)\; \_1\; +\; \backslash \backslash \; alpha\_2\; \backslash \backslash \; mathbf\; (x)\; \_2\; +\; \backslash \backslash \; ldots\; +\; \backslash \backslash \; alpha\_n\; \backslash \backslash \; mathbf\; (x)\; \_n$

The linear combination is called:

Basis. Dimension

Vectors $\backslash \backslash \; mathbf\; (x)\; \_1,\; \backslash \backslash \; mathbf\; (x)\; \_2,\; \backslash \backslash \; ldots,\; \backslash \backslash \; mathbf\; (x)\; \_n$ are called linearly dependentif there is a nontrivial linear combination of them equal to zero:

$\backslash \backslash \; alpha\_1\; \backslash \backslash \; mathbf\; (x)\; \_1\; +\; \backslash \backslash \; alpha\_2\; \backslash \backslash \; mathbf\; (x)\; \_2\; +\; \backslash \backslash \; ldots\; +\; \backslash \backslash \; alpha\_n\; \backslash \backslash \; mathbf\; (x)\; \_n\; \backslash u003d\; \backslash \backslash \; mathbf\; (0),\; \backslash \backslash \; quad\; \backslash \backslash \; |\; \backslash \backslash \; alpha\_1\; |\; +\; |\; \backslash \backslash \; alpha\_2\; |\; +\; \backslash \backslash \; ldots\; +\; |\; \backslash \backslash \; alpha\_n\; |\; \backslash \backslash \; neq\; 0.$

Otherwise, these vectors are called linearly independent.

This definition admits the following generalization: an infinite set of vectors from $V$ called linearly dependentif some the final a subset of it, and linearly independentif any of it the final the subset is linearly independent.

Basis properties:

• Any $n$ linearly independent elements $n$-dimensional space form basis this space.
• Any vector $\backslash \backslash \; mathbf\; (x)\; \backslash \backslash \; in\; V$ can be represented (uniquely) as a finite linear combination of basic elements:

$\backslash \backslash \; mathbf\; (x)\; \backslash u003d\; \backslash \backslash \; alpha\_1\; \backslash \backslash \; mathbf\; (x)\; \_1\; +\; \backslash \backslash \; alpha\_2\; \backslash \backslash \; mathbf\; (x)\; \_2\; +\; \backslash \backslash \; ldots\; +\; \backslash \backslash \; alpha\_n\; \backslash \backslash \; mathbf\; (x)\; \_n$.

Linear shell

Linear shell $\backslash \backslash \; mathcal\; V\; (X)$ subsets $X$ linear space $V$ - intersection of all subspaces $V$containing $X$.

The linear hull is a subspace $V$.

Linear shell is also called subspace generated by $X$... They also say that the linear hull $\backslash \backslash \; mathcal\; V\; (X)$ - space, stretched over a bunch of $X$.

Linear shell $\backslash \backslash \; mathcal\; V\; (X)$ consists of all kinds of linear combinations of various finite subsystems of elements from $X$... In particular, if $X$ is a finite set, then $\backslash \backslash \; mathcal\; V\; (X)$ consists of all linear combinations of elements $X$... Thus, the zero vector always belongs to the linear hull.

If $X$ is a linearly independent set, then it is a basis $\backslash \backslash \; mathcal\; V\; (X)$ and thus determines its dimension.

Examples of

• Zero space whose only element is zero.
• Space for all functions $X\; \backslash \backslash \; to\; F$ with finite support forms a vector space of dimension equal to the cardinality $X$.
• The field of real numbers can be viewed as a continual-dimensional vector space over the field of rational numbers.
• Any field is a one-dimensional space above itself.

Additional structures

see also

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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.
• Gelfand I.M. Lectures on linear algebra. 5th ed. - M .: Dobrosvet, MTsNMO, 1998 .-- 320 p. - ISBN 5-7913-0016-6.
• A. I. Kostrikin, Yu. I. Manin Linear Algebra and Geometry. 2nd ed. - M .: Nauka, 1986 .-- 304 p.
• A. I. Kostrikin Introduction to algebra. Part 2: Linear algebra. - 3rd. - M .: Nauka., 2004 .-- 368 p. - (University textbook).
• Maltsev A.I. Fundamentals of Linear Algebra. - 3rd. - M .: Nauka, 1970 .-- 400 p.

• Postnikov M.M. Linear Algebra (Lectures on Geometry. Semester II). - 2nd. - M .: Nauka, 1986 .-- 400 p.
• Strang G. Linear Algebra and Its Applications. - M .: Mir, 1980 .-- 454 p.
• Ilyin V.A., Poznyak E.G. Linear algebra. 6th ed. - Moscow: Fizmatlit, 2010 .-- 280 p. - ISBN 978-5-9221-0481-4.
• Halmos P. Finite-Dimensional Vector Spaces. - Moscow: Fizmatgiz, 1963 .-- 263 p.
• Faddeev D.K. Lectures on algebra. - 5th. - SPb. : Lan, 2007 .-- 416 p.
• Shafarevich I.R., Remizov A.O. Linear Algebra and Geometry. - 1st. - Moscow: Fizmatlit, 2009 .-- 511 p.
• Schreyer O., Sperner G. Introduction to linear algebra in geometric presentation \u003d Einfuhrung in die analytische Geometrie und Algebra / Olshansky G. (translated from German). - M. – L .: ONTI, 1934 .-- 210 p.

