Ring and vector space matrices. Linear vector space: appointment, authority. Vector line space

Lecture 6. Vector space.

Basic nutrition.

1. Vector linear space.

2. The basis is the expansion of space.

3. Orientation to space.

4. Deployment of a vector behind a basis.

5. Vector coordinates.

1. Vector linear space.

Anonymity, which is composed of the elements of any nature, in which linear operations are indicated: adding two elements, that multiplying an element by a number are called open spaces, And їх elements - vectors th space і are assigned as і, yak і vector quantities in geometry: . Vectors such abstract expanses, as a rule, cannot be conceived of with the greatest geometric vectors. Elements of abstract spaces can be functions, a system of numbers, matrices, etc., and in an okreme case, variable vectors. That's why it's customary to name vector open spaces .

vector space, for example, numberless number of nonary vectors that are indicated V1 , without coplanar vectors V2 , impersonal vector sizable (real space) V3 .

For this particular vipadka, it is possible to give a stepping stone to the vector expanse.

Appointment 1. Anonymous vector is called vector space, As a linear combination, whether there are any vectors in a multiplier, it is also a vector of that multiplier. The vectors themselves are called elements vector space.

It is more important both in the theoretical and in the applied perspective and in the more abstract (abstract) understanding of the vector space.

Appointment 2. Bezlich R elements, in which for any two elements and the sum is assigned and for any element width="68" is called vector(or linear) open space, like elements - vectors, like the operation of adding vectors and multiplying a vector by a number to satisfy the coming minds ( axioms) :

1) the addition is commutative, so gif width = "184" height = "25";

3) use such an element (zero vector), which for whatever https://pandia.ru/text/80/142/images/image003_99.gif" width="45". 99" height="27">;

5) for any number of vectors, such a number λ may be equal;

6) for whatever vectors and whatever numbers λ і µ fairness https://pandia.ru/text/80/142/images/image003_99.gif" λ і µ fair ;

8) https://pandia.ru/text/80/142/images/image003_99.gif" .

From the axioms that signify the vector space, exclaim the simplest evidence :

1. The vector space has more than one zero - the element is a zero vector.

2. A vector space has a single vector.

3. Up to the skin element vykonuetsya equanimity.

4. For any day number λ i of the zero vector.

5..gif" width="145" height="28">

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

Otzhe, fiyno, and impersonal of all geometric vectors in є linear (vector) space, so for the elements of which the multiplier is assigned to the addition and multiplication by the number, which satisfies the formulation of the axioms.

2. The basis is the expansion of space.

Іstotnimi concepts of vector space є understanding of the basis and rozmіrnіst.

Appointment. The collection of linearly independent vectors taken from the sing order basis what space. Vector. Warehouse basis for space, called basis .

The basis of the impersonal vectors, spread out on the dolnіy straight line, you can use one collinear straight vector .

Basis on the plane Let's name two non-collinear vectors on this plane, taken in the same order.

If the basis vectors are pairwise perpendicular (orthogonal), then the basis is called orthogonal, and if q vectors can be double, equal to one, then the basis is called orthonormal .

Largest number linearly independent vectors are called in space peace that space, i.e., the expansion of the space increases with the number of basic vectors in this space.

Otzhe, obviously praised to the dagi:

1. One-world space V1 is a straight line, and the basis is formed from one collinear vector https://pandia.ru/text/80/142/images/image028_22.gif" width="39".

3. Great expanse with trivial expanse V3 , the basis of which is formed from three non-coplanar vector_v.

It seems to me that the number of basis vectors on a straight line, on a plane, in real space varies with that, which in geometry is usually called the number of a straight line, a plane, a space. It is natural for this to lead to more blatant punishment.

Appointment. Vector space R called n- peaceful, as in the new world no more n linearly independent vectors and are assigned R n. Number n called peace space.

Vіdpovіdno up to rozmіrnostі open space podіlyayutsya kіntsevіі unlimited. The openness of the zero expanse beyond the appointments is considered equal to zero.

Respect 1. In the skin space, you can specify how many bases are needed, but all the bases of this space are added up from the same number of vectors.

