Linear shell of the vector system. Povyazanі vznachennya that power

Let the vector system out of vector space V over the field P.

Appointment 2: Linear shell L systems A nameless all linear combinations of vectors in the system A. Appointment L(A).

Can you show me what for any two systems Aі B,

A linearly through B then and only then, if . (one)

A equivalent B then and only then, if L(A)=L(B). (2)

The proof is strong from the previous authority

3 Linear shell of a vector system in the open space V.

Bringing

Let's take two vectors and that z L(A), what may come next in terms of vectors A: . Let's check the mentality of minds 1) and 2) according to the criterion:

So as a linear combination of vector systems A.

Oskilki tezh є linear combination of vector system A.

Let's look at the matrix now. Linear shell of a row matrix A called the row space of the matrix, it is designated L r (A). Linear shell of the matrix A called stovptsevim space and signified L c (A). To give respect, that with the row and column space of the matrix Aє subspaces of different arithmetic spaces P nі Pm obviously. Koristuyuchis hardennia (2), you can make such a visnovka:

Theorem 3: Just as one matrix is ​​taken from another lansy of elementary transformations, rows of spaces of such matrices are formed.

Suma ta peretin pіdprostorіv

Come on Lі M- two open spaces R.

sumoyu L+M called impersonal vectors x+y , de x ∈Lі y ∈M. It is obvious that there should be a linear combination of vectors L+M linger L+M, later L+Mє under space R(you can get away with space R).

Peretin L∩M pіdprostorіv Lі M called impersonal vectors, which lie at the same time subspaces Lі M(may add up to more than a zero vector).

Theorem 6.1. The sum of the volumes of the prevіlnyh pіdprostorіv Lі M end world linear space R more spaciousness of the sum of these subspaces and the spaceiness of the peretina of these subspaces:

dim L+dim M=dim(L+M)+dim(L∩M).

Bringing. Significantly F=L+Mі G=L∩M. Come on G g-peaceful pidprostir. We choose in a new basis. so yak G⊂Lі G⊂M, also the basis G you can add to the basis L i to the basis M. Give the basis to subspace L and give the basis to subspace M. Let's show that the vectors

(6.1) add the basis F=L+M. In order for the vectors (6.1) to add a basis to the space F stench can be but linearly independent and be a vector of open space F can be shown by a linear combination of vectors (6.1).

Let us bring the linear independence of vectors (6.1). Let the zero vector open F is represented by a linear combination of vectors (6.1) with various coefficients:

The last part (6.3) is the vector subspace L, and the right part is the vector subspace M. Otze vector

(6.4) lie within subspace G=L∩M. From the other side, vector v can be shown by a linear combination of basic vectors in subspace G:

(6.5) Z equal (6.4) and (6.5) may:

Ale vector and є basis subspace M, since the stench is linearly independent. Then (6.2) in the future I look:

Looking back at the linear independence of the basis of subspace L maybe:

The scores of all coefficients for equal (6.2) turned out to be zero, vectors

linearly independent. Ale be-yaky vector z h F(for the allocated amount of pidspace) you can give a sum x+y , de x ∈ Ly ∈ M. Into your hell x represented by a linear combination of vectors y - Linear combination of vectors. The same vectors (6.10) affect subspace F. We have taken into account that the vectors (6.10) satisfy the basis F=L+M.

Vivchayuchi basis pіdprostorіv Lі M that basis subspace F=L+M(6.10), maybe: dim L=g+l, dim M=g+m, dim (L+M)=g+l+m. Father:

dimL+dimM−dim(L∩M)=dim(L+M). ■

Direct sum of subspaces

Appointment 6.2. Expanse F is a direct sum of subspaces Lі M yakscho kozhen vector x space F you can use the same way of representations for the visual sumi x=y+z , de y ∈L and z ∈M.

Direct amount is indicated L⊕M. Say what it is F=L⊕M, then F spread out at the direct sum of their subspaces Lі M.

Theorem 6.2. In order to n- peaceful space R was a direct sum of subspaces Lі M, enough, schob peretin Lі M avenged only the zero element and the total expansion of R Lі M.

Bringing. We choose a real basis for the subspace L and a real basis for the subspace M. We conclude that

(6.11) R. Behind the mental theorem, the expansion of space R n increase the amount of subspace Lі M (n=l+m). It is sufficient to bring the linear independence of the elements (6.11). Let the zero vector open R is represented by a linear combination of vectors (6.11) with the following coefficients:

(6.13)Scales of the left part (6.13) as a vector subspace L and the rights of the part - by the vector subspace Mі L∩M=0 , then

(6.14) Ale vectors and є bases of subspaces Lі M obviously. Father, stinks are linearly independent. Todi

(6.15) It was established that (6.12) is true only for the mind (6.15), and then to bring the linear independence of the vectors (6.11). Otzhe stinks to satisfy the basis of R.

