Is the set of complex numbers a field? Complex number field

Def. A system of complex numbers is called a min field, which is an extension of the field of real numbers and in which there is an element i (i 2 -1=0)

Def. Algebra<ℂ, +, ∙, 0, 1, ℝ, ⊕, ⊙, i>is called a system of computer numbers if the following conditions (axioms) are met:

1. a,b∊ℂ∃!m∊ℂ: a+b=m

2. a,b,c∊ℂ (a+b)+c=a+(b+c)

3. a,b∊ℂa+b=b+a

4. ∃ 0∊ℂ a∊ℂ a+0=a

5. a∊ℂ ∃(-a)∊ℂ a+(-a)=0

6. a,b∊ℂ ∃! n∊ℂa∙b=n

7. a,b,c∊ℂ (a∙b)∙c=a∙(b∙c)

8. a,b∊ℂa∙b=b∙a

9. ∃1∊ℂ a∊ℂ a∙1=a

10. a∊ℂ ∃a -1 ∊ℂ a∙a -1 =1

11. a,b,c∊ℂ (a+b)c=ac+bc

12. - action field numbers

13. Rєℂ, a,b∊R a⊕b=a+b, a⊙b=a∙b

14. ∃i∊ℂ:i 2 +1=0

15. ℳ≠⌀ 1)ℳ⊂ℂ,R⊂ℳ 2) α,β∊ℳ⇒(α+β)∊ℳ and (α∙β)∊ℳ)⇒ℳ=ℂ

Holy numbers:

1. α∊ℂ∃! (a,b) ∊ R:α=a+b∙i

2. The field of comp numbers cannot be linearly ordered, i.e. α∊ℂ, α≥0 |+1, α 2 +1≥1, i 2 +1=0, 0≥1-impossible.

3. Fundamental theorem of algebra: The field ℂ of numbers is algebraically closed, that is, any plural number is positive. degrees over the field ℂ of numbers has at least one set. root

The following from the main alg. theorems: Any plurality of positive. degrees over the field of complex numbers can be divided into a product ... of the first degree with a positive coefficient.

Next: any quad level has 2 roots: 1) D>0 2 different. valid root 2)D=0 2-a dest. coincidence of root 3)D<0 2-а компл-х корня.

4. Axiom. the theory of complex numbers is categorical and consistent

Methodology.

In general education classes, the concept of a complex number is not considered; they are limited only to the study of real numbers. But in high school, schoolchildren already have a fairly mature mathematical education and are able to understand the need to expand the concept of number. From the point of view of general development, knowledge about complex numbers is used in natural sciences and technology, which is important for a student in the process of choosing a future profession. The authors of some textbooks include the study of this topic as mandatory in their textbooks on algebra and the beginnings of mathematical analysis for specialized levels, which is provided for by the state standard.

From a methodological point of view, the topic “Complex numbers” develops and deepens the concepts of polynomials and numbers laid down in the basic course of mathematics, in a certain sense completing the path of development of the concept of number in secondary school.

However, even in high school, many schoolchildren have poorly developed abstract thinking, or it is very difficult to imagine an “imaginary, imaginary” unit, to understand the differences between the coordinate and complex plane. Or, on the contrary, the student operates with abstract concepts in isolation from their real content.

After studying the topic “Complex numbers”, students should have a clear understanding of complex numbers, know the algebraic, geometric and trigonometric forms of a complex number. Students should be able to perform operations of addition, multiplication, subtraction, division, exponentiation, and root extraction on complex numbers; convert complex numbers from algebraic to trigonometric form, have an idea of ​​the geometric model of complex numbers

In the textbook for mathematical classes by N.Ya. Vilenkin, O.S. Ivashev-Musatov, S.I. Shvartsburd “Algebra and the beginnings of mathematical analysis”, the topic “Complex numbers” is introduced in the 11th grade. Study of the topic is offered in the second half of the 11th grade after the trigonometry section was studied in the 10th grade, and integral and differential equations, exponential, logarithmic and power functions, and polynomials in the 11th grade. In the textbook, the topic “Complex numbers and operations on them” is divided into two sections: Complex numbers in algebraic form; Trigonometric form of complex numbers. Consideration of the topic “Complex numbers and operations on them” begins with consideration of the issue of solving quadratic equations, equations of the third and fourth degree and, as a consequence, the need to introduce a “new number i” is revealed. The concepts of complex numbers and operations on them are immediately given: finding the sum, product and quotient of complex numbers. Next, a strict definition of the concept of a complex number, the properties of the operations of addition and multiplication, subtraction and division is given. The next paragraph talks about conjugate complex numbers and some of their properties. Next, we consider the issue of extracting square roots from complex numbers and solving quadratic equations with complex coefficients. The next paragraph discusses: the geometric representation of complex numbers; polar coordinate system and trigonometric form of complex numbers; multiplication, exponentiation and division of complex numbers in trigonometric form; Moivre's formula, application of complex numbers to the proof of trigonometric identities; extracting the root of a complex number; fundamental theorem of polynomial algebra; complex numbers and geometric transformations, functions of a complex variable.