Note 2. At n- to a peaceful vector space, the basis is called whether or not the ordered order is n linearly independent vectors.

3. Orientation to space.

For in order to orientate space, it is necessary to set a certain basis and voice it positively .

It can be shown that the impersonal basics of the space are divided into two classes, that they are divided into two submultiples, that they do not overlap.

a) all bases that belong to one submultiple (class) may however orientation (same-menu basis);

b) any two bases that lie life p_dmnozhin (classes), mayut protilezhnu orientation, ( different basis).

If one of the two classes of bases is positive, and the other is negative, then it seems that the expanse oriented .

Often, when orienting to space, one basis is called govern, and інші - livimi .

https://pandia.ru/text/80/142/images/image029_29.gif" width="61" height="24 src="> name rule, However, when the third vector is guarded, the shortest turn of the first vector is anti-year arrow(Fig. 1.8, a).

https://pandia.ru/text/80/142/images/image036_22.gif" width="16" height="24">

https://pandia.ru/text/80/142/images/image037_23.gif" width="15" height="23">

https://pandia.ru/text/80/142/images/image039_23.gif" width="13" height="19">

https://pandia.ru/text/80/142/images/image033_23.gif" width="16" height="23">

Rice. 1.8. Right basis (a) that left basis (b)

Ring out with a positive basis

The right (livy) basis can be assigned to the space, and for the additional rule of the “right” (“left”) screw or twisted.

By analogy with cim, the concept of right and left is introduced triplets non-communal vectors, which are due to ordering (Fig. 1.8).

In this way, in a wild trend, two ordered triples of unplanned vectors may have the same orientation (same) in the space V3 if the stench of offense is right, or if it is offensive, it is left, and the opposite orientation (different), if one of them is right, and the other is left.

Similar to fit and have space V2 (Squares).

4. Deployment of a vector behind a basis.

For the sake of simplicity, the mirroring can be seen on the example of a trivimir vector space R3 .

Come on - dovіlny vector tsgo space.

VECTOR SPACE (linear expanse), one of the fundamental understanding of algebra, zagalnyuyuche understanding of the totality of (free) vectors. In the vector space, the vectors are considered whether they are objects, if they can be added and multiplied by numbers; if necessary, so that the main powers of algebraic operations are the same as for vectors in elementary geometry. At the exact designated number, they are replaced by elements of the field K. The vector space over the field K is called the impersonal V with the operation of adding elements from V and the operation of multiplying elements from V by elements from the field K, which may lead to advancing power:

x + y \u003d y + x for whether x, y z V, so that V can be folded into an Abelian group;

λ(x + y) = λ χ + λy for any λ z K і x, y z V;

(λ + μ)х = λх + μх for any λ, μ z K і x z V;

(λ μ)х = λ(μх) for any λ, μ z K i x z V;

1x \u003d x for any x from V, here 1 means the unity of the field K.

Butts of the vector space є: multipliers L 1 L 2 і L 3 of all vectors in elementary geometry, apparently on a straight line, planes і in space with the outstanding operations of folding vectors and multiplying by a number; coordinate vector space K n , the elements of which є all rows (vectors) are n with elements from the field K, and the operations are given by formulas

impersonal F(M, K) of all functions assigned to a fixed multiplier M and take values ​​in the field To, with the most significant operations on functions:

Elements of the vector space e 1 ..., e n are called linearly independent, because of the equality λ 1 e 1 + ... n = 0 Є K. In the opposite direction, the elements e 1 , e 2 , ···> e n are called linearly fallow. If the vector space V has n + 1 elements e 1 ,..., e n+1 linearly indeterminate and n linearly independent elements, then V is called the n-world vector space, and n is the dimension of the vector space V Just as a vector space V for any natural n existing n linearly independent vectors, then V is called an infinite vector space. For example, the vector space L 1 , L 2 , L 3 і K n in the same way 1-, 2-, 3- and n-mіrnі; if M is impersonal, then the vector space F(M, K) is not limited.