Let x∈R. Laying out yoga behind the basis (6.11):

(6.16) Z (6.16) maybe:

(6.18)З (6.17) и (6.18) R you can show the sum of vectors x 1 ∈Lі x 2 ∈M. It is necessary to bring what is the only manifestation. Let the crime of manifestation (6.17) є th such manifestation:

(6.19) See (6.19) from (6.17), take

(6.20) Oskilki, i L∩M=0 , i . Otzhe i. ■

Theorem 8.4 about the expansion of sums of subspaces. As for the space of the last linear space, then the space of the sum of the space of the door space of the sum of their dimensions without the space of their peretina ( Grassmann's formula):

Right, high - the basis of the peretina. Supplemented by an ordered set of vectors to the subspace basis and an ordered set of vectors to the subspace basis. Such an addition is possible after Theorem 8.2. Three assignments of three sets of vectors are stored in orderings of a set of vectors. It is shown that these vectors are such that they make space. Indeed, whether a vector of which space is represented by a linear combination of vectors from an ordered set

Father, . Let us know what is linearly independent and that is the basis of space. The right foldable linear combination of these vectors and equals them to the zero vector: . All coefficients of such an arrangement are null: the subspace of the vector space with a bilinear form is the absence of all vectors orthogonal to the skin vector s. Tsya multiplier is a vector subspace, which sound is indicated.

Linear shell is also called subspace, spawn X. Call out to be identified. To say so, scho is a linear shell stretched over faceless X .

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Let's go - a system of vectors from . Linear shell vector systems is called impersonal of all linear combinations of vectors in the system, so

The dominance of the linear shell: Yakshcho, then for i.

Linear shell can be closed in terms of extension to linear operations (the operation of adding and multiplying by a number).

Pіdnіzhina pіdnіstі, scho mає vіst vіst zolivoєі іnіnіnієnіnі іnіnі opіrіy sladanіa і ііѕ mulіnіnіnі nіlі, is calledlinear subspace .

The linear shell of the vector system is a linear subspace.

The system of vectors is called the basis , as

Whether a vector can be shown by looking at a linear combination of basic vectors:

2. The system of vectors is linearly independent.

Lemma behind the basis is assigned unambiguously.

Vector folds with coefficients of vector layout behind the basis is called the coordinate vector of the vector at the base .

Appointment . This record supports that the coordinates of the vector lie in the basis.

Linear spaces

Appointment

Let it be given to impersonal elements of sufficient nature. Come on, for the elements of the multiplier, two operations are assigned: adding that multiplier to be speech number : i bezlіch closed chodo tsikh operations: . Let these operations be ordered by the axioms:

3. use zero vector with authority for ;

4. for dermal ischnuє zvorotny vector of power;

6. for , ;

7. for , ;

Todi such a multiplicity is called linear (vector) space, yoga elements are called vectors; scalars one). The space that consists of more than one zero vector is called trivial .

If in axioms 6 - 8 we allow the multiplication of i on complex scalars, then such a linear expanse is called comprehensive. For forgiveness, the mirroring should be more than enough space for us to look at it.

The linear space is a group of additional operations, moreover, a whitewash group.

It is elementary to bring the unity of the zero vector, and the unity of the vector, the reverse vector: , yogo ring out to mean.

The submultiplier of a linear space, which itself is a linear space (that is why the folding of a vector is closed and multiplied by a certain scalar), is called linear subspace space. Trivial spaces linear expanse are called the same and expanse, which is composed of one zero vector.

butt. Expanse of ordered triplets of real numbers

operations, which are equalities:

The geometrical interpretation is obvious: the vector in the space, "bindings" to the cob of coordinates, there can be tasks in the coordinates of your end. The little one shows a typical subspace: a flat, yak to pass through the cob of coordinates. More precisely, the elements are vectors that move the cob on the cob of coordinates and kіnci - at the points of the plane. The closure of such a multiplier of the random folding of vectors and their expansion 2) is obvious.

Vykhodyachi z tsієї geometrical ї іnpretatsії, often talk about the vector of enough linear space as about point to space. In other words, the point is called the "end of the vector". Crim of the clarity of associative spriynyattya, sim words do not hope for a formal sense: understanding the "end of the vector" is in the axiomatics of the linear expanse.

butt. Based on the same example, you can give a different interpretation of the vector space (I will put it before the speech, even in the very same words “vector” 3)) - it will indicate the set of “suviv” points to the space. Qi destruction - or parallel transfer be like a spacious figure - choose parallel planes.