In the textbook S.M. Nikolsky, M.K. Potapova, N.N. Reshetnikova, A.V. Shevkin “Algebra and the beginnings of mathematical analysis”, topic “Complex numbers are considered in grade 11 after studying all the topics, i.e. at the end of a school algebra course. The topic is divided into three sections: Algebraic form and geometric interpretation of complex numbers; Trigonometric form of complex numbers; Roots of polynomials, exponential form of complex numbers. The content of the paragraphs is quite voluminous; it contains many concepts, definitions, and theorems. The paragraph “Algebraic form and geometric interpretation of complex numbers” contains three sections: algebraic form of a complex number; conjugate complex numbers; geometric interpretation of a complex number. The paragraph “Trigonometric form of a complex number” contains the definitions and concepts necessary to introduce the concept of the trigonometric form of a complex number, as well as an algorithm for the transition from the algebraic form of notation to the trigonometric form of notation of a complex number. In the last paragraph “Roots of polynomials. Exponential form of complex numbers" contains three sections: roots of complex numbers and their properties; roots of polynomials; exponential form of a complex number.

The textbook material is presented in a small volume, but quite sufficient for students to understand the essence of complex numbers and master minimal knowledge about them. The textbook contains a small number of exercises and does not address the issue of raising a complex number to a power and the Moivre formula

In the textbook A.G. Mordkovich, P.V. Semenov “Algebra and the beginnings of mathematical analysis”, profile level, grade 10, the topic “Complex Numbers” is introduced in the second half of the 10th grade immediately after studying the topics “Real Numbers” and “Trigonometry”. This placement is not accidental: both the number circle and trigonometry formulas are actively used in the study of the trigonometric form of a complex number, the Moivre formula, and when extracting square and cube roots from a complex number. The topic “Complex numbers” is presented in Chapter 6 and is divided into 5 sections: complex numbers and arithmetic operations on them; complex numbers and the coordinate plane; trigonometric form of writing a complex number; complex numbers and quadratic equations; raising a complex number to a power, extracting the cube root of a complex number.

The concept of a complex number is introduced as an extension of the concept of number and the impossibility of performing certain operations in real numbers. The textbook presents a table with the main numerical sets and the operations allowed in them. The minimum conditions that complex numbers must satisfy are listed, and then the concept of an imaginary unit, the definition of a complex number, the equality of complex numbers, their sum, difference, product, and quotient are introduced.

From the geometric model of the set of real numbers we move on to the geometric model of the set of complex numbers. Consideration of the topic “Trigonometric form of writing a complex number” begins with the definition and properties of the modulus of a complex number. Next, we look at the trigonometric form of a complex number, the definition of the argument of a complex number, and the standard trigonometric form of a complex number.

Next, we study the extraction of the square root of a complex number and the solution of quadratic equations. And in the last paragraph, Moivre's formula is introduced and an algorithm for extracting the cube root of a complex number is derived.

Also in the textbook under review, in each paragraph, in parallel with the theoretical part, several examples are considered that illustrate the theory and give a more meaningful perception of the topic. Brief historical facts are given.

Definitions . Let a, b– real numbers, i– some symbol. A complex number is a notation of the form a+bi.

Addition And multiplication numbers on the set of complex numbers: (a+bi)+(c+di)=(a+c)+(b+d)i

(a+bi)(c+di)=(ac–bd)+(ad+bc)i. .

Theorem 1 . Set of complex numbers WITH with the operations of addition and multiplication it forms a field. Properties of addition

1) Commutativity b: (a+bi)+(c+di)=(a+c)+(b+d)i=(c+di)+(a+bi).

2) Associativity :[(a+bi)+(c+di)]+(e+fi)=(a+c+e)+(b+d+f)i=(a+bi)+[(c+di)+(e+fi)].

