Linear fallowness basis The size of the basis is replaced by the basis. Refining the basis of the vector space Upgrading the linear-independent system to the basis

Come on V vector space over the field R, S- a system of vectors V.

Value 1. The basis of the systems of vectors S be called so ordered linearly independent pidsystem B 1, B 2, ..., B R systems S, what a system vector S line combination of vectors B 1, B 2, ..., B R.

Value 2. By the rank of systems and vectors S the number of vectors in the basis of the system S... The rank of systems and vectors S symbol R= rang S.

Yaksho S = ( 0 ), then the system cannot be transferred to the basis and transfer, but rang S= 0.

stock 1. Let the system of vectors be given A 1 = (1,2), A 2 = (2,3), A 3 = (3,5), A 4 = (1.3). Vector A 1 , A 2 set the basis of the given system, some odors of the linear nezalezhni (div. Butt 3.1) that A 3 = A 1 + A 2 , A 4 = 3A 1 - A 2. The rank of the system of vectors in the road is two.

Theorem 1(Basis theorem). Nekhai S - Kintsev system of vectors z V, S ≠{0 }. Todi is just firm.

1 ° If it is linearly independent, the subsystem of the S system can be upgraded to the basis.

2 ° System S is a basic basis.

2 ° If there are two bases of the system and S, there are the same number of vectors, so that the rank of the system does not lie in the vibration basis.

4 ° Yaksho R= rang S, be it r linearly independent vectors establish the basis of the system S.

5 ° Yaksho R= rang S, Be-like k> r vectors in the system S lineage.

6 ° Be-like vector A€ S One rank to rotate linearly through the vector to the basis, that is, B 1, B 2, ..., B R is a basis of the system S, then

A = A1 B 1 + A2 B 2 +...+ ARB R; A1 , A2 , ..., AN€ P,(1)

І taka vistava dina.

By virtue of 5 °, the basis Maximally linear independent pidsystem systems S, and the rank of the system S the number of vectors in such a subsystem.

Vector submission A in viglyadі (1) be called Laying out vectors by vectors to the basis, And the numbers a1, a2 , ..., ar are called Vector coordinates A At the same basis.

Delivered. 1 ° Nekhai B 1, B 2, ..., B K- linearly independent pidsystem of the system S... Yaksho kozen vector systems S Linearly rotate through the vector of our subsystem, then for the value of the won є the basis of the system S.

Yaksho is the vector of the system S, which does not linearly rotate through the vector B 1, B 2, ..., B K, then meaningfully yogo through B K+1. Todi systems B 1, B 2, ..., B K, B K+1 - linearly square. Yaksho kozen vector systems S Linearly rotate through the vector of the central system, then the basis of the system S.

Yaksho is the vector of the system S, which line does not roll through B 1, B 2, ..., B K, B K+1, then mirkuvannya is repeatable. Prodding processes, or we will come to the basis of the system S, for the number of vectors in linear independent systems is increased per unit. Bo in the system S If there are a number of vectors, then another alternative cannot be continued indefinitely; S.

2 ° Come on S kintsev system of vectors S ≠{0 ). Todi in the system Sє vector B 1 ≠ 0, which will set up a line-independent system S... According to the first part of it, it is possible to upgrade to the basis of the system S... In such a rank, the system S maє basis.

3 ° It is acceptable that the system is S maє two bases:

B 1, B 2, ..., B R , (2)

C 1, C 2, ..., C S , (3)

Based on the basis, the system of vectors (2) is linearly independent and (2) Í S... Given the basis of the skin vector of the system (2) line combination of vectors in the system (3). Todi following the main theorem about two systems of vectors R £ S... Similarly, it should be S £ R... Three of two vapors' irregularities R = S.

4 ° Nekhai R= rang S, A 1, A 2, ..., A R- linear independent pidsystem S... It will be shown that is a basis of the systems S... If it’s not a basis, then according to the first part of the її it is possible to add to the basis and the basis is acceptable A 1, A 2, ..., A R, A R+1,..., A R+T, how to take revenge on more R

5 ° Yaksho K vector A 1, A 2, ..., A K (K > R) systems S- linearly independent, then, according to the first part, the system of vectors can be added to the basis and the basis is recognizable A 1, A 2, ..., A K, A K+1,..., A K+T, how to take revenge on more R vector Tse superinterpret what was reported in the third part.

