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.a+
=
+
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) Many 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 linear- 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 .
The basis of a two-dimensional space is formed by any two non-collinear vectors.
V) Any three non-coplanar vectors will be linearly independent, they form the basis of three-dimensional space R 3 .
G) As a basis for the space of polynomials of degree no higher than two, we can choose the following three vectors: ē 1 = x 2 ; ē 2 = x; ē 3 =1 .
(1 is a polynomial identically equal to one). This space will be three-dimensional.
Linear (vector) A space is a set V of arbitrary elements called vectors, in which the operations of adding vectors and multiplying a vector by a number are defined, i.e. any two vectors \mathbf(u) and (\mathbf(v)) are assigned a vector \mathbf(u)+\mathbf(v), called the sum of vectors \mathbf(u) and (\mathbf(v)), any vector (\mathbf(v)) and any number \lambda from the field of real numbers \mathbb(R) is associated with a vector \lambda\mathbf(v), called the product of the vector \mathbf(v) by the number \lambda ; so the following conditions are met:
1.
\mathbf(u)+ \mathbf(v)=\mathbf(v)+\mathbf(u)\,~\forall \mathbf(u),\mathbf(v)\in V(commutativity of addition);
2.
\mathbf(u)+(\mathbf(v)+\mathbf(w))=(\mathbf(u)+\mathbf(v))+\mathbf(w)\,~\forall \mathbf(u), \mathbf(v),\mathbf(w)\in V(associativity of addition);
3. there is an element \mathbf(o)\in V , called the zero vector, such that \mathbf(v)+\mathbf(o)=\mathbf(v)\,~\forall \mathbf(v)\in V;
4. for each vector (\mathbf(v)) there is a vector called opposite to the vector \mathbf(v) such that \mathbf(v)+(-\mathbf(v))=\mathbf(o);
5.
\lambda(\mathbf(u)+\mathbf(v))=\lambda \mathbf(u)+\lambda \mathbf(v)\,~\forall \mathbf(u),\mathbf(v)\in V ,~\forall \lambda\in \mathbb(R);
6.
(\lambda+\mu)\mathbf(v)=\lambda \mathbf(v)+\mu \mathbf(v)\,~ \forall \mathbf(v)\in V,~\forall \lambda,\mu\ in\mathbb(R);
7.
\lambda(\mu \mathbf(v))=(\lambda\mu)\mathbf(v)\,~ \forall \mathbf(v)\in V,~\forall \lambda,\mu\in \mathbb( R);
8.
1\cdot \mathbf(v)=\mathbf(v)\,~\forall \mathbf(v)\in V.
Conditions 1-8 are called axioms of linear space. The equal sign placed between the vectors means that the left and right sides of the equality represent the same element of the set V; such vectors are called equal.
In the definition of linear space, the operation of multiplying a vector by a number is introduced for real numbers. Such a space is called linear space over the field of real numbers, or, in short, real linear space. If in the definition, instead of the field \mathbb(R) of real numbers, we take the field complex numbers\mathbb(C) , then we get linear space over the field of complex numbers, or, in short, complex linear space. You can also select the \mathbb(Q) field as a numeric field rational numbers, in this case we obtain a linear space over the field of rational numbers. In what follows, unless otherwise stated, real linear spaces will be considered. In some cases, for brevity, we will talk about space, omitting the word linear, since all the spaces discussed below are linear.
Notes 8.1
1. Axioms 1-4 show that a linear space is a commutative group with respect to the operation of addition.
2. Axioms 5 and 6 determine the distributivity of the operation of multiplying a vector by a number in relation to the operation of adding vectors (axiom 5) or to the operation of adding numbers (axiom 6). Axiom 7, sometimes called the law of associativity of multiplication by a number, expresses the connection between two different operations: multiplying a vector by a number and multiplying numbers. The property defined by Axiom 8 is called the unitarity of the operation of multiplying a vector by a number.
3. Linear space is a non-empty set, since it necessarily contains a zero vector.
4. The operations of adding vectors and multiplying a vector by a number are called linear operations on vectors.
5. The difference between the vectors \mathbf(u) and \mathbf(v) is the sum of the vector \mathbf(u) with the opposite vector (-\mathbf(v)) and is denoted: \mathbf(u)-\mathbf(v)=\mathbf(u)+(-\mathbf(v)).