3) Existence neutral element :(a+bi)+(0 +0i)=(a+bi). Number 0 +0 i we will call zero and denote 0 .

4) Existence opposite element : (a+bi)+(–a–bi)=0 +0i=0 .

5) Commutativity of multiplication : (a+bi)(c+di)=(ac–bd)+(bc+ad)i=(c+di)(a+bi).

6) Associativity of multiplication :If z 1=a+bi, z 2=c+di, z 3=e+fi, That (z 1 z 2)z 3=z 1 (z 2 z 3).

7) Distributivity: If z 1=a+bi, z 2=c+di, z 3=e+fi, That z 1 (z 2+z 3)=z 1 z 2+z 1 z 3.

8) Neutral element for multiplication :(a+bi)(1+0i)=(a 1–b 0)+(a·0+b·1)i=a+bi.

9) Number 1 +0i=1 – unit.

9) Existence inverse element : " z¹ 0 $z –1 :zz –1 =1 .

Let z=a+bi. Real numbers a, called valid, A b - imaginary parts complex number z. Notations used: a=Rez, b=Imz.

If b=0 , That z=a+ 0i=a– real number. Therefore the set of real numbers R is part of the set of complex numbers C: R Í C.

Note: i 2=(0 +1i)(0+1i)=–1 +0i=–1 . Using this property of number i, as well as the properties of the operations proved in Theorem 1, you can perform operations with complex numbers according to the usual rules, replacing i 2 on - 1 .

Comment. The relations £, ³ (“less”, “greater”) are not defined for complex numbers.

2 Trigonometric notation .

The entry z = a+bi is called algebraic complex number form . Let's consider a plane with a selected Cartesian coordinate system. We will represent the number z point with coordinates (a, b). Then real numbers a=a+0i will be represented by axis points OX- it is called valid axis. Axis OY called imaginary axis, its points correspond to numbers of the form bi which are sometimes called purely imaginary . The entire plane is called complex plane .The number is called module numbers z: ,

Polar angle j called argument numbers z: j=argz.

The argument is determined up to a term 2kp; value for which – p< j £ p , called main importance argument. Numbers r, j are the polar coordinates of the point z. It's clear that a=r cosj, b=r sinj, and we get: z=a+b·i=r·(cosj+i sinj). trigonometric form writing a complex number.

Conjugate numbers . A complex number is called the conjugate of a numberz = a + bi . It's clear that . Properties : .

Comment. The sum and product of conjugate numbers are real numbers:

The concept of a complex number is primarily associated with the equation. There are no real numbers that satisfy this equation.

Thus, complex numbers arose as a generalization (extension) of the field of real numbers in attempts to solve arbitrary quadratic (and more general) equations by adding new numbers to it so that the extended set formed a number field in which the action of extracting the root would always be feasible.

Definition.A number whose square is - 1, usually denoted by the letteri and call imaginary unit.

Definition. Field of complex numbers C is called the minimal extension of the field of real numbers containing the root of the equation.

Definition. Field WITH called field of complex numbers, if it satisfies the following conditions:

Theorem. (On the existence and uniqueness of the field of complex numbers). There is only one, up to the designation of the root of the equation complex number field WITH .

Each element can be uniquely represented in the following form:

where , is the root of the equation i 2 +1=0.

Definition. Any element called complex number, the real number x is called real part number z and is denoted by , the real number y is called imaginary part number z and is denoted by .

Thus, a complex number is an ordered pair, a complex made up of real numbers x And y.

If X=0, then the number z= 0+iy=iy called purely imaginary or imaginary. If y=0, then the number z=x+ 0i=x is identified with a real number X.

Two complex numbers are considered equal if their real and imaginary parts are equal:

A complex number is equal to zero when its real and imaginary parts are equal to zero:

Definition. Two complex numbers having the same real part and whose imaginary parts are equal in absolute value but opposite in sign are called complex conjugate or simply conjugated.

Conjugate number z, denoted by . Thus, if , then .

1.3. Modulus and argument of a complex number.
Geometric representation of complex numbers

Geometrically, a complex number is depicted on a plane (Fig. 1) as a point M with coordinates ( x, y).

Definition. The plane on which complex numbers are depicted is called complex plane C, the Ox and Oy axes on which the real numbers are located and purely imaginary numbers , are called valid And imaginary axes respectively.

Point position can also be determined using polar coordinates r And φ , i.e. using the length of the radius vector and the inclination angle of the radius vector of the point M(x, y) to the positive real semi-axis Oh.