6 ° Nekhai B 1, B 2, ..., B R basis of the system S... For the basis, be a vector A € Sє line combination of vectors in the basis:

A = a1 B 1 + a2 B 2 + ... + ar B R.

Bringing on the identity of such a manifestation is permissible inappropriate, but more than one manifestation:

A = b1 B 1 + b2 B 2 + ... + br B R.

It is known by the time it is known

0 = (a1 - b1) B 1 + (a2 - b2) B 2 + ... + (ar - br) B R.

Oskilki basis B 1, B 2, ..., B R linearly independent system, all performance ai - bi = 0; I = 1, 2, ..., R... Also, ai = bi; I = 1, 2, ..., R that unity has been brought.

V.V. Golovizin Lectures on algebra and geometry. 5

Lectures on algebra and geometry. Semester 2

Lecture 23. The basis of the vector space.

A short change: the criterion of the lineage of systems and non-zero vectors, subsystems of systems and vectors; basis.

item 1. Criterion of linear deposition of a system of non-zero vectors.

Theorem. The system of non-zero vectors is linearly stale only if there is a vector of the system, but it is linearly rotated through the front vectors of the system.

Delivered. Don't let the system be stored from non-zero vectors and is not stale. The system can be seen from one vector:
... Because
, then the system
- Linear Square. Vector ... The system has been removed
is linearly square, then the forward vector is always present: ... І etc. prodovzhuєmo doti, docks not otrimamo line-up I will lay down the system
de. This number is obov'yazkovo known. Outgoing system
є lіnіyno fallow behind the mind.

Otzhe, for the inducement, the system was lined up in a line.
, moreover, the system
є Linear Square.

System
representing a zero vector is nontrivial, so. there is such a non-zero set of scalars
, scho

de scalar
.

Spravdy, inakshe, yaksho
, then mi mali b nontrivially the manifestation of a zero vector by a linearly independent system
, uh, it’s unfortunate.

Increased balance for a non-zero scalar
, we can see the vector :

,

Oskіlki zvorotne solidarity is more obvious, the theorem has been completed.

item 2. Pidsystems and vector vector space.

Viznachennya. Be not empty under multiplicity of systems and vectors
be called a subsystem of a chain of systems and vectors.

butt. Come on
- a system of 10 vectors. Todi systems and vectors:
;
,
- Pid systems of the chain of systems and vectors.

Theorem. If the vector system is in line with the depleted subsystem, then the vector system is also deserted.

Delivered. Let the system of vectors be given
do not care for the value of the subsystem
, de
є line fallow. Todi won represents a zero vector nontrivially:

de middle kofіtsієntіv
є I would like to have one not equal to zero. Alle to the offensive equality non-trivial manifestations of the zero vector:

signs, due to the value, vyplyaє lineage of the system
, Ch.d.

The theorem has been completed.

Slidstvo. Whether it is a line-independent system and vectors є line-independent system.

Delivered. Acceptable inappropriate. Do not go as the subsystem of the whole system is linear fallow. Todi with the theorem of the flowing line the depletion of the system of the system, how to supervise the minds.

Slidstvo brought.

p. 3. Systemi of arithmetic braces vector spaciousness of braces.

From the results of the foregoing section, as an example of the topic, the following theorem is used.

1) The system of stoppages є linearly fallow, if only one hundred of them wants to be in the system, which can rotate linearly through the other hundred of the central system.

2) The system is one hundred percent of the line-of-the-way, if the system is not one hundred percent of the system and does not rotate through one hundred percent of the system.

3) The system is stovpts_v, scho to avenge a zero stovpets, є lyn_yno fallow.

4) The system of stoppages, where to place two equal stoves in a line of fallow land.

5) The system of stovpts_v, scho to place two proportional stovptsi є lin_yno fallow.

6) The system of stoppages, which is to place a line of fallow subsystem, and a line of fallow.

7) Be-like a pidsystem of a line-independent system of stations є line-independent.