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

An excerpt characterizing Vector space

Kutuzov walked through the ranks, occasionally stopping and speaking a few kind words to the officers he knew from the Turkish war, and sometimes to the soldiers. Looking at the shoes, he several times sadly shook his head and pointed at them to the Austrian general with such an expression that, as it were, he did not reproach anyone for this, but he could not help but see how bad it was. The regimental commander ran ahead each time, fearing to miss the word of the commander-in-chief about the regiment. Behind Kutuzov, at such a distance that every weakly spoken word could be heard, walked about 20 of his suite. The gentlemen of the retinue talked among themselves and sometimes laughed. The handsome adjutant was closest to the commander-in-chief. It was Prince Bolkonsky. Next to him walked his comrade Nesvitsky, a high staff officer, extremely fat, with a kind and smiling handsome face and wet eyes; Nesvitsky could hardly restrain himself from laughing, aroused by the blackish hussar officer walking beside him. The hussar officer, without smiling, without changing the expression of his stopped eyes, looked with a serious face at the back of the regimental commander and imitated his every movement. Each time the regimental commander shuddered and bent forward, the hussar officer shuddered and bent forward in exactly the same way. Nesvitsky laughed and pushed others to look at the amusing man.
Kutuzov walked slowly and listlessly past a thousand eyes, which rolled out of their orbits, following the chief. After catching up with the 3rd company, he suddenly stopped. The retinue, not anticipating this stop, involuntarily moved towards him.
- Ah, Timokhin! - said the commander-in-chief, recognizing the captain with a red nose, injured for a blue overcoat.
It seemed that it was impossible to stretch out more than Timokhin stretched out, while the regimental commander made a remark to him. But at that moment of the commander-in-chief's address to him, the captain stretched out so that, it seemed, had the commander-in-chief looked at him for a few more time, the captain would not have resisted; and therefore Kutuzov, apparently understanding his position and wishing, on the contrary, every good to the captain, hastily turned away. A barely perceptible smile ran across Kutuzov's plump face, disfigured by a wound.
“Another Izmailovsky comrade,” he said. - Brave officer! Are you satisfied with him? - Kutuzov asked the regimental commander.
And the regimental commander, reflected as in a mirror, invisibly to himself, in a hussar officer, shuddered, walked forward and answered:
“I am very pleased, Your Excellency.
“We are all not without weaknesses,” said Kutuzov, smiling and moving away from him. - He had a commitment to Bacchus.
The regimental commander was frightened if he was to blame for this, and did not answer. The officer at that moment noticed the face of the captain with a red nose and a tucked up belly and so similarly mimicked his face and posture that Nesvitsky could not help laughing.
Kutuzov turned around. It was evident that the officer could control his face as he wanted: the minute Kutuzov turned around, the officer managed to make a grimace, and then take on the most serious, respectful and innocent expression.
The third company was the last, and Kutuzov pondered, apparently remembering something. Prince Andrew stepped out of the suite and said in French quietly:
- You ordered to remind about the demoted Dolokhov in this regiment.
- Where is Dolokhov? - asked Kutuzov.
Dolokhov, already dressed in a gray soldier's overcoat, did not wait to be summoned. The slender figure of a blond soldier with clear blue eyes emerged from the front. He went up to the commander-in-chief and made a guard.
- A claim? - Frowning slightly, asked Kutuzov.
“This is Dolokhov,” said Prince Andrey.
- A! - said Kutuzov. “I hope this lesson will correct you, serve well. The sovereign is merciful. And I will not forget you if you deserve it.
Blue, clear eyes looked at the commander-in-chief as boldly as at the regimental commander, as if by their expression were tearing the veil of convention that separated the commander-in-chief from the soldier so far.
“One thing I ask, your Excellency,” he said in his sonorous, firm, unhurried voice. “I ask you to give me a chance to make amends for my guilt and prove my loyalty to the Emperor and Russia.
Kutuzov turned away. His face flashed the same smile of eyes as when he turned away from Captain Timokhin. He turned away and winced, as if he wanted to express by this that everything that Dolokhov said to him, and everything that he could tell him, he has known for a long time, for a long time, that all this has already bored him and that all this is not at all what is needed ... He turned away and went to the wheelchair.
The regiment sorted out in companies and went to the assigned apartments not far from Braunau, where he hoped to put on shoes, dress and rest after difficult transitions.
“You don’t pretend to me, Prokhor Ignatyich? - said the regimental commander, bypassing the 3rd company that was moving towards the place and approaching the captain Timokhin who was walking in front of it. The regimental commander's face expressed uncontrollable joy after the happily served review. - The tsarist service ... you can't ... another time in the front you will cut off ... I'll apologize myself first, you know me ... Thank you very much! And he held out his hand to the company commander.
- Have mercy, General, but dare I! - answered the captain, blushing his nose, smiling and revealing with a smile the lack of two front teeth, knocked out by the butt under Ishmael.
- Yes, tell Mr. Dolokhov that I will not forget him, so that he was calm. Tell me, please, I still wanted to ask, what is he, how is he behaving? And that's it ...
- He's very good in service, your Excellency ... but the karakhter ... - Timokhin said.
- And what, what character? The regimental commander asked.
- He finds, your excellency, for days, - said the captain, - that he is clever and learned and kind. And then the beast. In Poland he killed a Jew, if you please know ...
- Well, yes, well, yes, - said the regimental commander, - we must all feel sorry for the young man in misfortune. After all, great connections ... So you that ...
“Yes, your Excellency,” said Timokhin, making him feel with a smile that he understands the boss's wishes.
- Yes Yes.
The regimental commander found Dolokhov in the ranks and held the horse back.
- Before the first case - epaulettes, - he told him.
Dolokhov looked around, said nothing and did not change the expression of his mockingly smiling mouth.
“Well, that's good,” continued the regimental commander. “People have a glass of vodka from me,” he added so that the soldiers could hear. - Thank you all! Thank God! - And he, having overtaken the company, drove up to another.
- Well, he's really a good man; you can serve with him, ”Timokhin said to the subaltern to the officer who was walking beside him.
- One word, red! ... (the regimental commander was nicknamed the king of hearts) - the subaltern officer said laughing.
The happy mood of the authorities after the review passed on to the soldiers. The company went on merrily. Soldiers' voices spoke from all sides.
- How did they say, Kutuzov crooked, about one eye?
- And then no! All the curve.
“Don't… brother, you’re bigger. Boots and rolls - I looked around ...
- How he, my brother, will look at my feet ... well! I think ...
- And then the other Austrian, with him was, as if smeared with chalk. Like flour, white. I tea, they clean the ammunition!
- What, Fedeshaw! ... did he say that when the guards began, you were standing closer? They said everything, Bunaparte himself stands in Brunov.
- Bunaparte is worth it! you lie, you fool! What he doesn't know! Now the Prussian is revolting. The Austrian, therefore, pacifies him. As he reconciles, then the war will open with Bunapart. And that, he says, is in Brunov Bunaparte! Then it is clear that he is a fool. Listen more.
- See the devil's lodgers! The fifth company, look, is already turning into the village, they will cook porridge, and we will not reach the place yet.
- Give me a crouton, devil.
- Did you give tobacco yesterday? That's that, brother. Well, on, God be with you.
- If only we made a halt, otherwise we won't eat another five miles.
- It was very pleasant how the Germans gave us carriages. You go, know: important!
- And here, brother, the people went completely wild. Everything there seemed to be a Pole, everything was of the Russian crown; and now, brother, a solid German has gone.
- Songbooks forward! The captain shouted.
And twenty people ran out in front of the company from different rows. The drummer sang turned around to face the songwriters, and, waving his hand, began to draw out a drawn-out soldier's song, which began: "Isn't it dawn, the sun was busy ..." and ended with the words: "Then, brothers, there will be glory to us with Kamensky father ..." This song was folded in Turkey and was sung now in Austria, only with the change that in place of the "Kamensky father" the words were inserted: "Kutuzov's father."
Tearing off these last words in a soldier's manner and waving his hands as if he were throwing something on the ground, the drummer, a dry and handsome soldier of about forty years old, sternly glanced at the songwriters and closed his eyes. Then, making sure that all eyes were fixed on him, he seemed to carefully raise some invisible, precious thing over his head with both hands, held it like that for several seconds and suddenly desperately threw it away:
Oh, you, my canopy, canopy!
"My new canopy ...", picked up twenty voices, and the spoon-maker, despite the weight of the ammunition, briskly jumped forward and went backwards in front of the company, moving his shoulders and threatening someone with spoons. The soldiers, swinging their arms to the beat of the song, walked with a spacious step, involuntarily falling into the leg. Behind the company came the sound of wheels, the crunching of springs and the stamping of horses.
Kutuzov with his retinue was returning to the city. The commander-in-chief gave a sign that the people should continue to march at ease, and on his face and on all the faces of his retinue, pleasure was expressed at the sound of the song, at the sight of a dancing soldier and merrily and briskly walking company soldiers. In the second row, from the right flank, from which the carriage overtook the companies, the blue-eyed soldier Dolokhov involuntarily caught the eye, who walked especially briskly and gracefully to the beat of the song and looked at the faces of those passing by with such an expression as if he pitied everyone who did not go at this time with the company. A hussar cornet from Kutuzov's retinue, mimicking the regimental commander, left the carriage and drove up to Dolokhov.
Hussar cornet Zherkov at one time in St. Petersburg belonged to that violent society, which was led by Dolokhov. Abroad Zherkov met Dolokhov as a soldier, but did not consider it necessary to recognize him. Now, after Kutuzov's conversation with the demoted one, he, with the joy of an old friend, turned to him:
- Friend of heart, how are you? - he said at the sound of the song, even the pace of his horse with the pace of the company.
- I am like? - answered Dolokhov coldly, - as you can see.
The lively song attached particular importance to the tone of cheeky gaiety with which Zherkov spoke, and the deliberate coldness of Dolokhov's answers.
- Well, how are you getting along with your superiors? Zherkov asked.
- Nothing, good people. How did you get into the headquarters?
- seconded, on duty.
They were silent.
“Letting the falcon out of the right sleeve,” the song said, involuntarily arousing a cheerful, cheerful feeling. Their conversation would probably have been different if they had not spoken at the sound of a song.
- Is it true, the Austrians were beaten? Dolokhov asked.
- And the devil knows them, they say.
- I'm glad, - Dolokhov answered shortly and clearly, as the song demanded.
- Well, come to us when in the evening, you will lay the Pharaoh, - said Zherkov.
- Or have you got a lot of money?
- Come.
- You can't. Zarok gave it. I don’t drink or play until it’s done.
- Well, before the first case ...
- It will be seen there.
They were silent again.
- You come in, if you need anything, everyone in the headquarters will help ... - said Zherkov.
Dolokhov chuckled.
“You better not worry. I will not ask what I need, I will take it myself.
- Well, I am so ...
- Well, I do.
- Goodbye.
- Be healthy…
... and high and far,
On the home side ...
Zherkov touched the horse with his spurs, which three times, hot, kicked him, not knowing where to start, coped and galloped, overtaking the company and overtaking the carriage, also in time to the song.