6. Two non-zero vectors \mathbf(u) and \mathbf(v) are called collinear (proportional) if there is a number \lambda such that \mathbf(v)=\lambda \mathbf(u). The concept of collinearity extends to any finite number of vectors. The zero vector \mathbf(o) is considered collinear with any vector.
Corollaries of the linear space axioms
1. There is only one zero vector in linear space.
2. In linear space, for any vector \mathbf(v)\in V there is a unique opposite vector (-\mathbf(v))\in V.
3. The product of an arbitrary space vector and the number zero is equal to the zero vector, i.e. 0\cdot \mathbf(v)=\mathbf(o)\,~\forall \mathbf(v)\in V.
4. The product of a zero vector by any number is equal to a zero vector, that is, for any number \lambda.
5. The vector opposite to a given vector is equal to the product given vector by number (-1), i.e. (-\mathbf(v))=(-1)\mathbf(v)\,~\forall \mathbf(v)\in V.
6. In expressions of the form \mathbf(a+b+\ldots+z)(sum of a finite number of vectors) or \alpha\cdot\beta\cdot\ldots\cdot\omega\cdot \mathbf(v)(the product of a vector and a finite number of factors) you can place the brackets in any order or not specify them at all.
Let us prove, for example, the first two properties. Uniqueness of the zero vector. If \mathbf(o) and \mathbf(o)" are two zero vectors, then by Axiom 3 we obtain two equalities: \mathbf(o)"+\mathbf(o)=\mathbf(o)" or \mathbf(o)+\mathbf(o)"=\mathbf(o), the left sides of which are equal according to Axiom 1. Consequently, the right sides are also equal, i.e. \mathbf(o)=\mathbf(o)". Uniqueness of the opposite vector. If the vector \mathbf(v)\in V has two opposite vectors (-\mathbf(v)) and (-\mathbf(v))", then by axioms 2, 3,4 we obtain their equality:
(-\mathbf(v))"=(-\mathbf(v))"+\underbrace(\mathbf(v)+(-\mathbf(v)))_(\mathbf(o))= \underbrace( (-\mathbf(v))"+\mathbf(v))_(\mathbf(o))+(-\mathbf(v))=(-\mathbf(v)).
The remaining properties are proved in a similar way.
Examples of linear spaces
1. Let us denote \(\mathbf(o)\) - a set containing one zero vector, with the operations \mathbf(o)+ \mathbf(o)=\mathbf(o) And \lambda \mathbf(o)=\mathbf(o). For the indicated operations, axioms 1-8 are satisfied. Consequently, the set \(\mathbf(o)\) is a linear space over any number field. This linear space is called null.
2. Let us denote V_1,\,V_2,\,V_3 - sets of vectors (directed segments) on a straight line, on a plane, in space, respectively, with the usual operations of adding vectors and multiplying vectors by a number. The fulfillment of axioms 1-8 of linear space follows from the course of elementary geometry. Consequently, the sets V_1,\,V_2,\,V_3 are real linear spaces. Instead of free vectors we can consider the corresponding sets of radius vectors. For example, a set of vectors on a plane that have a common origin, i.e. plotted from one fixed point of the plane is a real linear space. The set of radius vectors of unit length does not form a linear space, since for any of these vectors the sum \mathbf(v)+\mathbf(v) does not belong to the set under consideration.
3. Let us denote \mathbb(R)^n - a set of matrix-columns of sizes n\times1 with the operations of adding matrices and multiplying matrices by a number. Axioms 1-8 of linear space are satisfied for this set. The zero vector in this set is the zero column o=\begin(pmatrix)0&\cdots&0\end(pmatrix)^T. Consequently, the set \mathbb(R)^n is a real linear space. Similarly, a set of \mathbb(C)^n columns of size n\times1 with complex elements is a complex linear space. The set of column matrices with non-negative real elements, on the contrary, is not a linear space, since it does not contain opposite vectors.
4. Let us denote \(Ax=o\) - the set of solutions of the homogeneous system Ax=o linear algebraic equations with and unknowns (where A is the real matrix of the system), considered as a set of columns of size n\times1 with the operations of adding matrices and multiplying matrices by a number. Note that these operations are indeed defined on the set \(Ax=o\) . From Property 1 of solutions to a homogeneous system (see Section 5.5) it follows that the sum of two solutions of a homogeneous system and the product of its solution by a number are also solutions of a homogeneous system, i.e. belong to the set \(Ax=o\) . The axioms of linear space for columns are satisfied (see point 3 in examples of linear spaces). Therefore, the set of solutions of a homogeneous system is a real linear space.