Seemingly, with similar interpretations of the understanding of the vector, everything is not so simple. Try to appeal to yoga physical zmist - like to the object, what can valueі straight ahead- Call out the just answer of the suvorih mathematicians. The designation of the vector as an element of the vector space is already guessing episode s sepulcs from the famous fantastical explanation of Stanislav Lem (div. ☞HERE). Do not get hung up on formalisms, but follow this fuzzy object in your own private manifestations.

butt. Space to serve as natural corners: vector space . One of the ways to set up a subspace is to set up a boundary.

butt. An impersonal solution of the system of linear homogeneous equalities:

I create a linear space for space. Right, that's right

Solution of the system, then

Tezh solution for whatever. Yakscho

One more solution to the system, then

There will be її decisions.

Why is the solution of the system heterogeneous equals do not satisfy the linear subspace?

butt. Uzagalnyuyuchi far away, we can look at the expanse of "non-small" rows of abo sequences , which sounds like an object of mathematical analysis - when looking at sequences and rows. You can look at the rows (sequences) of "unskinned in the offending sides" - the stench is victorious in THEORY SIGNAL.

butt. An impersonal matrix with speech elements and operations of adding matrices and multiplications on speech numbers establishes a linear space.

In the space of square matrices, one can see two subspaces in order: subspace of symmetrical matrices and skew-symmetrical spaces of matrices. In addition, the skin is made up of multiple spaces: upper tricut, lower tricut and diagonal matrices.

butt. The impersonal polynomials of one variable step are exactly the same as the coefficients s do not satisfy linear space. Why? - To that, we won’t close it until we add: the sum of polynomials will not be a polynomial of the th step. Ale os bezlich polynomial step Not more

linear expanse utvoruє; only up to tsієї a multiplier is required to give a total zero polynomial 4) . Obvious subspaces є. In addition, the subspaces will be impersonal pairs and impersonal unpair polynomials of the step no more. The impersonality of different polynomials (without the exchange on the steps) also establishes the linear space.

butt. In order to understand the forward trend, there will be an expanse of polynomials for a number of changing steps, no more than with coefficients. For example, impersonal linear polynomials

make up the linear space. Anonymous homogeneous polynomials (forms) of a step (from advancing to the same multiplier of the same zero polynomial) - also a linear expanse.

From the point of view of the induced vision of the appointment, the faceless rows with the whole components

one can look at the operation of the component-by-component addition and multiplication by integer scalars, which is not a linear space. Tim is not less, all axioms 1 - 8 will be victorious, as it is permissible to multiply only by the number of scalars. We don’t accentuate respect for this object, but we can do it in discrete mathematics, for example, in ☞ CODING THEORY. Linear space above end fields take a look ☞ HERE.

Changes are isomorphic to the space of symmetric matrix order. The isomorphism is established in accordance with the following, as we can illustrate for the point:

The concept of isomorphism is introduced in order to study objects that are blamed on different algebras of algebra, but with “similar” powers of operations, to conduct on the butt of one srazka, in the practice of new results, which can be cheaply replicated. How can you take the same linear place “for the purpose”? - Div. ending point

Vektorne(otherwise linear) expanse- a mathematical structure, as a set of elements, called vectors, which are assigned to the operation of folding with one and multiplying by a number - a scalar. Tsі operatsії under the order of eight axioms. Scalars can be elements of a speech, complex, or be it some other field of numbers. A private view of a similar expanse is the most trivial Euclidean expanse, the vectors of which are victorious, for example, for the presentation of physical forces. In this case, it is important that the vector as an element of the vector space is not obligatorily responsible for the assignments of a seemingly straight line. The understanding of the concept of "vector" to the element of the vector expanse of be-like nature does not only call out the variance of terms, but allows us to understand or to induce the transmission of low results, fair expanses of a rather natural nature.

Vector spaces are the subject of the study of linear algebra. One of the main characteristics of vector space is its expansiveness. Expansion is the maximum number of linearly independent elements in the space, that, going into a rough geometric interpretation, the number of direct lines, invisible one through one for the additional operation of folding and multiplying by a scalar. Vector expanse can be filled with additional structures, for example, a norm or a scalar creation. Such expanses naturally appear in mathematical analysis, more importantly in seemingly infinite functional expanses. (English), where the role of vectors is played by functions . A lot of problems in the analysis need to be explained, so that the sequence of vectors converges to the next vector. Looking at such sources of possibilities in vector spaces with a complementary structure, in most cases - with a similar topology, which allows one to understand the proximity and continuity. Such topological vector expanses, zocrema, Banach and Hilbert, allow for greater twisting.

The first practices, which introduced the introduction of the concept of vector space, can be seen until the 17th century. The very same developments were taken away by analytical geometry, matrices, systems of linear alignments, Euclidean vectors.