Returning from the inspection, Kutuzov, accompanied by the Austrian general, went into his office and, having called the adjutant, ordered some papers related to the state of the arriving troops, and letters received from Archduke Ferdinand, who commanded the advanced army, to be submitted. Prince Andrey Bolkonsky entered the commander-in-chief's office with the required papers. In front of the plan spread out on the table sat Kutuzov and an Austrian member of the Hofkrigsrat.
"Ah ..." said Kutuzov, looking back at Bolkonsky, as if by this word inviting the adjutant to wait, and continued the conversation in French.
“I’m only saying one thing, General,” said Kutuzov with a pleasant grace of expression and intonation that made him listen attentively to every leisurely spoken word. It was evident that Kutuzov himself was listening to himself with pleasure. - I only say one thing, General, that if the matter depended on my personal desire, then the will of His Majesty Emperor Franz would have been fulfilled long ago. I would have joined the Archduke long ago. And believe me in my honor that for me personally to transfer the higher command of the army to a more knowledgeable and skillful general, which Austria is so abundant, and to give up all this heavy responsibility for me personally would be a joy. But circumstances are stronger than we are, General.
And Kutuzov smiled with such an expression as if he were saying: “You have every right not to believe me, and even I do not care whether you believe me or not, but you have no reason to tell me this. And that's the whole point. "
The Austrian general looked displeased, but he could not answer Kutuzov in the same tone.
“On the contrary,” he said in a grumpy and angry tone that so contradicted the flattering meaning of the words spoken, “on the contrary, your Excellency's participation in a common cause is highly valued by His Majesty; but we believe that a real slowdown deprives the glorious Russian troops and their commanders-in-chief of those laurels that they are used to reaping in battles, - he finished the apparently prepared phrase.
Kutuzov bowed without changing his smile.
- And I am so convinced, and based on the last letter that His Highness Archduke Ferdinand honored me with, I suppose that the Austrian troops, under the command of such a skillful assistant as General Mac, have now won a decisive victory and no longer need our help, - said Kutuzov.
The general frowned. Although there was no positive news of the defeat of the Austrians, there were too many circumstances to confirm the general unfavorable rumors; and therefore the assumption of Kutuzov about the victory of the Austrians was very similar to a mockery. But Kutuzov smiled meekly, all with the same expression that said that he had the right to assume this. Indeed, the last letter he received from Mac's army informed him of the victory and the most advantageous strategic position of the army.
“Give me this letter here,” said Kutuzov, addressing Prince Andrey. - If you please see. - And Kutuzov, with a mocking smile at the ends of his lips, read in German to the Austrian general the following passage from the letter of Archduke Ferdinand: “Wir haben vollkommen zusammengehaltene Krafte, nahe an 70,000 Mann, um den Feind, wenn er den Lech passirte, angreifen und schl konnen. Wir konnen, da wir Meister von Ulm sind, den Vortheil, auch von beiden Uferien der Donau Meister zu bleiben, nicht verlieren; mithin auch jeden Augenblick, wenn der Feind den Lech nicht passirte, die Donau ubersetzen, uns auf seine Communikations Linie werfen, die Donau unterhalb repassiren und dem Feinde, wenn er sich gegen unsere treue Allirte mit ganzer Macht, wenden wollte Wir werden auf solche Weise den Zeitpunkt, wo die Kaiserlich Ruseische Armee ausgerustet sein wird, muthig entgegenharren, und sodann leicht gemeinschaftlich die Moglichkeit finden, dem Feinde das Schicksal zuzubereiten. So erdient [We have a quite concentrated force, about 70,000 people, so that we can attack and defeat the enemy in the event of a crossing over Leh. Since we already own Ulm, we can retain the benefit of commanding both banks of the Danube, therefore, every minute, if the enemy does not cross Lech, cross the Danube, rush to its communication line, below cross the Danube and the enemy, if he decides to turn all his strength on our faithful allies, not to allow his intention to be fulfilled. Thus, we will cheerfully await the time when the imperial Russian army is completely ready, and then together we will easily find an opportunity to prepare the enemy the fate he deserves. "]

