Algebraic theory of vector spaces. Linear vector space: definition, properties. Basis and dimension of space

Material from Wikipedia - the free encyclopedia

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 the real, complex, or any other number field. A special case of such a space is the usual three-dimensional Euclidean space, whose vectors are used, for example, to represent physical forces. It should be noted that a vector as an element of vector space does not necessarily have to be specified in the form of a directed segment. Generalizing the concept of “vector” to an element of a vector space of any nature not only does not cause confusion of terms, but also makes it possible to understand or even predict a number of results that are valid for spaces of arbitrary nature.

Vector spaces are the subject of linear algebra. One of the main characteristics of a vector space is its dimension. Dimension represents the maximum number of linearly independent elements of space, that is, resorting to a rough geometric description, the number of directions that cannot be expressed through each other through only the operations of addition and multiplication by a scalar. The vector space can be endowed with additional structures, such as a norm or an inner product. Such spaces appear naturally in mathematical analysis, primarily in the form of infinite-dimensional function spaces ( English), where the functions . 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 additional structure, in most cases a suitable topology, which allows us 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 began to develop.

Definition

Linear, or vector space $V\left(F\right)$ over the field $F$ - this is an ordered four $(V,F,+,\cdot)$ , Where

$V$ - a non-empty set of elements of arbitrary nature, which are called vectors;
$F$ - (algebraic) field whose elements are called scalars;
Operation defined addition vectors $V\times V\to V$ , which associates each pair of elements $\mathbf(x), \mathbf(y)$ sets $V$ $V$ called them amount and designated $\mathbf(x) + \mathbf(y)$ ;
Operation defined multiplying vectors by scalars $F\times V\to V$ , matching each element $\lambda$ fields $F$ and each element $\mathbf(x)$ sets $V$ the only element of the set $V$ , denoted $\lambda\cdot\mathbf(x)$ or $\lambda\mathbf(x)$ ;

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 $\mathbb(R)^2$ 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

A vector space is an Abelian group under addition.
Neutral element $\mathbf(0) \in V$
$0\cdot\mathbf(x) = \mathbf(0)$ for anyone $\mathbf(x) \in V$ .
For anyone $\mathbf(x) \in V$ opposite element $-\mathbf(x)\in V$ is the only thing that follows from group properties.
$1\cdot\mathbf(x) = \mathbf(x)$ for anyone $\mathbf(x) \in V$ .
$(-\alpha)\cdot\mathbf(x) = \alpha\cdot(-\mathbf(x)) = -(\alpha\mathbf(x))$ for any $\alpha \in F$ And $\mathbf(x) \in V$ .
$\alpha\cdot \mathbf(0) = \mathbf(0)$ for anyone $\alpha \in F$ .

Related definitions and properties

Subspace

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

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

The last two statements are equivalent to the following:

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

In particular, a vector space consisting of only one zero vector is a subspace of any space; every 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 $K_i\quad|\quad i \in 1\ldots N$ is defined as a set containing all possible sums of elements K_i: \sum_(i=1)^N (K_i):= $\mathbf(x)_1 + \mathbf(x)_2 + \ldots + \mathbf(x)_N\quad|\quad \mathbf(x)_i \in K_i\quad (i\in 1\ldots N)$.
- The sum of a finite family of subspaces is again a subspace.

Linear combinations

Final sum of the form

$\alpha_1\mathbf(x)_1 + \alpha_2\mathbf(x)_2 + \ldots + \alpha_n\mathbf(x)_n$

The linear combination is called:

Basis. Dimension

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

Otherwise these vectors are called linearly independent.

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

Properties of the basis:

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

\mathbf(x) = \alpha_1\mathbf(x)_1 + \alpha_2\mathbf(x)_2 + \ldots + \alpha_n\mathbf(x)_n

Linear shell

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

The linear span is a subspace $V$ .

Linear shell is also called subspace generated $X$ . It is also said that the linear shell $\mathcal V(X)$ - space, stretched over a bunch of $X$ .