Definition. Module complex number is the length of the vector representing the complex number on the coordinate (complex) plane.

The modulus of a complex number is denoted by or by the letter r and is equal to the arithmetic value of the square root of the sum of the squares of its real and imaginary parts.

Complex number z called expression where A And V– real numbers, i– imaginary unit or special sign.

In this case, the following agreements are fulfilled:

1) with the expression a+bi you can perform arithmetic operations according to the rules that are accepted for literal expressions in algebra;

5) the equality a+bi=c+di, where a, b, c, d are real numbers, occurs if and only if a=c and b=d.

The number 0+bi=bi is called imaginary or purely imaginary.

Any real number a is a special case of a complex number, because it can be written in the form a=a+ 0i. In particular, 0=0+0i, but then if a+bi=0, then a+bi=0+0i, therefore, a=b=0.

Thus, a complex number a+bi=0 if and only if a=0 and b=0.

From the agreements follow the laws of transformation of complex numbers:

(a+bi)+(c+di)=(a+c)+(b+d)i;

(a+bi)-(c+di)=(a-c)+(b-d)i;

(a+bi)+(c+di)=ac+bci+adi-bd=(ac-bd)+(bc+ad)i;

We see that the sum, difference, product and quotient (where the divisor is not equal to zero) of complex numbers is, in turn, a complex number.

Number A called real part of a complex number z(denoted by ), V– the imaginary part of the complex number z (denoted by ).

A complex number z with zero real part is called. purely imaginary, with zero imaginary – purely real.

Two complex numbers are called. equal if their real and imaginary parts coincide.

Two complex numbers are called. conjugated, if they have substances. the parts coincide, but the imaginary parts differ in signs. , then its conjugate.

The sum of conjugate numbers is the number of substances, and the difference is a purely imaginary number. The operations of multiplication and addition of numbers are naturally defined on the set of complex numbers. Namely, if and are two complex numbers, then the sum is: ; work: .

Let us now define the operations of subtraction and division.

Note that the product of two complex numbers is the number of substances.

(since i=-1). This number is called. square modulus numbers. Thus, if a number is , then its modulus is a real number.

Unlike real numbers, the concepts of “more” and “less” are not introduced for complex numbers.

Geometric representation of complex numbers. Real numbers are represented by points on the number line:

Here is the point A means the number –3, dot B– number 2, and O- zero. In contrast, complex numbers are represented by points on the coordinate plane. For this purpose, we choose rectangular (Cartesian) coordinates with the same scales on both axes. Then the complex number a+ bi will be represented by a dot P with abscissa a and ordinate b(rice.). This coordinate system is called complex plane.

Module complex number is the length of the vector OP, representing a complex number on the coordinate ( comprehensive) plane. Modulus of a complex number a+ bi denoted | a+ bi| or letter r and is equal to:

Conjugate complex numbers have the same modulus. __

Argument complex number is the angle between the axis OX and vector OP, representing this complex number. Hence, tan = b / a .

Trigonometric form of a complex number. Along with writing a complex number in algebraic form, another form is also used, called trigonometric.

Let the complex number z=a+bi be represented by the vector OA with coordinates (a,b). Let's denote the length of the OA vector by beech r: r=|OA|, and the angle it forms with the positive direction of the Ox axis by the angle φ.

Using the definitions of the functions sinφ=b/r, cosφ=a/r, the complex number z=a+bi can be written as z=r(cosφ+i*sinφ), where , and the angle φ is determined from the conditions

Trigonometric form of a complex number z is its representation in the form z=r(cosφ+i*sinφ), where r and φ are real numbers and r≥0.

Indeed, the number r is called module complex number and is denoted by |z|, and the angle φ is the argument of the complex number z. The argument φ of a complex number z is denoted by Arg z.

Operations with complex numbers represented in trigonometric form:

This is famous Moivre's formula.

8 .Vector space. Examples and simplest properties of vector spaces. Linear dependence and independence of a system of vectors. Basis and rank of the final system of vectors

Vector space - a mathematical concept that generalizes the concept of the set of all (free) vectors of ordinary three-dimensional space.