Linear (vector) space is the set V of arbitrary elements, called vectors, in which the operations of adding vectors and multiplying a vector by a number are defined, i.e. any two vectors \\ mathbf (u) and (\\ mathbf (v)) are assigned the vector \\ mathbf (u) + \\ mathbf (v), called the sum of vectors \\ mathbf (u) and (\\ mathbf (v)), any vector (\\ mathbf (v)) and any number \\ lambda from the field of real numbers \\ mathbb (R) is mapped to the vector \\ lambda \\ mathbf (v), called the product of the vector \\ mathbf (v) and the number \\ lambda; so the following conditions are met:

1. \\ mathbf (u) + \\ mathbf (v) \u003d \\ mathbf (v) + \\ mathbf (u) \\, ~ \\ forall \\ mathbf (u), \\ mathbf (v) \\ in V (commutative addition);
2. \\ mathbf (u) + (\\ mathbf (v) + \\ mathbf (w)) \u003d (\\ mathbf (u) + \\ mathbf (v)) + \\ mathbf (w) \\, ~ \\ forall \\ mathbf (u), \\ mathbf (v), \\ mathbf (w) \\ in V (associativity of addition);
3. there is an element \\ mathbf (o) \\ in V, called the zero vector, such that \\ mathbf (v) + \\ mathbf (o) \u003d \\ mathbf (v) \\, ~ \\ forall \\ mathbf (v) \\ in V;
4.for each vector (\\ mathbf (v)) there is a vector called the opposite of the vector \\ mathbf (v) such that \\ mathbf (v) + (- \\ mathbf (v)) \u003d \\ mathbf (o);
5. \\ lambda (\\ mathbf (u) + \\ mathbf (v)) \u003d \\ lambda \\ mathbf (u) + \\ lambda \\ mathbf (v) \\, ~ \\ forall \\ mathbf (u), \\ mathbf (v) \\ in V , ~ \\ forall \\ lambda \\ in \\ mathbb (R);
6. (\\ lambda + \\ mu) \\ mathbf (v) \u003d \\ lambda \\ mathbf (v) + \\ mu \\ mathbf (v) \\, ~ \\ forall \\ mathbf (v) \\ in V, ~ \\ forall \\ lambda, \\ mu \\ \\ lambda (\\ mu \\ mathbf (v)) \u003d (\\ lambda \\ mu) \\ mathbf (v) \\, ~ \\ forall \\ mathbf (v) \\ in V, ~ \\ forall \\ lambda, \\ mu \\ in \\ mathbb ( R);
7. 1 \\ cdot \\ mathbf (v) \u003d \\ mathbf (v) \\, ~ \\ forall \\ mathbf (v) \\ in V;
8. Conditions 1-8 are called.

axioms of linear space ... The equal sign placed between the vectors means that the same element of the set V is represented on the left and right sides of the equality, such vectors are called equal.In the definition of a linear space, the operation of multiplying a vector by a number is introduced for real numbers. Such a space is called

linear space over the field of real (real) numbers , or, in short,real linear space ... If in the definition, instead of the field \\ mathbb (R) of real numbers, we take the field of complex numbers \\ mathbb (C), then we getlinear space over the field of complex numbers complex linear spacereal linear space {!LANG-d86c8ad5b4bdf14b0d63e130db22dd25!}... The field \\ mathbb (Q) of rational numbers can also be chosen as a number field, and we obtain a linear space over the field of rational numbers. Further, unless otherwise stated, real linear spaces will be considered. In some cases, for brevity, we will talk about space, omitting the word linear, since all the spaces considered below are linear.

Remarks 8.1

1. Axioms 1-4 show that a linear space is a commutative group with respect to addition.

2. Axioms 5 and 6 determine the distributivity of the operation of multiplying a vector by a number with respect to the operation of adding vectors (axiom 5) or to the operation of adding numbers (axiom 6). Axiom 7, sometimes called the law of associativity of multiplication by a number, expresses the connection between two different operations: multiplying a vector by a number and multiplying numbers. The property defined by Axiom 8 is called the unitarity of the operation of multiplying a vector by a number.

3. Linear space is a non-empty set, since it necessarily contains a zero vector.

4. Operations of addition of vectors and multiplication of a vector by a number are called linear operations on vectors.

5. The difference of vectors \\ mathbf (u) and \\ mathbf (v) is the sum of the vector \\ mathbf (u) with the opposite vector (- \\ mathbf (v)) and is denoted: \\ mathbf (u) - \\ mathbf (v) \u003d \\ mathbf (u) + (- \\ mathbf (v)).

6. Two nonzero vectors \\ mathbf (u) and \\ mathbf (v) are called collinear (proportional) if there is a number \\ lambda such that \\ mathbf (v) \u003d \\ lambda \\ mathbf (u)... Collinearity applies to any finite number of vectors. The null vector \\ mathbf (o) is considered collinear with any vector.

Consequences of the axioms of linear space

1. There is only one zero vector in linear space.

2. In the linear space for any vector \\ mathbf (v) \\ in V there is a unique opposite vector (- \\ mathbf (v)) \\ in V.

3. The product of an arbitrary space vector by the number zero is equal to the zero vector, that is, 0 \\ cdot \\ mathbf (v) \u003d \\ mathbf (o) \\, ~ \\ forall \\ mathbf (v) \\ in V.

4. The product of a zero vector by any number is equal to the zero vector, that is, for any number \\ lambda.

5. The vector opposite to the given vector is equal to the product of the given vector and the number (-1), i.e. (- \\ mathbf (v)) \u003d (- 1) \\ mathbf (v) \\, ~ \\ forall \\ mathbf (v) \\ in V.

6. In expressions like \\ mathbf (a + b + \\ ldots + z) (the sum of a finite number of vectors) or \\ alpha \\ cdot \\ beta \\ cdot \\ ldots \\ cdot \\ omega \\ cdot \\ mathbf (v) (product of a vector by a finite number of factors), you can place parentheses in any order, or not at all.

Let us prove, for example, the first two properties. Uniqueness of the zero vector. If \\ mathbf (o) and \\ mathbf (o) "are two zero vectors, then by Axiom 3 we obtain two equalities: \\ mathbf (o) "+ \\ mathbf (o) \u003d \\ mathbf (o)" or \\ mathbf (o) + \\ mathbf (o) "\u003d \\ mathbf (o), the left-hand sides of which are equal by axiom 1. Consequently, the right-hand sides are also equal, that is, \\ mathbf (o) \u003d \\ mathbf (o) "... The uniqueness of the opposite vector. If the vector \\ mathbf (v) \\ in V has two opposite vectors (- \\ mathbf (v)) and (- \\ mathbf (v)) ", then by axioms 2, 3,4 we obtain their equality:

(- \\ mathbf (v)) "\u003d (- \\ mathbf (v))" + \\ underbrace (\\ mathbf (v) + (- \\ mathbf (v))) _ (\\ mathbf (o)) \u003d \\ underbrace ( (- \\ mathbf (v)) "+ \\ mathbf (v)) _ (\\ mathbf (o)) + (- \\ mathbf (v)) \u003d (- \\ mathbf (v)).