Alone, it is possible, here it is necessary to clarify the understanding of the proportional standards.

Viznachennya. Two non-zero hundredths
called proportional, when there is a scalar
, such, scho
abo

,
, …,
.

butt. System
In line with the fallow land, that is why there are two hundred proportions.

Respect. We already know (div. Lecture 21), that the designer is back to zero, as the system of one hundred percent (rows) є is linearly fallen. Nadal will be informed that it is true and reliable: if the designer is not expensive, then the system of the first row and the system of the first row will be fallen asleep.

item 4. Vector space basis.

Viznachennya. Vector system
vector space over the field is called a generic (set up) system of vectors in the vector space, if you represent any vector, tobto. there is such a set of scalars
scho.

Viznachennya. The system of vectors in the vector space is called a minimal system, which is generated, as when seen from the whole system, any vector won’t cease to be a system that is inherent.

Respect. From the point of view of the system of vectors, but it’s not minimal, then we want one vector of the system, when you see any of the systems, the system of vectors, it’s gone, as if it’s going to be born sooner.

Lemma.

If, in a linear fallow and rock-juvenile system of vectors, one of the vectors is linearly twisting through the other, then it can be seen from the system and the system of vectors, which has become redundant, if it is.

Delivered. Come on system
It is linearly fallen, and it’s not just one of the vectors, and one of the vectors is linearly spinning through the vectors of the whole system.

For the value and for the sake of simplicity, let’s write it down.

So yak
- If the system is rooted, then
there is such a set of scalars
, scho

.

Zvidsi otrimumo,

tobto. be a vector x linearly rotate through the vectors of the system
, and tse means, wona є by the system, by, ch.t.d.

Naslidok 1. Linearly fallen that rock-juvenile system of vectors is not minimal.

Delivered. Immediately, the vaping from the lemma and the designation of the minimal system and vectors, which is the case.

Naslіdok 2. Minimal system of vectors, which is common, linearly square.

Delivered. Allowing inappropriateness, it comes to supernaturalness due to 1.

Viznachennya. p align = "justify"> The system of vectors in the vector space is called the maximal linearly independent system, even if the system of any vector is added to the whole system, it becomes linearly fallow.

Respect. From the point of view of the next step, the system is linearly independent, but at the maximum, then there is a vector, when added to the system, the system is linearly independent.

Viznachennya. The basis of a vector space over a field K is an ordered system of its vectors, which represent any vector of a vector space in one way.

In other words, it seems, the system of vectors
vector space V over the field K is called its basis, which
isnu single set of scalars
, such, scho.

Theorem. (About chotiri equal to the basis.)

Come on
- Ordered a system of vectors in vector space. Todi are strong enough:

1. System
є basis.

2. System
є Linearly square system of vectors, scho breed.

3. System
є maximum linearly square system of vectors.

4. System
є minimal system of vectors

Delivered.

Come on system of vectors
є basis. On the basis of the basis at once, we need to bring the system of vectors to the system of vectors in the vector space, so we need to bring the line of independence.

Acceptable, scho given the system vector of line fallow. Today, two manifestations of a zero vector are trivial and non-trivial, but to be supervised over a given basis.

Come on system of vectors
є Linear Square that breed. We need to make sure that the system is linearly independent є as much as possible.

Acceptable inappropriate. Don't worry, the system of vectors is not maximal. Todi, through the respected vische, there is a vector that can be added to the whole system, and the system of vectors is rendered out to become linearly square. However, on the other hand, the additions to the system and the vector can be represented by the viewer line combination out-of-order systems and vectors through those that are in the system that are inherently.

I can’t recognize that in a new, expanded system of vectors, one of the vectors is linearly rotated through the other vectors of the system. This is a system of vectors є linear fallow. We've gotten some rubbing.

Come on system of vectors
vector space є maximum linear square. Evidently, it’s a minimal system, it’s a breed.

a) The sphatku is brought, now, by the system, by the way.