For vectors in three-dimensional space, the rules for adding vectors and multiplying them by real numbers are indicated. Applicable to any vectors x, y, z and any numbers α, β these rules satisfy following conditions:

1) X+at=at+X(commutativity 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,where 1 is the field unit

6) α (βx)=(αβ )X(associativity of multiplication), where the product αβ is the product of scalars

7) (α +β )X=αх+βх(distributive property relative to the numerical factor);

8) α (X+at)=αх+αу(distributive property relative to the vector multiplier).

A vector (or linear) space is a set R, consisting of elements of any nature (called vectors), in which the operations of adding elements and multiplying elements by real numbers that satisfy conditions 1-8 are defined.

Examples of such spaces are the set of real numbers, the set of vectors on the plane and in space, matrices, etc.

Theorem “The simplest properties of vector spaces”

1. There is only one zero vector in a vector space.

2. In vector space, any vector has a unique opposite to it.

4. .

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Let 0 be the zero vector of the vector space V. Then . Let be another zero vector. Then . Let's take in the first case , and in the second - . Then and , whence it follows that , etc.

First we will prove that the product of a zero scalar and any vector is equal to a zero vector.

Let . Then, applying the vector space axioms, we obtain:

With respect to addition, a vector space is an Abelian group, and the cancellation law is valid in any group. Applying the law of reduction, the last equality implies 0*x=0

Now we prove statement 4). Let be an arbitrary vector. Then

It immediately follows that the vector (-1)x is opposite to the vector x.

Let now x=0. Then, applying the vector space axioms, we obtain:

Let us assume that . Since , where K is a field, then . Let's multiply the equality on the left by :, which implies either 1*x=0 or x=0

Linear dependence and independence of a system of vectors. A set of vectors is called a vector system.

A system of vectors is called linearly dependent if there are numbers that are not all equal to zero at the same time, such that (1)

A system of k vectors is called linearly independent if equality (1) is possible only for , i.e. when the linear combination on the left side of equality (1) is trivial.

Notes:

1. One vector also forms a system: at linearly dependent, and linearly independent at.

2. Any part of a system of vectors is called a subsystem.

Properties of linearly dependent and linearly independent vectors:

1. If a system of vectors includes a zero vector, then it is linearly dependent.

2. If a system of vectors has two equal vectors, then it is linearly dependent.

3. If a system of vectors has two proportional vectors, then it is linearly dependent.

4. A system of k>1 vectors is linearly dependent if and only if at least one of the vectors is a linear combination of the others.

5. Any vectors included in a linearly independent system form a linearly independent subsystem.

6. A system of vectors containing a linearly dependent subsystem is linearly dependent.

7. If a system of vectors is linearly independent, and after adding a vector to it it turns out to be linearly dependent, then the vector can be expanded into vectors , and, moreover, in a unique way, i.e. the expansion coefficients can be found uniquely.

Let us prove, for example, the last property. Since the system of vectors is linearly dependent, there are numbers that are not all equal to 0, which. In this equality. In fact, if , then. This means that a nontrivial linear combination of vectors is equal to the zero vector, which contradicts the linear independence of the system. Consequently, and then, i.e. a vector is a linear combination of vectors. It remains to show the uniqueness of such a representation. Let's assume the opposite. Let there be two expansions and , and not all coefficients of the expansions are respectively equal to each other (for example, ).

Then from the equality we get .

Therefore, a linear combination of vectors is equal to the zero vector. Since not all of its coefficients are equal to zero (at least), this combination is nontrivial, which contradicts the condition of linear independence of vectors. The resulting contradiction confirms the uniqueness of the expansion.

Rank and basis of the vector system. The rank of a system of vectors is the maximum number of linearly independent vectors of the system.

The basis of the vector system is called the maximal linearly independent subsystem of a given system of vectors.

Theorem. Any system vector can be represented as a linear combination of system basis vectors. (Any system vector can be expanded into basis vectors.) The expansion coefficients are determined uniquely for a given vector and a given basis.

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Let the system have a basis.

1 case. Vector - from the basis. Therefore, it is equal to one of the basis vectors, say . Then = .

Case 2. The vector is not from the basis. Then r>k.

Let's consider a system of vectors. This system is linearly dependent, since it is a basis, i.e. maximal linearly independent subsystem. Consequently, there are numbers with 1, with 2, ..., with k, with, not all equal to zero, such that

It is obvious that (if c = 0, then the basis of the system is linearly dependent).

Let us prove that the expansion of the vector with respect to the basis is unique. Let's assume the opposite: there are two expansions of the vector with respect to the basis.

Subtracting these equalities, we get

Taking into account the linear independence of the basis vectors, we obtain

Consequently, the expansion of the vector in terms of the basis is unique.