The set \(Ax=b\) of solutions to the inhomogeneous system Ax=b,~b\ne o , on the contrary, is not a linear space, if only because it does not contain a zero element (x=o is not a solution to the inhomogeneous system).
5. Let us denote M_(m\times n) - a set of matrices of size m\times n with the operations of adding matrices and multiplying matrices by a number. Axioms 1-8 of linear space are satisfied for this set. The zero vector is a zero matrix O of appropriate sizes. Therefore, the set M_(m\times n) is a linear space.
6. Let us denote P(\mathbb(C)) - the set of polynomials of one variable with complex coefficients. Operations of addition of many terms and multiplication of a polynomial by a number considered as a polynomial zero degree, are defined and satisfy axioms 1-8 (in particular, the zero vector is a polynomial that is identically equal to zero). Therefore, the set P(\mathbb(C)) is a linear space over the field of complex numbers. The set P(\mathbb(R)) of polynomials with real coefficients is also a linear space (but, of course, over the field of real numbers). The set P_n(\mathbb(R)) of polynomials of degree at most n with real coefficients is also a real linear space. Note that the operation of addition of many terms is defined on this set, since the degree of the sum of polynomials does not exceed the degrees of the terms.
The set of polynomials of degree n is not a linear space, since the sum of such polynomials may turn out to be a polynomial of a lower degree that does not belong to the set in question. The set of all polynomials of degree no higher than n with positive coefficients is also not a linear space, since when such a polynomial is multiplied by negative number we obtain a polynomial that does not belong to this set.
7. Let us denote C(\mathbb(R)) - the set of real functions defined and continuous on \mathbb(R) . Sum (f+g) functions f,g and the product \lambda f of the function f and the real number \lambda are determined by the equalities:
(f+g)(x)=f(x)+g(x),\quad (\lambda f)(x)=\lambda\cdot f(x) for all x\in \mathbb(R)
These operations are indeed defined on C(\mathbb(R)) , since the sum of continuous functions and the product of a continuous function by a number are continuous functions, i.e. elements of C(\mathbb(R)) . Let us check the fulfillment of the axioms of linear space. Since the addition of real numbers is commutative, it follows that the equality f(x)+g(x)=g(x)+f(x) for any x\in \mathbb(R) . Therefore f+g=g+f, i.e. axiom 1 is satisfied. Axiom 2 follows similarly from the associativity of addition. The zero vector is the function o(x), identically equal to zero, which, of course, is continuous. For any function f the equality f(x)+o(x)=f(x) holds, i.e. Axiom 3 is true. The opposite vector for the vector f will be the function (-f)(x)=-f(x) . Then f+(-f)=o (axiom 4 is true). Axioms 5, 6 follow from the distributivity of the operations of addition and multiplication of real numbers, and axiom 7 - from the associativity of multiplication of numbers. The last axiom is satisfied, since multiplication by one does not change the function: 1\cdot f(x)=f(x) for any x\in \mathbb(R), i.e. 1\cdot f=f . Thus, the considered set C(\mathbb(R)) with the introduced operations is a real linear space. Similarly, it is proved that C^1(\mathbb(R)),C^2(\mathbb(R)), \ldots, C^m(\mathbb(R))- sets of functions that have continuous derivatives of the first, second, etc. orders, respectively, are also linear spaces.
Let us denote the set of trigonometric binomials (often \omega\ne0 ) with real coefficients, i.e. many functions of the form f(t)=a\sin\omega t+b\cos\omega t, Where a\in \mathbb(R),~b\in \mathbb(R). The sum of such binomials and the product of a binomial by a real number are trigonometric binomials. The linear space axioms for the set under consideration are satisfied (since T_(\omega)(\mathbb(R))\subset C(\mathbb(R))). Therefore, many T_(\omega)(\mathbb(R)) with the usual operations of addition and multiplication by a number for functions, it is a real linear space. The zero element is the binomial o(t)=0\cdot\sin\omega t+0\cdot\cos\omega t, identically equal to zero.
The set of real functions defined and monotone on \mathbb(R) is not a linear space, since the difference of two monotone functions may turn out to be a non-monotone function.