It is great, through the line of independence, the system
do not take revenge on the null vector. Let's get a good non-zero vector. Dodamo yogo to the given systems and vectors:
... The system of non-zero vectors, which has arrived, is linearly fallow, because The output system of vectors is maximum linearly independent. This means that the whole system has a vector, but it turns linearly through the front. At the out-of-the-box line-independent systems
However, it is not possible to rotate through the front, the same, linearly turn through the front vector x. In such a rank, the system
imagine a non-zero vector. Lingering respect, well, the system, zoosuly, representing і a zero vector, tobto. system
є for the juvenile.

b) Now we are talking about minimalism. Acceptable inappropriate. Even one of the vectors in the system and there may be visions from the system and the system of vectors, which has been lost, as and earlier would be a kind of juvenile system, and, also, outward from the system, and the vector may also be linearly wandering through the vector system of the system, but not in the future.

Come on system of vectors
vector space є minimal system, scho roozhuє. Todi vona represents a vector vector to the vastness. We need to bring the unity of the manifestation.

Acceptable inappropriate. Let a certain vector x rotate linearly through the vectors of the given system in two different ways:

In recognition of the same rationality, we will recognize:

Vnasledok Slidstva 2, system
є Linear Square, Tobto. representing a zero vector is only trivial, so all the performance of the linear combination is due to zero:

In such a rank, be it a vector x to rotate linearly through the vectors of a given system in one way, etc.

The theorem has been completed.

p. 5. The size of the vector is wide.

Theorem 1. (About the number of vectors in linear independent systems of vectors.) The number of vectors in any linear systems of vectors does not change the number of vectors in any systems of vectors and vector space.

Delivered. Come on
a fairly linear system of vectors,
- the system is pretty good. Acceptable, scho.

Because
rocking system, won’t represent be-like vector to space, spring і vector ... It comes to the whole system. I will recognize linearly fallen asleep for the juvenile system of vectors:
... Todi you know vector
the system of systems, which linearly rotates through the forward vector of the system, and this, due to lemia, can be seen from the system, moreover, the system of vectors, which has been overshadowed, as previously generated.

... Because qia system of rojuє, then won representє vector
and, joining this system to the whole system, I will know how to line up the juvenile system:.

Let everything repeat itself. There is a vector in the whole system, which rotates linearly through the front, and the vector cannot be since Outgoing system
linearly square vector do not swing linearly through the vector
... This means that only one of the vectors
... I see it from the system, it will be recognized, if it is re-numbered, the system will be generated by the system. By prodding the process, through the crocs we can recognize the system of vectors
since for our pripuschennya. Otzhe, tsya system, how kind, represents a vector, how to supervise the minds of the linear independence of the system
.

Theorem 1 has been completed.

Theorem 2. (About the number of vectors in the basis.) In the skin basis of the vector space there is one the same number of vectors.

Delivered. Come on
і
- two good bases of vector space. Be-like a basis є linearly independent system of vectors, which is the basis.

Because If the system is linearly independent, and if the other is in general, then, according to Theorem 1,
.

Similarly, the other system is linearly independent, and the persha is very common, then. Zvidsi next
, Ch.d.

Theorem 2 is completed.

The Qia theorem allows us to introduce the same value.

Viznachennya. The size of a vector space V over a field K is the number of vectors in its basis.

Designation:
abo
.

p. 6. Taking the basis of the vector space.

Viznachennya. The vector space is called Kintsev, as it is in the Volodya Kintsevo system of vectors, which is common.

Respect. Mi vivchatimo deprive the vector space. Unimportant to those who already have a lot of knowledge about the basis of the endless vector space, but it’s not the same as the basis of such a space. Efforts earlier to reject the power of the bully have been rejected from the pre-empted, which is the basis of isnu. The food curtailment theorem is approaching.

Theorem. (About the basis of the endless vector space.) Any kind of endless vector space is a basis.

Delivered. Behind the sinking is a set of vectors in a given endless vector space V:
.

Impressively immediately, that the system of vectors is empty, tobto. do not take revenge on the zhodnogo vector, for the value of vvazhayut, but the vector is not space є is zero, so that.
... At the same time, the basis of the zero vector space is an empty basis and the size of the basis is zero.