The number of vectors in any basis of the system is the same and equal to the rank of the system of vectors.

Axioms of the field. Field of complex numbers. Trigonometric notation for a complex number.

A complex number is a number of the form , where and are real numbers, the so-called imaginary unit. The number is called real part ( ) complex number, the number is called imaginary part ( ) complex number.

A bunch of or complex numbers usually denoted by a “bold” or thickened letter

Complex numbers are represented by complex plane:

The complex plane consists of two axes:
– real axis (x)
– imaginary axis (y)

The set of real numbers is a subset of the set of complex numbers

Actions with complex numbers

In order to add two complex numbers, you need to add their real and imaginary parts.

Subtracting Complex Numbers

The action is similar to addition, the only peculiarity is that the subtrahend must be put in brackets, and then the parentheses must be opened in the standard way, changing the sign

Multiplying complex numbers

open the brackets according to the rule for multiplying polynomials

Division of complex numbers

The division of numbers is carried out by multiplying the denominator and numerator by the conjugate expression of the denominator.

Complex numbers have many properties inherent in real numbers, of which we note the following, called main.

1) (a + b) + c = a + (b + c) (addition associativity);

2) a + b = b + a (commutativity of addition);

3) a + 0 = 0 + a = a (existence of a neutral element by addition);

4) a + (−a) = (−a) + a = 0 (existence of the opposite element);

5) a(b + c) = ab + ac ();

6) (a + b)c = ac + bc (distributivity of multiplication relative to addition);

7) (ab)c = a(bc) (associativity of multiplication);

8) ab = ba (commutativity of multiplication);

9) a∙1 = 1∙a = a (existence of a neutral element under multiplication);

10) for anyone a≠ 0 such a thing exists b, What ab = ba = 1 (existence of an inverse element);

11) 0 ≠ 1 (no name).

A set of objects of arbitrary nature on which the operations of addition and multiplication are defined, possessing the indicated 11 properties (which in this case are axioms), is called field.

The field of complex numbers can be understood as an extension of the field of real numbers in which the polynomial has a root

Any complex number (except zero) can be written in trigonometric form:
, where is it modulus of a complex number, A - complex number argument.

Modulus of a complex number is the distance from the origin to the corresponding point in the complex plane. Simply put, module is the length radius vector, which is indicated in red in the drawing.

The modulus of a complex number is usually denoted by: or

Using the Pythagorean theorem, it is easy to derive a formula for finding the modulus of a complex number: . This formula is correct for any meanings "a" and "be".

Argument of a complex number called corner between positive semi-axis the real axis and the radius vector drawn from the origin to the corresponding point. The argument is not defined for singular: .

The argument of a complex number is standardly denoted: or

Let φ = arg z. Then, by definition of the argument, we have:

Ring of matrices over the field of real numbers. Basic operations on matrices. Properties of operations.

Matrix size m´n, where m is the number of rows, n is the number of columns, is called a table of numbers arranged in a certain order. These numbers are called matrix elements. The location of each element is uniquely determined by the number of the row and column at the intersection of which it is located. The elements of the matrix are denoted by a ij, where i is the row number and j is the column number.

Definition. If the number of matrix columns is equal to the number of rows (m=n), then the matrix is ​​called square.

Definition. View matrix:

= E,

called identity matrix.

Definition. If a mn = a nm, then the matrix is ​​called symmetrical.

Example. - symmetric matrix

Definition. Square matrix of the form called diagonal matrix.

Multiplying a matrix by a number

Multiplying a matrix by a number(designation: ) consists in constructing a matrix, the elements of which are obtained by multiplying each element of the matrix by this number, that is, each element of the matrix is ​​equal to

Properties of multiplying matrices by a number:

· eleven A = A;

· 2. (λβ)A = λ(βA)

· 3. (λ+β)A = λA + βA

· 4. λ(A+B) = λA + λB

Matrix addition

Matrix addition is the operation of finding a matrix, all elements of which are equal to the pairwise sum of all corresponding elements of the matrices and, that is, each element of the matrix is ​​equal

Properties of matrix addition:

· 1.commutativity: A+B = B+A;

· 2.associativity: (A+B)+C =A+(B+C);

· 3.addition with zero matrix: A + Θ = A;

· 4.existence of an opposite matrix: A + (-A) = Θ;

All properties of linear operations repeat the axioms of linear space and therefore the theorem is valid:

The set of all matrices of the same size m x n with elements from the field P(the field of all real or complex numbers) forms a linear space over the field P (each such matrix is ​​a vector of this space). However, first of all, in order to avoid terminological confusion, matrices in ordinary contexts are avoided without the need (which is not present in the most common standard applications) and a clear clarification of the use of the term to be called vectors.