Linear shell $\mathcal V(X)$ consists of all possible linear combinations of various finite subsystems of elements from $X$ . In particular, if $X$ is a finite set, then $\mathcal V(X)$ 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 $\mathcal V(X)$ and thereby determines its dimension.

Examples

A null space whose only element is zero.
Space of all functions $X\to F$ with finite support forms a vector space of dimension equal to the cardinality $X$ .
The field of real numbers can be considered as a continuum-dimensional vector space over the field of rational numbers.
Any field is a one-dimensional space above itself.

Additional structures

Write a review about the article "Vector space"

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.
Kostrikin A. I., Manin Yu. I. Linear algebra and geometry. 2nd ed. - M.: Nauka, 1986. - 304 p.
Kostrikin A.I. Introduction to algebra. Part 2: Linear algebra. - 3rd. - M.: Nauka., 2004. - 368 p. - (University textbook).
Maltsev A. I. Basics 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. - M.: Fizmatlit, 2010. - 280 p. - ISBN 978-5-9221-0481-4.
Halmos P. Finite-Dimensional Vector Spaces. - M.: Fizmatgiz, 1963. - 263 p.
Faddeev D.K. Lectures on algebra. - 5th. - St. Petersburg. : Lan, 2007. - 416 p.
Shafarevich I. R., Remizov A. O. Linear algebra and geometry. - 1st. - M.: Fizmatlit, 2009. - 511 p.
Schreyer O., Sperner G. Introduction to linear algebra in geometric presentation = Einfuhrung in die analytische Geometrie und Algebra / Olshansky G. (translation 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 whom he knew from the Turkish war, and sometimes to the soldiers. Looking at the shoes, he sadly shook his head several times and pointed them out to the Austrian general with such an expression that he didn’t seem to blame anyone for it, but he couldn’t help but see how bad it was. Each time the regimental commander ran ahead, afraid to miss the commander-in-chief's word regarding the regiment. Behind Kutuzov, at such a distance that any faintly spoken word could be heard, walked about 20 people in his retinue. The gentlemen of the retinue talked among themselves and sometimes laughed. The handsome adjutant walked closest to the commander-in-chief. It was Prince Bolkonsky. Next to him walked his comrade Nesvitsky, a tall staff officer, extremely fat, with a kind and smiling handsome face and moist eyes; Nesvitsky could hardly restrain himself from laughing, excited by the blackish hussar officer walking next to him. The hussar officer, without smiling, without changing the expression of his fixed eyes, looked with a serious face at the back of the regimental commander and imitated his every movement. Every time the regimental commander flinched and bent forward, in exactly the same way, in exactly the same way, the hussar officer flinched and bent forward. Nesvitsky laughed and pushed others to look at the funny man.
Kutuzov walked slowly and sluggishly past thousands of eyes that rolled out of their sockets, watching their boss. Having caught 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 the red nose, who suffered for his blue overcoat.
It seemed that it was impossible to stretch out more than Timokhin stretched out, while the regimental commander reprimanded him. But at that moment the commander-in-chief addressed him, the captain stood up straight so that it seemed that if the commander-in-chief had looked at him for a little longer, the captain would not have been able to stand it; and therefore Kutuzov, apparently understanding his position and wishing, on the contrary, all the best for the captain, hastily turned away. A barely noticeable smile ran across Kutuzov’s plump, wound-disfigured face.
“Another Izmailovo comrade,” he said. - Brave officer! Are you happy with it? – Kutuzov asked the regimental commander.
And the regimental commander, reflected as in a mirror, invisible to himself, in a hussar officer, shuddered, came 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 devotion to Bacchus.
The regimental commander was afraid that he was to blame for this, and did not answer anything. The officer at that moment noticed the captain’s face with a red nose and a tucked belly and imitated his face and pose so closely that Nesvitsky could not stop laughing.
Kutuzov turned around. It was clear that the officer could control his face as he wanted: the minute Kutuzov turned around, the officer managed to make a grimace, and after that take on the most serious, respectful and innocent expression.
The third company was the last, and Kutuzov thought about it, apparently remembering something. Prince Andrei stepped out from his retinue and said quietly in French:
– You ordered a reminder about Dolokhov, who was demoted, in this regiment.
-Where is Dolokhov? – asked Kutuzov.
Dolokhov, already dressed in a soldier’s gray overcoat, did not wait to be called. The slender figure of a blond soldier with clear blue eyes stepped out from the front. He approached the commander-in-chief and put him on guard.
- Claim? – Kutuzov asked, frowning slightly.
“This is Dolokhov,” said Prince Andrei.
- A! - said Kutuzov. “I hope this lesson will correct you, serve well.” The Lord is merciful. And I will not forget you if you deserve it.
Blue, clear eyes looked at the commander-in-chief as defiantly as at the regimental commander, as if with their expression they were tearing apart the veil of convention that so far separated the commander-in-chief from the soldier.
“I ask one thing, Your Excellency,” he said in his sonorous, firm, unhurried voice. “Please give me a chance to make amends for my guilt and prove my devotion to the Emperor and Russia.”
Kutuzov turned away. The same smile in his eyes flashed across his face as when he turned away from Captain Timokhin. He turned away and winced, as if he wanted to express that everything that Dolokhov told him, and everything that he could tell him, he had known for a long, long time, that all this had already bored him and that all this was not at all what he needed . He turned away and headed towards the stroller.
The regiment disbanded in companies and headed to assigned quarters not far from Braunau, where they hoped to put on shoes, dress and rest after difficult marches.
– You don’t lay claim to me, Prokhor Ignatyich? - said the regimental commander, driving around the 3rd company moving towards the place and approaching Captain Timokhin, who was walking in front of it. The regimental commander’s face expressed uncontrollable joy after a happily completed review. - The royal service... it’s impossible... another time you’ll end it at the front... I’ll apologize first, you know me... I thanked you very much! - And he extended his hand to the company commander.
- For mercy's sake, general, do I dare! - answered the captain, turning red with 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 can be calm. Yes, please tell me, I kept wanting to ask how he is, how he is behaving? And that's all...
“He is very serviceable in his service, Your Excellency... but the charterer...” said Timokhin.
- What, what character? – asked the regimental commander.
“Your Excellency finds, for days,” said the captain, “that he is smart, learned, and kind.” It's a beast. He killed a Jew in Poland, if you please...
“Well, yes, well,” said the regimental commander, “we still need to feel sorry for the young man in misfortune.” After all, great connections... So you...
“I’m listening, Your Excellency,” Timokhin said, smiling, making it feel like he understood the boss’s wishes.
- Yes Yes.
The regimental commander found Dolokhov in the ranks and reined in his horse.
“Before the first task, epaulets,” 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. “The people each have a glass of vodka from me,” he added so that the soldiers could hear. – Thank you everyone! God bless! - And he, overtaking the company, drove up to another.
“Well, he’s really a good man; “You can serve with him,” said subaltern Timokhin to the officer walking next to him.
“One word, the king of hearts!... (the regimental commander was nicknamed the king of hearts),” the subaltern officer said, laughing.
The happy mood of the authorities after the review spread to the soldiers. The company walked cheerfully. Soldiers' voices were talking from all sides.
- What did they say, crooked Kutuzov, about one eye?
- Otherwise, no! Totally crooked.
- No... brother, he has bigger eyes than you. Boots and tucks - I looked at everything...
- How can he, my brother, look at my feet... well! Think…
- And the other Austrian, with him, was as if smeared with chalk. Like flour, white. I tea, how they clean ammunition!
- What, Fedeshow!... did he say that when the fighting began, you stood closer? They all said that Bunaparte himself stands in Brunovo.
- Bunaparte is worth it! he's lying, you fool! What he doesn’t know! Now the Prussian is rebelling. The Austrian, therefore, pacifies him. As soon as he makes peace, then war will open with Bunaparte. Otherwise, he says, Bunaparte is standing in Brunovo! That's what shows that he's a fool. Listen more.
- Look, damn the lodgers! The fifth company, look, is already turning into the village, they will cook porridge, and we still won’t reach the place.