The vector space V and U over the field K are called isomorphic, so that φ : V -> U is mutually unique, so that φ(x+y) = φ(x) + φ(y) for either x, y z V and φ (λx) = λ φ(x) for any λ z K i x z V. Isomorphic vector spaces are algebraically indistinguishable. The classification of finite vector spaces up to isomorphism is given to their diversity: whether there be an n-dimensional vector space over the field Do is isomorphic to the coordinate vector space Do n . Marvel at the same expanse of Hilbert, Linear Algebra.

Let R - field. Elements a, b, ... н R we will name scalars.

Appointment 1. class V objects (elements) , , , ... of sufficient nature are called vector space over the field Р, and the elements of the class V are called vectors even though V is closed, but the operation “+” is the operation of multiplication by scalars from P (that is, for any , нV + н V; "aÎ R aÎV), and vykonuyutsya so mind:

A 1: Algebra - Abelian group;

A 2: for whether a, bÎР, for whether or not ÎV, a(b)=(ab)-relevant associative law;

A 3: for whatever a, bÎP, for whatever ÎV, vikonuetsya (a+b)= a+ b;

A 4: for any a z P, for any s V, we win a(+)=a+a(increased distributive laws);

A 5: whether or not V is victorious 1 = , de 1 - the unity of the field P - the power of unitarity.

The elements of the field P are called scalars, and the elements of the multiplier V are called vectors.

Respect. Multiplying a vector by a scalar is not a binary operation on the multiplier V, but the scaling is PV®V.

Let's take a look at vector spaces.

example 1. Zero (zero-world) vector expanse - expanse V 0 =() - which is composed of one zero-vector.

For whatever aОР a=. Let's reconsider the validity of the axioms of vector space.

Respectfully, the zero-dimensional space over the field R. Thus, the zero-dimensional space over the field rational numbers i above the field day numbers vvazhayutsya raznimi, hoch add up from a single zero-vector.

butt 2. The field P is itself a vector space over the field P. Let V=P. Let's reconsider the validity of the axioms of vector space. Since P is a field, then P is an additive group and A1 wins. Looking back at the zdіysnennostі in R asociativnostі mnozhennja vykonuєtsya A 2 . Axioms A 3 and A 4 win due to the fact that R is distributive and multiplied freely. Shards in the field R is a single element 1, the power of unitarity A 5 . In this order, the field P is a vector space over the field P.

example 3. Arithmetic n-dimensional vector space.

Let R - field. Considerably impersonal V = P n = ((a 1, a 2, …, a n) ½ a i P, i = 1, ..., n). Let's introduce on the multiplier V the operation of adding vectors and multiplying a vector by a scalar according to the following rules:

"= (a 1 , a 2 , … , a n), = (b 1 , b 2 , … , b n) О V, "aО P += (a 1 + b 1 , a 2 + b 2 , … , a n + bn) (1)

a=(aa 1 , aa 2 , … , aa n) (2)

Elements and multiplies V are called n-world vectors. Two n-world vectors are called equal, since their two-dimensional components (coordinates) are equal. It can be shown that V is a vector space over the field P. Since the operation of folding a vector into and multiplying a vector by a scalar is known, V is a closed choice of these operations. Since the addition of elements from V is reduced to the addition of elements of the field P, and P is an additive Abelian group, then і V is an additive Abelian group. Moreover, = , de 0 is the zero of the field Р, -= (-a 1, -a 2, ..., -a n). In this rank, A1 wins. The scalings of the multiplication of the element V by the element P are reduced to the multiplication of the elements of the field P, then:

A 2 wins due to the associativity of the multiplier on P;

A 3 and A 4 are concatenated by the distributive multiplication of how folding on P;

And 5 wins, because 1 P is a neutral element that can be multiplied by R.

Appointment 2. The impersonal V = P n with operations defined by formulas (1) and (2) is called an arithmetic n-dimensional vector space over the field Р.

Let's take a look at the sequence that is formed by the elements of the action simple field GF(q) (a^, a......a p). Such a sequence is called l-by

consistency over the field GF)