8. Let us denote \mathbb(R)^X - the set of real functions defined on the set X with the operations:
(f+g)(x)=f(x)+g(x),\quad (\lambda f)(x)=\lambda\cdot f(x)\quad \forall x\in X
It is a real linear space (the proof is the same as in the previous example). In this case, the set X can be chosen arbitrarily. In particular, if X=\(1,2,\ldots,n\), then f(X) is an ordered set of numbers f_1,f_2,\ldots,f_n, Where f_i=f(i),~i=1,\ldots,n Such a set can be considered a matrix-column of dimensions n\times1 , i.e. many \mathbb(R)^(\(1,2,\ldots,n\)) coincides with the set \mathbb(R)^n (see point 3 for examples of linear spaces). If X=\mathbb(N) (recall that \mathbb(N) is the set natural numbers), then we get a linear space \mathbb(R)^(\mathbb(N))- many number sequences \(f(i)\)_(i=1)^(\infty). In particular, the set of convergent number sequences also forms a linear space, since the sum of two convergent sequences converges, and when all terms of a convergent sequence are multiplied by a number, we obtain a convergent sequence. In contrast, the set of divergent sequences is not a linear space, since, for example, the sum of divergent sequences may have a limit.
9. Let us denote \mathbb(R)^(+) - the set of positive real numbers in which the sum a\oplus b and the product \lambda\ast a (the notations in this example differ from the usual ones) are defined by the equalities: a\oplus b=ab,~ \lambda\ast a=a^(\lambda), in other words, the sum of elements is understood as a product of numbers, and multiplication of an element by a number is understood as raising to a power. Both operations are indeed defined on the set \mathbb(R)^(+) , since the product of positive numbers is a positive number and any real degree a positive number is a positive number. Let's check the validity of the axioms. Equalities
a\oplus b=ab=ba=b\oplus a,\quad a\oplus(b\oplus c)=a(bc)=(ab)c=(a\oplus b)\oplus c
show that axioms 1 and 2 are satisfied. The zero vector of this set is one, since a\oplus1=a\cdot1=a, i.e. o=1 . The opposite vector for a is the vector \frac(1)(a) , which is defined since a\ne o . In fact, a\oplus\frac(1)(a)=a\cdot\frac(1)(a)=1=o. Let's check the fulfillment of axioms 5, 6,7,8:
\begin(gathered) \mathsf(5))\quad \lambda\ast(a\oplus b)=(a\cdot b)^(\lambda)= a^(\lambda)\cdot b^(\lambda) = \lambda\ast a\oplus \lambda\ast b\,;\hfill\\ \mathsf(6))\quad (\lambda+ \mu)\ast a=a^(\lambda+\mu)=a^( \lambda)\cdot a^(\mu)=\lambda\ast a\oplus\mu\ast a\,;\hfill\\ \mathsf(7)) \quad \lambda\ast(\mu\ast a) =(a^(\mu))^(\lambda)=a^(\lambda\mu)=(\lambda\cdot \mu)\ast a\,;\hfill\\ \mathsf(8))\quad 1\ast a=a^1=a\,.\hfill \end(gathered)
All axioms are satisfied. Consequently, the set under consideration is a real linear space.
10. Let V be a real linear space. Let us consider the set of linear scalar functions defined on V, i.e. functions f\colon V\to \mathbb(R), taking real values and satisfying the conditions:
f(\mathbf(u)+\mathbf(v))=f(u)+f(v)~~ \forall u,v\in V(additivity);
f(\lambda v)=\lambda\cdot f(v)~~ \forall v\in V,~ \forall \lambda\in \mathbb(R)(homogeneity).
Linear operations on linear functions are specified in the same way as in paragraph 8 of examples of linear spaces. The sum f+g and the product \lambda\cdot f are defined by the equalities:
(f+g)(v)=f(v)+g(v)\quad \forall v\in V;\qquad (\lambda f)(v)=\lambda f(v)\quad \forall v\ in V,~ \forall \lambda\in \mathbb(R).
The fulfillment of the linear space axioms is confirmed in the same way as in paragraph 8. Therefore, the set linear functions defined on a linear space V is a linear space. This space is called conjugate to the space V and is denoted by V^(\ast) . Its elements are called covectors.
For example, many linear forms n variables, considered as a set of scalar functions of a vector argument, is a linear space conjugate to the space \mathbb(R)^n.
If you notice an error, typo or have any suggestions, write in the comments.