The rest of the properties are proved similarly.

Examples of linear spaces

1. Denote \\ (\\ mathbf (o) \\) - a set containing one zero vector with operations \\ mathbf (o) + \\ mathbf (o) \u003d \\ mathbf (o) and \\ lambda \\ mathbf (o) \u003d \\ mathbf (o)... Axioms 1-8 are fulfilled for the indicated operations. Consequently, the set \\ (\\ mathbf (o) \\) is a linear space over any number field. This linear space is called zero.

2. We denote by V_1, \\, V_2, \\, V_3 - sets of vectors (directed segments) on a straight line, on a plane, in space, respectively, with the usual operations of vector addition and vector multiplication by a number. The fulfillment of axioms 1-8 of linear space follows from the course in elementary geometry. Therefore, the sets V_1, \\, V_2, \\, V_3 are real linear spaces. Instead of free vectors, one can consider the corresponding sets of radius vectors. For example, a set of vectors on a plane having a common origin, i.e. deferred from one fixed point of the plane, is a real linear space. The set of radius vectors of unit length does not form a linear space, since for any of these vectors the sum \\ mathbf (v) + \\ mathbf (v) does not belong to the set under consideration.

3. Let \\ mathbb (R) ^ n denote the set of n \\ times1 column matrices with the operations of matrix addition and matrix multiplication by a number. Axioms 1-8 of the linear space for this set are satisfied. The zero vector in this set is the zero column o \u003d \\ begin (pmatrix) 0 & \\ cdots & 0 \\ end (pmatrix) ^ T... Therefore, the set \\ mathbb (R) ^ n is a real linear space. Similarly, the set \\ mathbb (C) ^ n of n \\ times1 columns with complex elements is a complex linear space. The set of column matrices with non-negative real elements, on the contrary, is not a linear space, since it does not contain opposite vectors.

4. Denote \\ (Ax \u003d o \\) - the set of solutions of the homogeneous system Ax \u003d o of linear algebraic equations with and unknowns (where A is the real matrix of the system), considered as a set of columns of size n \\ times1 with the operations of matrix addition and matrix multiplication by the number ... Note that these operations are indeed defined on the set \\ (Ax \u003d o \\). Property 1 of solutions to a homogeneous system (see Section 5.5) implies that the sum of two solutions of a homogeneous system and the product of its solution by a number are also solutions of a homogeneous system, i.e. belong to the set \\ (Ax \u003d o \\). The axioms of the linear space for columns are satisfied (see item 3 in the examples of linear spaces). Therefore, the set of solutions to a homogeneous system is a real linear space.

The set \\ (Ax \u003d b \\) of solutions to the inhomogeneous system Ax \u003d b, ~ b \\ ne o, on the contrary, is not a linear space, if only because it does not contain a zero element (x \u003d o is not a solution to the inhomogeneous system).

5. Let M_ (m \\ times n) be the set of matrices of size m \\ times n with operations of matrix addition and matrix multiplication by a number. Axioms 1-8 of the linear space for this set are satisfied. The zero vector is a zero matrix O of appropriate sizes. Therefore, the set M_ (m \\ times n) is a linear space.

6. Let P (\\ mathbb (C)) be the set of polynomials of one variable with complex coefficients. The operations of addition of many terms and multiplication of a polynomial by a number considered as a polynomial of degree zero are defined and satisfy Axioms 1-8 (in particular, the zero vector is a polynomial that is identically equal to zero). Therefore, the set P (\\ mathbb (C)) is a linear space over the field of complex numbers. The set P (\\ mathbb (R)) of polynomials with real coefficients is also a linear space (but, of course, over the field of real numbers). The set P_n (\\ mathbb (R)) of polynomials of degree at most n with real coefficients is also a real linear space. Note that the operation of addition of many terms is defined on this set, since the degree of the sum of polynomials does not exceed the powers of the terms.

The set of polynomials of degree n is not a linear space, since the sum of such polynomials may turn out to be a polynomial of lesser degree that does not belong to the set under consideration. The set of all polynomials of degree at most l, with positive coefficients, is also not a linear space, since multiplying such a polynomial by a negative number gives us a polynomial that does not belong to this set.

7. Let C (\\ mathbb (R)) denote the set of real functions defined and continuous on \\ mathbb (R). The sum (f + g) of the functions f, g and the product \\ lambda f of the function f by the real number \\ lambda are defined by the equalities:

(f + g) (x) \u003d f (x) + g (x), \\ quad (\\ lambda f) (x) \u003d \\ lambda \\ cdot f (x) for all x \\ in \\ mathbb (R)

These operations are indeed defined on C (\\ mathbb (R)), since the sum of continuous functions and the product of a continuous function by a number are continuous functions, i.e. elements of C (\\ mathbb (R)). Let us check the fulfillment of the axioms of the linear space. The commutativity of the addition of real numbers implies the equality f (x) + g (x) \u003d g (x) + f (x) for any x \\ in \\ mathbb (R). Therefore, f + g \u003d g + f, i.e. axiom 1 is satisfied. Axiom 2 follows similarly from the associativity of addition. The zero vector is the function o (x), identically equal to zero, which, of course, is continuous. For any function f, the equality f (x) + o (x) \u003d f (x) holds, i.e. axiom 3 is valid. The opposite vector for the vector f is the function (-f) (x) \u003d - f (x). Then f + (- f) \u003d o (axiom 4 holds). Axioms 5, 6 follow from the distributivity of operations of addition and multiplication of real numbers, and axiom 7 - from the associativity of multiplication of numbers. The last axiom is fulfilled, since multiplication by one does not change the function: 1 \\ cdot f (x) \u003d f (x) for any x \\ in \\ mathbb (R), i.e. 1 \\ cdot f \u003d f. Thus, the set C (\\ mathbb (R)) under consideration with the introduced operations is a real linear space. It can be proved similarly that C ^ 1 (\\ mathbb (R)), C ^ 2 (\\ mathbb (R)), \\ ldots, C ^ m (\\ mathbb (R)) - sets of functions having continuous derivatives of the first, second, etc. orders, respectively, are also linear spaces.