- Give me a cracker, damn it.
- Did you give me tobacco yesterday? That's it, brother. Well, here we go, God be with you.
“At least they made a stop, otherwise we won’t eat for another five miles.”
– It was nice how the Germans gave us strollers. When you go, know: it’s important!
“And here, brother, the people have gone completely rabid.” Everything there seemed to be a Pole, everything was from the Russian crown; and now, brother, he’s gone completely German.
– Songwriters forward! – the captain’s cry was heard.
And twenty people ran out from different rows in front of the company. The drummer began to sing and turned to face the songwriters, and, waving his hand, began a drawn-out soldier’s song, which began: “Isn’t it dawn, the sun has broken...” and ended with the words: “Then, brothers, there will be glory for us and Kamensky’s father...” This song was composed in Turkey and was now sung in Austria, only with the change that in place of “Kamensky’s father” the words were inserted: “Kutuzov’s father.”
Having torn off these last words like a soldier and waving his hands, as if he was throwing something to the ground, the drummer, a dry and handsome soldier of about forty, looked sternly at the soldier songwriters and closed his eyes. Then, making sure that all eyes were fixed on him, he seemed to carefully lift with both hands some invisible, precious thing above his head, held it like that for several seconds and suddenly desperately threw it:
Oh, you, my canopy, my canopy!
“My new canopy...”, twenty voices echoed, and the spoon holder, despite the weight of his ammunition, quickly jumped forward and walked backwards in front of the company, moving his shoulders and threatening someone with his spoons. The soldiers, waving their arms to the beat of the song, walked with long strides, involuntarily hitting their feet. From behind the company the sounds of wheels, the crunching of springs and the trampling of horses were heard.
Kutuzov and his retinue were returning to the city. The commander-in-chief gave a sign for the people to continue walking freely, and pleasure was expressed on his face and on all the faces of his retinue at the sounds of the song, at the sight of the dancing soldier and the soldiers of the company walking cheerfully and briskly. In the second row, from the right flank, from which the carriage overtook the companies, one involuntarily caught the eye of a blue-eyed soldier, Dolokhov, who especially briskly and gracefully walked to the beat of the song and looked at the faces of those passing with such an expression, as if he felt sorry for everyone who did not go at this time with the company. A hussar cornet from Kutuzov's retinue, imitating the regimental commander, fell behind the carriage and drove up to Dolokhov.
The hussar cornet Zherkov at one time in St. Petersburg belonged to that violent society 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 man, he turned to him with the joy of an old friend:
- Dear friend, how are you? - he said at the sound of the song, matching the step of his horse with the step of the company.
- I am like? - Dolokhov answered coldly, - as you see.
The lively song gave particular significance to the tone of cheeky gaiety with which Zherkov spoke and the deliberate coldness of Dolokhov’s answers.
- Well, how do you get along with your boss? – asked Zherkov.
- Nothing, good people. How did you get into the headquarters?
- Seconded, on duty.
They were silent.
“She released a falcon from her right sleeve,” said the song, involuntarily arousing a cheerful, cheerful feeling. Their conversation would probably have been different if they had not spoken to the sound of a song.
– Is it true that the Austrians were beaten? – asked Dolokhov.
“The devil knows them,” they say.
“I’m glad,” Dolokhov answered briefly and clearly, as the song required.
“Well, come to us in the evening, you’ll pawn the Pharaoh,” said Zherkov.
– Or do you have a lot of money?
- Come.
- It is forbidden. I made a vow. I don’t drink or gamble until they make it.
- Well, on to the first thing...
- We'll see there.
Again they were silent.
“You come in if you need anything, everyone at headquarters will help...” said Zherkov.
Dolokhov grinned.
- You better not worry. I won’t ask for anything I need, I’ll take it myself.
- Well, I’m so...
- Well, so am I.
- Goodbye.
- Be healthy…
... and high and far,
On the home side...
Zherkov touched his spurs to the horse, which, getting excited, kicked three times, not knowing which one to start with, managed and galloped off, overtaking the company and catching up with the carriage, also to the beat of the song.