Come on, non-zero vector space
Kintseva system, which generates non-null vectors. Iakshcho vona is linearly independent, everything is brought up, tk. linear system of vectors in the vector space is a common basis. If a system of vectors is given in line with fallow, then one of the vectors of the whole system will rotate linearly through that system, but it is possible to see it from the system, and the system of vectors, which will probably be lost, due to Lemi No. 5, will sooner.

We re-number the system of vectors, which has been lost:
... Dalі mіrkuvannya repeat itself. While the system is linearly independent, it’s the basis. Well, then I know there is a vector in the system, which can be seen, but the system, which has been lost, will become a juvenile.

By repetitive processes, we can get rid of the empty vector system, because in extreme disagreement, we come to the system, which is generated, from one nonzero vector, which is linearly square, but, also, a basis. To that, at the very least, it comes to the linear independent systems of vectors, which are very common, tobto. to the basis.

The theorem has been completed.

Lemma. Come on. Todi:

1. Be-yaka vector system є lіnіyno fallow.

2. Whether the system of vectors is linearly independent as a basis.

Delivered. one). As a matter of fact, the number of vectors in the basis is both the basis itself and the system, so the number of vectors in any linear independent system cannot be overridden.

2). Like a system of vectors, whether it is linearly independent, the system of vectors is a maximum, but also a basis.

The lemma has been completed.

Theorem (About the addition to the basis) Whether the system of vectors in the vector space is linearly independent, it can be extended to the basis of the wide space.

Delivered. Let the vector space of size n that
deyaka linear-square system of yogo vectors. Todi
.

Yaksho
, Which is in front of the front lemy, the system is a basis and brings nothing.

Yaksho
Todi tsya system is not a maximal linear independent system (іnakhe won bula b basis, but it’s not a pity, because). Also, you know, there is a vector
, such, scho system
- linearly square.

Yaksho, now, then the system
є basis.

Yaksho
, everything will repeat itself. The process of updating the system cannot be continued indefinitely, because on the skin, the system of vectors in the space can be recognized linearly, and behind the forefront, the number of vectors in such systems cannot outweigh the space in space. Otzhe, at least in the smallest detail, we will come to the basis of the vastness.

The theorem has been completed.

p. 7. butt.

1. Nehay K - a sufficient field, - arithmetic vector space of 100%. Todi. For the support of the system, the system is 100% spacious.