Let us denote the set of trigonometric binomials (often \\ omega \\ ne0) with real coefficients, i.e. many functions of the form f (t) \u003d a \\ sin \\ omega t + b \\ cos \\ omega twhere a \\ in \\ mathbb (R), ~ b \\ in \\ mathbb (R)... The sum of such binomials and the product of a binomial by a real number is a trigonometric binomial. The axioms of the linear space for the set under consideration are satisfied (since T _ (\\ omega) (\\ mathbb (R)) \\ subset C (\\ mathbb (R))). Therefore the set T _ (\\ omega) (\\ mathbb (R)) with the usual operations for functions of addition and multiplication by a number is a real linear space. The zero element is the binomial o (t) \u003d 0 \\ cdot \\ sin \\ omega t + 0 \\ cdot \\ cos \\ omega t, identically equal to zero.

The set of real functions defined and monotone on \\ mathbb (R) is not a linear space, since the difference of two monotone functions may turn out to be a non-monotone function.

8. Let \\ mathbb (R) ^ X denote the set of real functions defined on the set X with the operations:

(f + g) (x) \u003d f (x) + g (x), \\ quad (\\ lambda f) (x) \u003d \\ lambda \\ cdot f (x) \\ quad \\ forall x \\ in X

It is a real linear space (the proof is the same as in the previous example). Moreover, the set X can be chosen arbitrarily. In particular, if X \u003d \\ (1,2, \\ ldots, n \\), then f (X) is an ordered set of numbers f_1, f_2, \\ ldots, f_nwhere f_i \u003d f (i), ~ i \u003d 1, \\ ldots, n Such a set can be considered a n \\ times1 column matrix, i.e. a bunch of \\ mathbb (R) ^ (\\ (1,2, \\ ldots, n \\)) coincides with the set \\ mathbb (R) ^ n (see item 3 for examples of linear spaces). If X \u003d \\ mathbb (N) (recall that \\ mathbb (N) is the set of natural numbers), then we obtain the linear space \\ mathbb (R) ^ (\\ mathbb (N)) - many numerical sequences \\ (f (i) \\) _ (i \u003d 1) ^ (\\ infty)... In particular, the set of converging number sequences also forms a linear space, since the sum of two converging sequences converges, and multiplying all members of the converging sequence by a number, we obtain a converging sequence. In contrast, the set of divergent sequences is not a linear space, since, for example, the sum of divergent sequences may have a limit.

9. Let \\ mathbb (R) ^ (+) denote the set of positive real numbers, in which the sum of a \\ oplus b and the product \\ lambda \\ ast a (the notation in this example differs from the usual) are defined by the equalities: a \\ oplus b \u003d ab, ~ \\ lambda \\ ast a \u003d a ^ (\\ lambda)in other words, the sum of elements is understood as the product of numbers, and the multiplication of an element by a number is understood as exponentiation. Both operations are indeed defined on the set \\ mathbb (R) ^ (+), since the product of positive numbers is a positive number and any real power of a positive number is a positive number. Let us check the validity of the axioms. Equality

a \\ oplus b \u003d ab \u003d ba \u003d b \\ oplus a, \\ quad a \\ oplus (b \\ oplus c) \u003d a (bc) \u003d (ab) c \u003d (a \\ oplus b) \\ oplus c

show that axioms 1, 2 are fulfilled. The zero vector of this set is one, since a \\ oplus1 \u003d a \\ cdot1 \u003d a, i.e. o \u003d 1. The opposite vector for a is \\ frac (1) (a), which is defined since a \\ ne o. Indeed, a \\ oplus \\ frac (1) (a) \u003d a \\ cdot \\ frac (1) (a) \u003d 1 \u003d o... Let us check the fulfillment of axioms 5, 6,7,8:

\\ begin (gathered) \\ mathsf (5)) \\ quad \\ lambda \\ ast (a \\ oplus b) \u003d (a \\ cdot b) ^ (\\ lambda) \u003d a ^ (\\ lambda) \\ cdot b ^ (\\ lambda) \u003d \\ lambda \\ ast a \\ oplus \\ lambda \\ ast b \\,; \\ hfill \\\\ \\ mathsf (6)) \\ quad (\\ lambda + \\ mu) \\ ast a \u003d a ^ (\\ lambda + \\ mu) \u003d a ^ ( \\ lambda) \\ cdot a ^ (\\ mu) \u003d \\ lambda \\ ast a \\ oplus \\ mu \\ ast a \\,; \\ hfill \\\\ \\ mathsf (7)) \\ quad \\ lambda \\ ast (\\ mu \\ ast a) \u003d (a ^ (\\ mu)) ^ (\\ lambda) \u003d a ^ (\\ lambda \\ mu) \u003d (\\ lambda \\ cdot \\ mu) \\ ast a \\,; \\ hfill \\\\ \\ mathsf (8)) \\ quad 1 \\ ast a \u003d a ^ 1 \u003d a \\,. \\ Hfill \\ end (gathered)

All axioms are fulfilled. Therefore, the set under consideration is a real linear space.

10. Let V be a real linear space. Consider the set of linear scalar functions defined on V, i.e. functions f \\ colon V \\ to \\ mathbb (R), taking real values \u200b\u200band satisfying the conditions:

f (\\ mathbf (u) + \\ mathbf (v)) \u003d f (u) + f (v) ~~ \\ forall u, v \\ in V (additivity);

f (\\ lambda v) \u003d \\ lambda \\ cdot f (v) ~~ \\ forall v \\ in V, ~ \\ forall \\ lambda \\ in \\ mathbb (R) (uniformity).

Linear operations on linear functions are specified in the same way as in item 8 of examples of linear spaces. The sum f + g and the product \\ lambda \\ cdot f are defined by the equalities:

(f + g) (v) \u003d f (v) + g (v) \\ quad \\ forall v \\ in V; \\ qquad (\\ lambda f) (v) \u003d \\ lambda f (v) \\ quad \\ forall v \\ The fulfillment of the axioms of a linear space is confirmed in the same way as in Section 8. Therefore, the set of linear functions defined on a linear space V is a linear space. This space is called the dual to the space V and is denoted by V ^ (\\ ast). Its elements are called covectors.

For example, the set of linear forms of n variables, considered as the set of scalar functions of a vector argument, is a linear space dual to the space \\ mathbb (R) ^ n.

If you notice an error, typo, or have suggestions, write in the comments.

Lecture 6. Vector space.

Main questions.

1. Vector linear space.

2. Basis and dimension of space.

3. Orientation of space.

4. Decomposition of the vector in basis.

5. Vector coordinates.

1. Vector linear space.

A set consisting of elements of any nature in which linear operations are defined: addition of two elements and multiplication of an element by a number are called

spaces , and their elements areof this space and are designated in the same way as vector quantities in geometry:. vectors such abstract spaces, as a rule, have nothing to do with ordinary geometric vectors. Elements of abstract spaces can be functions, a system of numbers, matrices, etc., and in a particular case, ordinary vectors. Therefore, such spaces are usually called Vectors vector spaces vector spaces are, .

eg , the set of colli-nonary vectors, denotedV , the set of coplanar vectors1 set of vectors of ordinary (real space) , the set of coplanar vectors2 , for this particular case, the following definition of a vector space can be given. , the set of coplanar vectors3 .