Matrix multiplication

Matrix multiplication(notation: , less often with a multiplication sign) - is an operation of calculating a matrix, each element of which is equal to the sum of the products of the elements in the corresponding row of the first factor and column of the second.

The number of columns in the matrix must match the number of rows in the matrix, in other words, the matrix must be agreed upon with matrix. If the matrix has dimension , - , then the dimension of their product is .

Properties of matrix multiplication:

· 1.associativity (AB)C = A(BC);

· 2.non-commutativity (in the general case): AB BA;

· 3. the product is commutative in the case of multiplication with the identity matrix: AI = IA;

· 4.distributivity: (A+B)C = AC + BC, A(B+C) = AB + AC;

· 5.associativity and commutativity with respect to multiplication by a number: (λA)B = λ(AB) = A(λB);

Matrix Transpose.

Finding the inverse matrix.

A square matrix is ​​invertible if and only if it is non-singular, that is, its determinant is not equal to zero. For non-square matrices and singular matrices, there are no inverse matrices.

Matrix rank theorem

The rank of matrix A is the maximum order of a non-zero minor

The minor that determines the rank of the matrix is ​​called the Basis minor. The rows and columns that form the BM are called basic rows and columns.

Designations: r(A), R(A), Rang A.

Comment. Obviously, the rank of a matrix cannot exceed the smaller of its dimensions.

For any matrix, its minor, row and column ranks are the same.

Proof. Let the minor rank of the matrix A equals r . Let us show that the row rank is also equal to r . To do this, we can assume that the invertible minor M order r is in the first r rows of the matrix A . It follows that the first r matrix rows A linearly independent and a set of minor rows M linearly independent. Let a -- length string r , composed of elements i th rows of the matrix, which are located in the same columns as the minor M . Since the lines are minor M form the base in k r , That a -- linear combination of minor strings M . Subtract from i -th line A the same linear combination of the first r matrix rows A . If you end up with a string containing a non-zero element in column number t , then consider minor M 1 order r+1 matrices A by adding the th row of the matrix to the rows of the minor A and to the columns of the minor th column of the matrix A (they say it's minor M 1 received bordering the minor M by using i -th line and t th matrix column A ). By our choice t , this minor is invertible (it is enough to subtract from the last row of this minor the linear combination of the first ones chosen above r rows, and then expand its determinant along the last row to make sure that this determinant coincides with the determinant of the minor, up to a non-zero scalar factor M . A-priory r such a situation is impossible and, therefore, after the transformation i -th line A will become zero. In other words, the original i -th line is a linear combination of the first r matrix rows A . We showed that the first r rows form the basis of a set of matrix rows A , that is, string rank A equals r . To prove that the column rank is r , it is enough to swap “rows” and “columns” in the above reasoning. The theorem has been proven.

This theorem shows that there is no point in distinguishing between the three ranks of a matrix, and in what follows, by the rank of a matrix we will understand the row rank, remembering that it is equal to both the column and minor ranks (notation r(A) -- matrix rank A ). Note also that from the proof of the rank theorem it follows that the rank of a matrix coincides with the dimension of any invertible minor of the matrix such that all minors bordering it (if they exist at all) are degenerate.

Kronecker-Capelli theorem

A system of linear algebraic equations is consistent if and only if the rank of its main matrix is ​​equal to the rank of its extended matrix, and the system has a unique solution if the rank is equal to the number of unknowns, and an infinite number of solutions if the rank is less than the number of unknowns.

Necessity

Let the system be cooperative. Then there are numbers such that . Therefore, the column is a linear combination of the columns of the matrix. From the fact that the rank of a matrix will not change if a row (column) is deleted or added from the system of its rows (columns), which is a linear combination of other rows (columns), it follows that .

Adequacy

Let . Let's take some basic minor in the matrix. Since, then it will also be the basis minor of the matrix. Then, according to the basis minor theorem, the last column of the matrix will be a linear combination of the basis columns, that is, the columns of the matrix. Therefore, the column of free terms of the system is a linear combination of the columns of the matrix.

Consequences

· The number of main variables of the system is equal to the rank of the system.