Returning from the review, Kutuzov, accompanied by the Austrian general, went into his office and, calling the adjutant, ordered to be given some papers related to the state of the arriving troops, and letters received from Archduke Ferdinand, who commanded the advanced army. Prince Andrei Bolkonsky entered the commander-in-chief's office with the required papers. Kutuzov and an Austrian member of the Gofkriegsrat sat in front of the plan laid out on the table.
“Ah...” said Kutuzov, looking back at Bolkonsky, as if with this word he was inviting the adjutant to wait, and continued the conversation he had started in French.
“I’m just saying one thing, General,” Kutuzov said with a pleasant grace of expression and intonation, which forced you to listen carefully to every leisurely spoken word. It was clear that Kutuzov himself enjoyed listening to himself. “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 my honor that for me personally to transfer the highest command of the army to a more knowledgeable and skilled general than me, of which Austria is so abundant, and to relinquish all this heavy responsibility would be a joy for me personally. But circumstances are stronger than us, General.
And Kutuzov smiled with an expression as if he was saying: “You have every right not to believe me, and even I don’t care at all 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 dissatisfied, but could not help but respond to Kutuzov in the same tone.
“On the contrary,” he said in a grumpy and angry tone, so contrary to the flattering meaning of the words he spoke, “on the contrary, your Excellency’s participation in the common cause is highly valued by His Majesty; but we believe that the present slowdown deprives the glorious Russian troops and their commanders-in-chief of the laurels that they are accustomed to reaping in battles,” he finished his apparently prepared phrase.
Kutuzov bowed without changing his smile.
“And I am so convinced and, based on the last letter with which His Highness Archduke Ferdinand honored me, I assume that the Austrian troops, under the command of such a skillful assistant as General Mack, have now won a decisive victory and no longer need our help,” said Kutuzov.
The general frowned. Although there was no positive news about the defeat of the Austrians, there were too many circumstances that confirmed the general unfavorable rumors; and therefore Kutuzov’s assumption about the victory of the Austrians was very similar to ridicule. But Kutuzov smiled meekly, still with the same expression, which 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, turning to Prince Andrei. - 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 a letter from Archduke Ferdinand: “Wir haben vollkommen zusammengehaltene Krafte, nahe an 70,000 Mann, um den Feind, wenn er den Lech passirte, angreifen und schlagen zu 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, seine Absicht alabald vereitelien. 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 er verdient.” [We have quite concentrated forces, about 70,000 people, so that we can attack and defeat the enemy if he crosses Lech. Since we already own Ulm, we can retain the benefit of command of both banks of the Danube, therefore, every minute, if the enemy does not cross the Lech, cross the Danube, rush to his communication line, and below cross the Danube back to the enemy, if he decides to turn all his power on our faithful allies, prevent his intention from being fulfilled. Thus, we will cheerfully await the time when the imperial Russian army is completely ready, and then together we will easily find the opportunity to prepare for the enemy the fate he deserves.”]