Viznachennya. Element system xh ..., xch line space V is called a lineage fallow, as long as the numbers a ", ..., otq are known, not all equal to zero, and so, if the equality (1) is displayed only for a] = ... = aq = 0, then the system of elements xj ,. .., x9 are called linearly square. These are just things. Theorem 1. The system of elements X \, ..., xq (q ^ 2) is linearly stale for a whole lot if you want one of these elements to be possible in a linear combination of others. It is admissible that the system of elements xb ..., xq is linearly stale. Noticeably for the value, for the equality (1) the value of the value is displayed as a9. Transferring all supplements, except for the rest, to the right part, by sending them to otq F About, the element xq is a linear combination of elements xi, ..., xq: Back, as one element of transferring livu part, we can accept the line combination from the zero efficiency (-1 F 0). Otzhe, the system of elements Xi, _____ xq is linearly fallen. Theorem 2. Don't worry, the system of elements X |, ..., X9 is linearly independent і у = a \ X \ +. + Aqxq. Todi kofitsinti ori, ..., aq are based on the element of the same rank. m Nekhai todі Linear deposit Basis Rosemir Replace the basis of the star. From the line of independence of elements X |, ..., xq vyplyaє, so a (i, from the same, a Theorem 3. The system of elements, so to locate a line depleted subsystem, line depleted. " xg + l, ..., хт lіnіyno olezhnі.Todi there is a lіnіyna combination of these elements such, but not all functions come ", ..., aq to bring zero. a whole line of space, like elements in |, ..., line-on-line and leather element from V can be represented in the view of the line combination. Nekhai a.b.c - three non-coplanar vectors із Vj (Fig. 6) .Todi ordered three-lines - different bases Nekhai z = (v! ... en) - basis for the space V. Todi for any element x z V there is a set of numbers ..., З such, but by virtue of Theorems 2 the numbers, ..., З are the coordinates of the element x in the basis - the values ​​are uniquely. We are happy to see the coordinates of the elements with the simplest actions with them. Do not, for any number but in such a rank, when the elements are folded, the coordinates are stored, and when the element is multiplied by the number of the whole coordinate, it is multiplied by the whole number. The coordinates of an element are often manually recorded. For example, n is the coordinate value of the element in the basis. The system of elements X |, ..., x can be stored, according to the basis with, and the coordinates of the elements X |, ..., x9 are clearly visible on the basis of the following: Theorem 4. The system of elements x \, ..., xq is linearly fallen to if only the system of coordinate stations is in line with the baseline. * Let me tell you why I would like to have one of the functions A * change from zero. Recorded during the report of Zvidsi through the single distribution of the element on the basis of the viplivay, the level of the density of the basis. ?). Tse means that the coordinate system is linearly fallen. As soon as one observes the parity (2), then, when conducting peace at the vortex order, the formula (1) is obsessed. Tim by ourselves, turning to zero deyakoi non-trivial (I want one of the factors to come from zero) of the linear combination of elements in the linear space, which is not trivial to the combination of coordinates Theorem 5. Let the basis of the linear space V be built up from n elements. Todi every system of elements, de t> n, is linearly stale. For, well, maybe, * 3 I'll look at Theorem 3 to see how it looks like Nekhai Xj, ..., xn + | - enough elements for the vastness of V. It is possible to put the skin element behind the basis and write down the coordinates of the elements ........... in the matrix view, in the same number of coordinates of the element. Otrimanimo matrix with rows of n + 1 hundred percent - For those, where the rank of matrix K does not change the number of rows, one hundred matrices K (ïx n + 1) are lined with fallow. Oscillations of the coordinates of the coordinates of the elements, then, according to Theorems 4, the system of elements X | ..... х „+ | it is also linearly fallow. Slidstvo. All bases of the linear space V are stored from the same number of elements. Let the basis be stored in n elements, and the basis in n elements. By the way, with the theorems we can recognize that n ^ n ". Tim by ourselves, n = n. The number of elements in the basis of space V is called the number of elements in the basis of the space. Application 1. The basis of the coordinate space. .., it is not linearly independent: it can be readily recognizable, which means, In addition, whether it is an element E, = ... s R "can be written in the line combination of elements Tim by ourselves, the size of the space R. A single line system is a non-null solution, which is a fundamental solution system (FSR). FSR is the basis of the linear space of the solution of the one-sided system. The size of the whole line of space for the road to the number of elements of the FSR, tobto. n - p de g is the rank of the matrix of coefficients of the one-row system, an is the number of non-home ones. Appendix 3. The size of the line-to-space Mn of the large step is not visible to the road n + 1. 4 So, as any polynomial / * (() step is not to see, it is possible to reach the display of the line-neatness of the element at =. 0, can be recognized, scho "o = 0. 5 Zak. 750 Prodifferentsiiєmo parity (3) by t: Knowing POSITION t = 0, recognizable, SCHO 0 | = 0. Continuing the process, the last change has changed in that, well oo =" I = ... = a „= 0. Ce. means that the system of elements y | = 1, ..., en4) = * is linearly independent. Otzhe, shukana size of doors n + 1. Pleasure. It’s far from a whole lot of space to get involved everywhere, otherwise it’s unimaginable, as the vastness of the line-to-space V is far away. It is clear that if W is in the n-vimir linear space V, then dim W ^ n. It will be shown that in the n-vimir linear space V is in the linear space, be it in the n-vimir linear space to ^ n. It is easy to turn over to the fact that the shell is small up to. Based on the value Theorem b (on the basis of a new basis). Let the system of elements in the linear space V in the space and in the linear space in. Todi in the space V there are elements a * + 1, ..., as well, but the system a is the basis V. it is linearly fallen, then as in a non-trivial linear combination ... there will be a combination of functions in the case of a linear independence of the system, and it is possible to write it for any element | s values. Ale through the mind is ill-advised. In addition, there is an element a * + i € V such that the system ai, ..., ab, a * + | will be linearly square. When up to + 1 = n, the system is the basis for the space V. When up to + 1, then the system a next repeat in front of the world. In such a way, be-yaku given a linearly independent system of elements can be added to the basis of the whole space V. Butt. Add two vectors to the system | = (1,2,0,1), aj = (-1,1.1,0) the space R4 to the basis space. However, in the space of R4 the vector aj = (and it will be shown that the system of vectors ai.aj.aj, a4 is the basis of R4. , also, і vectors at ag.az, а ^ lіnіyno nezalezhnі> A subtitle іnідхід vikoristyutsya і in a zeal vypad: if you add the system to the linear independent elements to the basis of space, the matrix Lіnіostіnі Basic forms The rank of the possessed matrix is ​​suitable for the item. It is fair to approach the basis of the matrix. Theorem 7. Nekhai - the line space of the linear space of V, Todi. from the basis of the village. The proof of the chain of power is carried out from the protolezhny., ... "e" n at the basis of the s. basis. Otzhe, pri If det S = 0, it is not so. 2. Yaksho ..., і ..., are the coordinates of the element x in the bases with і c "as if matrix I will write down the values ​​of the equality, it will change at the fairness of the quality 2. 3. S-1 - the matrix for the transition from the basis from to the basis from.