Definition 1.

The set of vectors is calledvector space {!LANG-9216be99275dee1a8b2ed4dba33782eb!} if a linear combination of any vectors of a set is also a vector of this set. The vectors themselves are called elements vector space.

More important both in theoretical and applied terms is the general (abstract) concept of vector space.

Definition 2.A bunch of R elements, in which for any two elements the sum is determined and for any element https://pandia.ru/text/80/142/images/image006_75.gif "width \u003d" 68 "height \u003d" 20 "\u003e called vector(or linear) space , and its elements are vectors if the operations of addition of vectors and multiplication of a vector by a number satisfy the following conditions ( axioms) :

1) addition is commutative, ie gif "width \u003d" 184 "height \u003d" 25 "\u003e;

3) there is such an element (zero vector) that for any https://pandia.ru/text/80/142/images/image003_99.gif "width \u003d" 45 "height \u003d" 20 "\u003e. Gif" width \u003d " 99 "height \u003d" 27 "\u003e;

5) for any vectors and and any number λ, equality holds;

6) for any vectors and any numbers λ and µ the equality is valid https://pandia.ru/text/80/142/images/image003_99.gif "width \u003d" 45 height \u003d 20 "height \u003d" 20 "\u003e and any numbers λ and µ fair ;

8) https://pandia.ru/text/80/142/images/image003_99.gif "width \u003d" 45 "height \u003d" 20 "\u003e.

From the axioms defining a vector space follow the simplest consequences :

1. In vector space there is only one zero - element - zero vector.

2. In vector space, each vector has a single opposite vector.

3. For each element, equality holds.

4. For any real number λ and zero vector https://pandia.ru/text/80/142/images/image017_45.gif "width \u003d" 68 "height \u003d" 25 "\u003e.

5..gif "width \u003d" 145 "height \u003d" 28 "\u003e

6..gif "width \u003d" 15 "height \u003d" 19 src \u003d "\u003e. Gif" width \u003d "71" height \u003d "24 src \u003d"\u003e a vector is called that satisfies the equality https://pandia.ru/text/80 /142/images/image026_26.gif "width \u003d" 73 "height \u003d" 24 "\u003e.

So, indeed, the set of all geometric vectors is a linear (vector) space, since for the elements of this set the actions of addition and multiplication by a number are defined, which satisfy the formulated axioms.

2. Basis and dimension of space.

The essential concepts of a vector space are the concepts of basis and dimension.

Definition.The set of linearly independent vectors, taken in a certain order, through which any vector of space is linearly expressed, is called basis this space. Vectors. The basis of the space is called basic .

The basis of a set of vectors located on an arbitrary straight line can be considered one collinear this straight vector.

Basis on the planelet's name two non-collinear vectors on this plane, taken in a certain order https://pandia.ru/text/80/142/images/image029_29.gif "width \u003d" 61 "height \u003d" 24 "\u003e.

If the basis vectors are pairwise perpendicular (orthogonal), then the basis is called orthogonal , and if these vectors have length equal to one, then the basis is called orthonormal .

The largest number of linearly independent vectors of the space is called dimension of this space, that is, the dimension of the space coincides with the number of basis vectors of this space.

So, in accordance with these definitions:

1. One-dimensional space , the set of coplanar vectors1 is a straight line, and the basis consists of one collinearvectors https://pandia.ru/text/80/142/images/image028_22.gif "width \u003d" 39 "height \u003d" 23 src \u003d "\u003e.

3. Ordinary space is three-dimensional space , the set of coplanar vectors3 whose basis consists of three non-coplanar vectors.

From here we see that the number of basis vectors on a straight line, on a plane, in real space, coincides with what in geometry is usually called the number of dimensions (dimension) of a straight line, plane, space. Therefore, it is natural to introduce a more general definition.

Definition.Vector space R called n - dimensional if it contains no more n linearly independent vectors and denoted R n ... Number n is called dimension space.

In accordance with the dimension, the spaces are divided into finite-dimensional and endless ... The dimension of the zero space is considered to be zero by definition.

Remark 1. In each space, you can specify as many bases as you like, but in this case all the bases of a given space consist of the same number of vectors.

Remark 2. AT n - dimensional vector space, a basis is any ordered set n linearly independent vectors.

3. Orientation of space.

Let the basis vectors in the space , the set of coplanar vectors3 have common start and ordered, that is, it is indicated which vector is considered the first, which is the second and which is the third. For example, in the basis the vectors are ordered according to the index.

For to orient the space, you need to set some basis and declare it positive .

It can be shown that the set of all bases of the space falls into two classes, that is, into two disjoint subsets.

a) all bases belonging to one subset (class) have the same orientation (bases of the same name);

b) any two bases belonging to various subsets (classes) have the opposite orientation, ( opposite bases).

If one of the two classes of bases of a space is declared positive, and the other negative, then they say that this space oriented .

Often, when orienting a space, some bases are called right and others - left .

https://pandia.ru/text/80/142/images/image029_29.gif "width \u003d" 61 "height \u003d" 24 src \u003d "\u003e called right if, when observing from the end of the third vector, the shortest rotation of the first vector https://pandia.ru/text/80/142/images/image033_23.gif "width \u003d" 16 "height \u003d" 23 "\u003e is carried out counterclock-wise (Fig. 1.8, a).

https://pandia.ru/text/80/142/images/image036_22.gif "width \u003d" 16 "height \u003d" 24 "\u003e

https://pandia.ru/text/80/142/images/image037_23.gif "width \u003d" 15 "height \u003d" 23 "\u003e

https://pandia.ru/text/80/142/images/image039_23.gif "width \u003d" 13 "height \u003d" 19 "\u003e

https://pandia.ru/text/80/142/images/image033_23.gif "width \u003d" 16 "height \u003d" 23 "\u003e

Figure: 1.8. Right basis (a) and left basis (b)

Usually the right basis of the space is declared a positive basis

The right (left) basis of space can also be determined using the rule of the "right" ("left") screw or gimbal.

By analogy with this, the concept of right and left triplets non-compliant vectors that must be ordered (Figure 1.8).

Thus, in the general case, two ordered triples of non-complementary vectors have the same orientation (are of the same name) in the space , the set of coplanar vectors3 if they are both right or both left, and - the opposite orientation (opposite), if one of them is right and the other is left.

The same is done in the case of the space , the set of coplanar vectors2 (plane).

4. Decomposition of the vector in basis.

For simplicity of reasoning, we will consider this question using the example of a three-dimensional vector space R3 .

Let https://pandia.ru/text/80/142/images/image021_36.gif "width \u003d" 15 "height \u003d" 19 "\u003e be an arbitrary vector of this space.