Appointment

Line or vector space V (F) (\displaystyle V\left(F\right)) over the field F (\displaystyle F)- tse ordered chetverka (V, F, +, ⋅) (\displaystyle (V,F,+,\cdot)), de

• V (\displaystyle V)- Empty, impersonal elements of a rather natural nature, as they are called vectors;
• F (\displaystyle F)- field, the elements of which are called scalars;
• Operation assigned dodavannya vector_v V × V → V (\displaystyle V\times V\to V), setting the skin pair of elements x , y (\displaystyle \mathbf (x) ,\mathbf (y) ) faceless V (\displaystyle V) V (\displaystyle V), ranks їх bag and signify x + y (\displaystyle \mathbf(x) +\mathbf(y) );
• Operation assigned multiplication of vectors on scalars F × V → V (\displaystyle F\times V\to V), what spіvstavlyaє skin element λ (\displaystyle\lambda) fields F (\displaystyle F) and skin element x (\displaystyle \mathbf (x) ) faceless V (\displaystyle V) the single element of the multiplier V (\displaystyle V), what is signified λ ⋅ x (\displaystyle \lambda \cdot \mathbf (x) ) or λ x (\displaystyle \lambda \mathbf (x) );

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

The simplest power

1. The vector space is an abelian group by addition.
2. Neutral element 0 ∈ V (\displaystyle \mathbf (0) \in V)
3. 0 ⋅ x = 0 (\displaystyle 0\cdot \mathbf(x) =\mathbf(0) ) for whatever.
4. For be-whom x ∈ V (\displaystyle \mathbf(x) \in V) prolapse element − x ∈ V (\displaystyle -\mathbf (x) \in V) We are united, that we sing out of the group authorities.
5. 1 ⋅ x = x (\displaystyle 1\cdot \mathbf(x) =\mathbf(x) ) for whoever x ∈ V (\displaystyle \mathbf(x) \in V).
6. (−α) ⋅ x = α ⋅ (− x) = − (α x) (\displaystyle (-\alpha)\cdot \mathbf (x) =\alpha \cdot(-\mathbf(x))=- \ alpha\mathbf(x))) for whatever i x ∈ V (\displaystyle \mathbf(x) \in V).
7. α ⋅ 0 = 0 (\displaystyle \alpha \cdot \mathbf(0) =\mathbf(0) ) for whoever α ∈ F (\displaystyle \alpha \in F).

Povyazanі vznachennya that power

Pіdprosіr

Algebraic designation: Linear subspace or vector subspace is a non-empty subset K (\displaystyle K) linear space V (\displaystyle V) so what K (\displaystyle K) the most linear expanse in relation to singing V (\displaystyle V) diam folding that multiplication by a scalar. A lot of subspaces ring out to mean like Lat (V) (\displaystyle \mathrm (Lat) (V)). Sob pіdnіzhina bula pіdprostorom, nebhіdno i enough, shob

The remaining two statements are equivalent to the offensive:

For whatever vectors x , y ∈ K (\displaystyle \mathbf (x) ,\mathbf (y) \in K) vector α x + β y (\displaystyle \alpha \mathbf(x) +\beta \mathbf(y) ) also lying down K (\displaystyle K) for whatever α , β ∈ F (\displaystyle \alpha ,\beta \in F).

Zokrema, vector expanse, which is composed of more than a zero vector, is a subspace of any kind of space; be a kind of space for yourself. Under the space that two people don’t get along with, they call hold on or non-trivial.

The power of subspaces

Linear combinations

Kintseva sum mind

Linear combination is called:

Basis. Romіrnіst

Vectors x 1 , x 2 , … , x n (\displaystyle \mathbf (x) _(1),\mathbf (x) _(2),\ldots ,\mathbf (x) _(n)) called linear fallow yakscho іsnuє їх non-trivial linear combination, the value of which is closer to zero; tobto

for deyakyh coefficients α 1 , α 2 , … , α n ∈ F (\displaystyle \alpha _(1),\alpha _(2),\ldots ,\alpha _(n)\in F,) moreover, I want one of the coefficients α i (\displaystyle \alpha _(i)) vіdminny vіd zero.

In another way, the vectors are called linearly independent.

Dane vyznachennya posledaє nadne zagalnennya: impersonal vector_v z V (\displaystyle V) called linear fallow yakscho linearly fallow deyak kіntseve yoga multiplier, i linearly independent yakscho be-yak yogo kіntseve submultiple is linearly independent.

Dominance basis:

Linear shell

Linear shell submultiple X (\displaystyle X) linear space V (\displaystyle V)- retin u V (\displaystyle V) what to revenge X (\displaystyle X).

Linear shell with subspace V (\displaystyle V).

Linear shell is also called subspace, spawn X (\displaystyle X). To say so, scho is a linear shell V (X) (\displaystyle (\mathcal (V))(X))- space, pull on faceless X (\displaystyle X).