It is called Kintsevomirnim, because it’s a Kintsev system of vectors, so it’s born.

Respect. Mi vivchatimo deprive the vector space. Unimportant to those who already know a lot about the basis of the endless vector space, we have a lot of ideas about such a space. Effortlessly earlier, we have eliminated the boules from the posed, but the basis is simple. The food is closing.

Theorem. (About the basis of the endless vector space.)

Be a kintseviy vector space is a basis.

Delivered. Behind the mind is the Kintsev system, which is the origin of the given Kintsevomir vector space V:.

Impressively immediately, that the system of vectors is empty, tobto. do not take revenge on the vector vector, then for the value of the vector, vvazhayut, so the vector space є is zero, tobto. ... At the same time, the basis of the zero vector space is an empty basis and the value of the value is equal to zero.

Well, the system is square, everything is done, because linear system of vectors in the vector space is a common basis.

If a system of vectors is given in line with the fallow, then one of the vectors of the whole system will rotate linearly through that system, and it is possible to see it from the system, and the system of vectors, which will be lost, will be generated earlier.

We re-numbered the system of vectors, which has been overshadowed:. Dalі mіrkuvannya repeat itself.

While the system is linearly independent, it’s the basis. Well, then I know there is a vector in the system, which can be seen, but the system, which has been lost, will become a juvenile.

By repetitive processes, we can get rid of the empty vector system, because in extreme disagreement, we come to the system, which is generated, from one nonzero vector, which is linearly square, but, also, a basis. To that, at the very least, it comes to the linear independent systems of vectors, which are very common, tobto. to the basis, p.t.d.

The theorem has been completed.

Lemma. (About systems of vectors in the n-dimensional vector space.)

Come on. Todi:

1. Be-yaka vector system є lіnіyno fallow.

2. Whether the system of vectors is linearly independent as a basis.

Delivered. one). As a matter of fact, the number of vectors in the basis and the basis is a kind of system, so the number of vectors in any linear independent system is unsuitable for changing, so that. be it a system to take revenge on the vector, є linіyno fallow.

2). Like a system of vectors, whether it is linearly independent, the system of vectors is a maximum, but also a basis.

The lemma has been completed.

Theorem (About the addition to the basis) Whether the system of vectors in the vector space is linearly independent, it can be extended to the basis of the wide space.

Delivered. Do not have vector space of dimension n that the system of vectors is linearly independent. Todi.

Yaksho, then from the front lemma, the system is a basis and bring nothing.

Also, this system is not a maximal square system (the basis, which is unfortunate, because). Otzhe, znaydetsya vector, such a system - linearly square.

Yaksho, now, then the system є basis.

Well, everything will repeat itself. The process of updating the system cannot be continued indefinitely, because on the skin, the system of vectors in the space can be recognized linearly, and behind the forefront, the number of vectors in such systems cannot outweigh the space in space. Otzhe, on the other hand, on the basis of a small deydemo basis of the given space., Ch.t.d.

Viznachennya. Basis

arithmetic vector space of one hundred and fifty n is called canonical and natural.