Lecture 6. Vector space.
Basic questions.
1. Vector linear space.
2. Basis and dimension of space.
3. Space orientation.
4. Decomposition of a vector by basis.
5. Vector coordinates.
1. Vector linear space.
A set consisting of elements of any nature in which they are defined linear operations: Adding two elements and multiplying an element by a number are called spaces, and their elements are vectors this space and are designated in the same way as vector quantities in geometry: . Vectors Such abstract spaces, as a rule, have nothing in common with ordinary geometric vectors. Elements of abstract spaces can be functions, a system of numbers, matrices, etc., and in a particular case, ordinary vectors. Therefore, such spaces are usually called vector spaces .
Vector spaces are, For example, a set of collinear vectors, denoted V1 , set of coplanar vectors V2 , set of vectors of ordinary (real space) V3 .
For this particular case, we can give the following definition of a vector space.
Definition 1. The set of vectors is called vector space, if a linear combination of any vectors of a set is also a vector of this set. The vectors themselves are called elements vector space.
More important, both theoretically and appliedly, is the general (abstract) concept of vector space.
Definition 2. Many R elements, in which the sum is determined for any two elements and for any element https://pandia.ru/text/80/142/images/image006_75.gif" width="68" height="20"> called vector(or linear) space, and its elements are vectors, if the operations of adding vectors and multiplying a vector by a number satisfy the following conditions ( axioms) :
1) addition is commutative, i.e..gif" width="184" height="25">;
3) there is such an element (zero vector) that for any https://pandia.ru/text/80/142/images/image003_99.gif" width="45" height="20">.gif" width=" 99" height="27">;
5) for any vectors and and any number λ the equality holds;
6) for any vectors and any numbers λ
And µ
the equality is true: https://pandia.ru/text/80/142/images/image003_99.gif" width="45 height=20" height="20"> and any numbers λ
And µ
fair 
;
8) https://pandia.ru/text/80/142/images/image003_99.gif" width="45" height="20">.
From the axioms defining vector space, the simplest consequences :
1. There is only one zero in a vector space - the element - the zero vector.
2. In vector space, each vector has a single opposite vector.
3. For each element the equality is satisfied.
4. For anyone real number λ and zero vector https://pandia.ru/text/80/142/images/image017_45.gif" width="68" height="25">.
5..gif" width="145" height="28">
6..gif" width="15" height="19 src=">.gif" width="71" height="24 src="> is a vector that satisfies the equality https://pandia.ru/text/80 /142/images/image026_26.gif" width="73" height="24">.
So, indeed, the set of all geometric vectors is a linear (vector) space, since for the elements of this set the actions of addition and multiplication by a number are defined that satisfy the formulated axioms.
2. Basis and dimension of space.
The essential concepts of a vector space are the concepts of basis and dimension.
Definition. A set of linearly independent vectors, taken in a certain order, through which any vector of space can be linearly expressed, is called basis this space. Vectors. The components of the basis of space are called basic .
The basis of a set of vectors located on an arbitrary line can be considered one collinear vector to this line.
Basis on the plane let's call two non-collinear vectors on this plane, taken in a certain order https://pandia.ru/text/80/142/images/image029_29.gif" width="61" height="24">.
If the basis vectors are pairwise perpendicular (orthogonal), then the basis is called orthogonal, and if these vectors have a length equal to one, then the basis is called orthonormal .
Largest number linearly independent vectors of space are called dimension of this space, i.e. the dimension of the space coincides with the number of basis vectors of this space.
So, according to these definitions:
1. One-dimensional space V1 is a straight line, and the basis consists of one collinear vector https://pandia.ru/text/80/142/images/image028_22.gif" width="39" height="23 src="> .
3. Ordinary space is three-dimensional space V3 , whose basis consists of three non-coplanar vectors
From here we see that the number of basis vectors on a straight line, on a plane, in real space coincides with what in geometry is usually called the number of dimensions (dimension) of a straight line, plane, space. Therefore, it is natural to introduce a more general definition.
Definition. Vector space R called n– dimensional if there are no more than n linearly independent vectors and is denoted R n. Number n called dimension space.
In accordance with the dimension of the space are divided into finite-dimensional And infinite-dimensional. The dimension of the null space is considered equal to zero by definition.
Note 1. In each space you can specify as many bases as you like, but all the bases of a given space consist of the same number of vectors.