Chapter 3. Linear vector spaces

Topic 8. Linear vector spaces

Definition of linear space. Examples of linear spaces

In Section 2.1, the operation of addition of free vectors from R 3 and the operation of multiplying vectors by real numbers, and also lists the properties of these operations. The extension of these operations and their properties to a set of objects (elements) of arbitrary nature leads to a generalization of the concept of a linear space of geometric vectors from R 3 defined in §2.1. Let us formulate the definition of a linear vector space.

Definition 8.1. A bunch of , the set of coplanar vectors elements x , at , z , ... is called linear vector space, if:

there is a rule that every two elements x and at of , the set of coplanar vectors matches the third element from , the set of coplanar vectorscalled sum x and at and designated x + at ;

there is a rule that for each element x and matches any real number an element from , the set of coplanar vectorscalled product of element x by the number and designated x .

Moreover, the sum of any two elements x + at and work x any element by any number must satisfy the following requirements - axioms of linear space:

1 °. x + at = at + x (addition commutability).

2 °. ( x + at ) + z = x + (at + z ) (addition associativity).

3 °. There is an element 0 called zerosuch that

x + 0 = x , x .

4 °. For anyone x there is an element (- x ) called opposite for x such that

x + (– x ) = 0 .

5 °. ( x ) = ()x , x , , R.

6 °. x = x , x .

7 °. () x = x + x , x , , R.

8 °. ( x + at ) = x + y , x , y , R.

Elements of linear space will be called vectors regardless of their nature.

From the axioms 1 ° –8 ° it follows that in any linear space , the set of coplanar vectors the following properties are true:

1) there is only one zero vector;

2) for each vector x there is only one opposite vector (- x ), and (- x ) \u003d (- l) x ;

3) for any vector x the equality 0 × x = 0 .

Let us prove, for example, property 1). Suppose that in space , the set of coplanar vectors there are two zeros: 0 1 and 0 2. Putting in the axiom 3 ° x = 0 1 , 0 = 0 2, we get 0 1 + 0 2 = 0 1 . Similarly, if x = 0 2 , 0 = 0 1, then 0 2 + 0 1 = 0 2. Taking into account axiom 1 °, we obtain 0 1 = 0 2 .

Let us give examples of linear spaces.

1. The set of real numbers forms a linear space R... The axioms 1 ° –8 ° are obviously fulfilled in it.

2. The set of free vectors of a three-dimensional space, as shown in §2.1, also forms a linear space, denoted R 3. The zero of this space is the zero vector.

The set of vectors on the plane and on the line are also linear spaces. We will denote them R 1 and R 2 respectively.

3. Generalization of spaces R 1 , R 2 and R 3 serves as space R n, n Ncalled arithmetic n-spacewhose elements (vectors) are ordered collections n arbitrary real numbers ( x 1 ,…, x n), i.e.

R n = {(x 1 ,…, x n) | x i R, i = 1,…, n}.

It is convenient to use the notation x = (x 1 ,…, x n), wherein x i called i-th coordinate(component) vector x .

For x , at R n and R we define addition and multiplication by a number by the following formulas:

x + at = (x 1 + y 1 ,…, x n+ y n);

x = (x 1 ,…, x n).

Zero space element R n is the vector 0 \u003d (0, ..., 0). Equality of two vectors x = (x 1 ,…, x n) and at = (y 1 ,…, y n) from R n, by definition, means the equality of the corresponding coordinates, i.e. x = at Û x 1 = y 1 &… & x n = y n.

The fulfillment of axioms 1 ° –8 ° is obvious here.

4. Let C [ a ; b ] Is the set of real continuous on the segment [ a; b] functions f: [a; b] R.

The sum of functions f and g of C [ a ; b ] is called the function h = f + gdefined by the equality

h = f + g Û h(x) = (f + g)(x) = f(x) + g(x), " x Î [ a; b].

Product of function f Î C [ a ; b ] to the number a Î R is defined by the equality

u = f Û u(x) = (f)(x) = f(x), " x Î [ a; b].

Thus, the introduced operations of addition of two functions and multiplication of a function by a number transform the set C [ a ; b ] into a linear space whose vectors are functions. The axioms 1 ° –8 ° in this space obviously hold. The zero vector of this space is the identically zero function, and the equality of the two functions f and g means, by definition, the following:

f = g f(x) = g(x), " x Î [ a; b].

A vector (linear) space is a set of vectors (elements) with real components, in which the operations of adding vectors and multiplying a vector by a number are defined that satisfy certain axioms (properties)

1) x+ at= at+ x (addition permutation);

2)(x+ at)+ z= x+(y+ z) (associativity of addition);

3) there is a zero vector 0 (or zero vector) satisfying the condition x+ 0 = x: for any vector x;

4) for any vector x there is an opposite vector at such that x+ at = 0 ,

5) 1 x= x,

6) a(bx)=(ab) x (associativity of multiplication);

7) (a+ b) x= aх+ bx (distribution property with respect to a numerical factor);

8) a(x+ at)= aх+ ay(distribution property with respect to a vector factor).

A linear (vector) space V (P) over a field P is a nonempty set V. Elements of the set V are called vectors, and elements of the field P are called scalars.

Simplest properties.

1. A vector space is an abelian group (a group in which the group operation is commutative. The group operation in abelian groups is usually called "addition" and is denoted by +)

2. The neutral element is the only one that follows from the group properties for any.

3. For any, the opposite element is unique, which follows from the group properties.

4. (- 1) x \u003d - x for any x є V.

5. (- α) x \u003d α (–x) \u003d - (αx) for any α є P and x є V.

Expression a 1 e 1+ a 2 e 2+ … + a n e n (1) is called a linear combination of vectors e 1, e 2, ..., e n with coefficients a 1, a 2,..., a n. Linear combination (1) is called nontrivial if at least one of the coefficients a 1, a 2, ..., a n nonzero. Vectors e 1, e 2, ..., e n are called linearly dependent if there is a nontrivial combination (1), which is a zero vector. Otherwise (that is, if only a trivial combination of vectors e 1, e 2, ..., e n is equal to zero vector) vectors e 1, e 2, ..., e n are called linearly independent.

Dimension of space - the maximum number of LZ vectors contained in it.

Vector space called n-dimensional (or has “dimension n "), if there are n linearly independent elements e 1, e 2, ..., e n, and any n+ 1 elements are linearly dependent (generalized condition B). Vector space are called infinite-dimensional if for any natural n exists n linearly independent vectors. Any n linearly independent vectors of the n-dimensional Vector space form the basis of this space. If e 1, e 2, ..., e n - basis Vector space, then any vector xof this space can be represented uniquely as a linear combination of basis vectors: x= a 1 e 1+ a 2 e 2+... + a n e n.
Moreover, the numbers a 1, a 2, ..., a n are called the coordinates of the vector x in this basis.