Chapter 3. Linear vector spaces

Topic 8. Linear vector spaces

Definition of linear space. Examples of linear spaces

In §2.1 the operation of adding 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 V elements X , at , z ,... called linear vector space, If:

there is a rule that every two elements x And at from V matches the third element from V, called amount X And at and designated X + at ;

there is a rule that each element x and matches any real number with an element from V, called product of the element X per number and designated x .

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

1°. X + at = at + X (commutativity of addition).

2°. ( X + at ) + z = X + (at + z ) (associativity of addition).

3°. There is an element 0 , called zero, such 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.

We will call the elements of linear space vectors regardless of their nature.

From axioms 1°–8° it follows that in any linear space V the following properties are valid:

1) there is a single zero vector;

2) for each vector x there is only one opposite vector (– X ) , and (– X ) = (– l) X ;

3) for any vector X the equality 0× is true X = 0 .

Let us prove, for example, property 1). Let us assume that in space V there are two zeros: 0 1 and 0 2. Putting 3° in the axiom X = 0 1 , 0 = 0 2, we get 0 1 + 0 2 = 0 1 . Likewise, 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. Axioms 1°–8° are obviously satisfied in it.

2. The set of free vectors in 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 space Rn, n N, called arithmetic n-dimensional space, whose elements (vectors) are ordered collections n arbitrary real numbers ( x 1 ,…, x n), i.e.

Rn = {(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 Rn And R We define addition and multiplication by a number using the following formulas:

X + at = (x 1 + y 1 ,…, x n+ y n);

x = (x 1 ,…, x n).

The zero element of space Rn is a vector 0 = (0,…, 0). Equality of two vectors X = (x 1 ,…, x n) And at = (y 1 ,…, y n) from Rn, 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] – set of real continuous ones on the interval [ a; b] functions f: [a; b] R.

Sum of functions f And g from C [ a ; b] is called a function h = f + g, defined by equality

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

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

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

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

f = g f(x) = g(x), " x Î [ a; b].

4.3.1 Definition of linear space

Let ā , , - elements of some set ā , , L and λ , μ - real numbers, λ , μ R..

The set L is calledlinear orvector space, if two operations are defined:

1 0 . Addition. Each pair of elements of this set is associated with an element of the same set, called their sum

ā + =

2°.Multiplying by a number. Any real number λ and element ā L matches an element of the same set λ ā L and the following properties are satisfied:

1. ā+= + ā;

2. ā+(+ )=(ā+ )+ ;

3. exists zero element
, such that ā +=ā ;

4. exists opposite element -
such that ā +(-ā )=.

If λ , μ - real numbers, then:

5. λ(μ , ā)= λ μ ā ;

6. 1ā= ā;

7. λ(ā +)= λ ā+λ ;

8. (λ+ μ ) ā=λ ā + μ ā

Elements of linear space ā, , ... are called vectors.

Exercise. Show yourself that these sets form linear spaces:

1) A set of geometric vectors on a plane;

2) Many geometric vectors in three-dimensional space;

3) A set of polynomials of some degree;

4) A set of matrices of the same dimension.

4.3.2 Linearly dependent and independent vectors. Dimension and basis of space

Linear combination vectors ā 1 , ā 2 , …, ā n Lis called a vector of the same space of the form:

Where λ i are real numbers.

Vectors ā 1 , .. , ā n are calledlinearly independent, if their linear combination is a zero vector if and only if all λ i are equal to zero, that is

λ i =0

If the linear combination is a zero vector and at least one of λ i is different from zero, then these vectors are called linearly dependent. The latter means that at least one of the vectors can be represented as a linear combination of other vectors. Indeed, even if, for example,
. Then,
, Where

.

A maximally linearly independent ordered system of vectors is called basis space L. The number of basis vectors is called dimension space.

Let's assume that there is n linearly independent vectors, then the space is called n-dimensional. Other space vectors can be represented as a linear combination n basis vectors. Per basis n- dimensional space can be taken any n linearly independent vectors of this space.

Example 17. Find the basis and dimension of these linear spaces:

a) a set of vectors lying on a line (collinear to some line)

b) a set of vectors belonging to the plane

c) a set of vectors of three-dimensional space

d) a set of polynomials of degree no higher than two.

Solution.

A) Any two vectors lying on a straight line will be linearly dependent, since the vectors are collinear
, That
, λ - scalar. Consequently, the basis of a given space is only one (any) vector different from zero.

Usually this space is designated R, its dimension is 1.

b) any two non-collinear vectors
will be linearly independent, and any three vectors on the plane will be linearly independent. For any vector , there are numbers  And  such that
. The space is called two-dimensional, denoted by R 2 .