Note 2. IN n– in a dimensional vector space, a basis is any ordered collection n linearly independent vectors.
3. Space orientation.
Let the basis vectors in space V3 have general beginning And ordered, i.e. it is indicated which vector is considered the first, which is considered the second and which is considered the third. For example, in the basis the vectors are ordered according to indexation. |
|
For that to orient space, it is necessary to set some basis and declare it positive .
It can be shown that the set of all bases of space falls into two classes, that is, into two disjoint subsets.
a) all bases belonging to one subset (class) have the same orientation (bases of the same name);
b) any two bases belonging to various subsets (classes), have the opposite orientation, ( different names bases).
If one of the two classes of bases of a space is declared positive and the other negative, then it is said that this space oriented .
Often, when orienting space, some bases are called right, and others - left .
https://pandia.ru/text/80/142/images/image029_29.gif" width="61" height="24 src="> are called right, if, when observing from the end of the third vector, the shortest rotation of the first vector https://pandia.ru/text/80/142/images/image033_23.gif" width="16" height="23"> is carried out counterclockwise(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) and left basis (b)
Usually the right basis of the space is declared to be a positive basis
The right (left) basis of space can also be determined using the rule of a “right” (“left”) screw or gimlet.
By analogy with this, the concept of right and left is introduced threes non-coplanar vectors that must be ordered (Fig. 1.8).
Thus, in the general case, two ordered triplets of non-coplanar vectors have the same orientation (the same name) in space V3 if they are both right or both left, and - the opposite orientation (opposite) if one of them is right and the other is left.
The same is done in the case of space V2 (plane).
4. Decomposition of a vector by basis.
For simplicity of reasoning, let us consider this question using the example of a three-dimensional vector space R3 .
Let https://pandia.ru/text/80/142/images/image021_36.gif" width="15" height="19"> be an arbitrary vector of this space.
A vector (linear) space is a set of vectors (elements) with real components, in which the operations of adding vectors and multiplying a vector by a number are defined that satisfy certain axioms (properties)
1)x+at=at+X(commutability of addition);
2)(X+at)+z=x+(y+z) (associativity of addition);
3) there is a zero vector 0 (or null vector) satisfying the condition x+ 0 =x: for any vector x;
4) for any vector X there is an opposite vector at such that X+at = 0 ,
5) 1 x=X,
6) a(bx)=(ab)X(associativity of multiplication);
7) (a+b)X=ah+bx(distributive property relative to the numerical factor);
8) a(X+at)=ah+ay(distributive property relative to the vector multiplier).
A linear (vector) space V(P) over a field P is a non-empty set V. The elements of the set V are called vectors, and the elements of the field P are called scalars.
The simplest properties.
1. A vector space is an Abelian group (a group in which the group operation is commutative. The group operation in Abelian groups is usually called “addition” and is denoted by the + sign)
2. The neutral element is the only one that follows from the group properties for any .
3. For any, the opposite element is the only one that follows from group properties.
4.(–1) x = – x for any x є V.
5.(–α) x = α(–x) = – (αx) for any α є P and x є V.
Expression a 1 e 1+a 2 e 2+ … +a n e n(1) is called a linear combination of vectors e 1 , e 2 ,..., e n with odds a 1 , a 2,..., a n . Linear combination (1) is called nontrivial if at least one of the coefficients a 1 , a 2 ,..., a n different from zero. Vectors e 1 , e 2 ,..., e n are called linearly dependent if there is a non-trivial combination (1), which is a zero vector. Otherwise (that is, if only a trivial combination of vectors e 1 , e 2 ,..., e n equal to the zero vector) vectors e 1 , e 2 ,..., e n are called linearly independent.
The dimension of space is the maximum number of LZ vectors contained in it.
Vector space is called n-dimensional (or has “dimension n"),
if it exists n linearly independent elements e 1 , e 2 ,..., e n , and any n+ 1
elements are linearly dependent (generalized condition B). Vector space are called infinite-dimensional if in it for any natural n exists n linearly independent vectors. Any n linearly independent n-dimensional vectors Vector space form the basis of this space. If e 1 , e 2 ,..., e n- basis Vector space, then any vector X this space can be represented the only way as a linear combination of basis vectors: x=a 1 e 1+a 2 e 2+...
+a n e n.
At the same time, the numbers a 1 , a 2, ..., a n are called vector coordinates X in this basis.