· A consistent system will be defined (its solution is unique) if the rank of the system is equal to the number of all its variables.

The theorem on the basis minor.

Theorem. In an arbitrary matrix A, each column (row) is a linear combination of the columns (rows) in which the basis minor is located.

Thus, the rank of an arbitrary matrix A is equal to the maximum number of linearly independent rows (columns) in the matrix.

If A is a square matrix and detA = 0, then at least one of the columns is a linear combination of the remaining columns. The same is true for strings. This statement follows from the property of linear dependence when the determinant is equal to zero.

7. SLU solution. Cramer method, matrix method, Gauss method.

Cramer's method.

This method is also applicable only in the case of systems of linear equations, where the number of variables coincides with the number of equations. In addition, it is necessary to introduce restrictions on the system coefficients. It is necessary that all equations be linearly independent, i.e. no equation would be a linear combination of the others.

To do this, it is necessary that the determinant of the system matrix does not equal 0.

Indeed, if any equation of the system is a linear combination of the others, then if you add elements of another row to the elements of one row, multiplied by some number, using linear transformations you can get a zero row. The determinant in this case will be equal to zero.

Theorem. (Cramer's Rule):

Theorem. System of n equations with n unknowns

if the determinant of the system matrix is ​​not equal to zero, it has a unique solution and this solution is found according to the formulas:

x i = D i /D, where

D = det A, and D i is the determinant of the matrix obtained from the system matrix by replacing column i with a column of free terms b i.

D i =

Matrix method for solving systems of linear equations.

The matrix method is applicable to solving systems of equations where the number of equations is equal to the number of unknowns.

The method is convenient for solving low-order systems.

The method is based on the application of the properties of matrix multiplication.

Let the system of equations be given:

Let's compose the matrices: A = ; B = ; X = .

The system of equations can be written: A×X = B.

Let's make the following transformation: A -1 ×A×X = A -1 ×B, because A -1 ×A = E, then E×X = A -1 ×B

X = A -1 ×B

To apply this method, it is necessary to find the inverse matrix, which may be associated with computational difficulties when solving high-order systems.

Definition. A system of m equations with n unknowns in general form is written as follows:

, (1)

where a ij are coefficients, and b i are constants. The solutions of the system are n numbers, which, when substituted into the system, turn each of its equations into an identity.

Definition. If a system has at least one solution, then it is called joint. If a system does not have a single solution, then it is called non-joint.

Definition. The system is called certain, if it has only one solution and uncertain, if more than one.

Definition. For a system of linear equations of the form (1), the matrix

A = is called the matrix of the system, and the matrix

A * =
called the extended matrix of the system

Definition. If b 1, b 2, …,b m = 0, then the system is called homogeneous. a homogeneous system is always consistent.

Elementary transformations of systems.

Elementary transformations include:

1) Adding to both sides of one equation the corresponding parts of the other, multiplied by the same number, not equal to zero.

2) Rearranging the equations.

3) Removing from the system equations that are identities for all x.

The Gauss method is a classical method for solving a system of linear algebraic equations (SLAE). This is a method of sequential elimination of variables, when, using elementary transformations, a system of equations is reduced to an equivalent triangular system, from which all other variables are found sequentially, starting with the last (by number) variables

Let the original system look like this

The matrix is ​​called the main matrix of the system - a column of free terms.

Then, according to the property of elementary transformations over rows, the main matrix of this system can be reduced to echelon form (the same transformations must be applied to the column of free terms):

Then the variables are called main variables. All others are called free.

If at least one number is , where , then the system under consideration is inconsistent, i.e. she doesn't have a single solution.

Let it be for anyone.

Let's move the free variables beyond the equal signs and divide each of the system equations by its coefficient at the leftmost ( , where is the line number):

If we give all possible values ​​to the free variables of system (2) and solve the new system with respect to the main unknowns from bottom to top (that is, from the lower equation to the upper), then we will obtain all solutions to this SLAE. Since this system was obtained by elementary transformations over the original system (1), then according to the equivalence theorem under elementary transformations, systems (1) and (2) are equivalent, that is, their sets of solutions coincide.

Consequences:
1: If in a joint system all variables are main, then such a system is definite.

2: If the number of variables in a system exceeds the number of equations, then such a system is either uncertain or inconsistent.

Algorithm

The algorithm for solving SLAEs using the Gaussian method is divided into two stages.

At the first stage, the so-called direct move is carried out, when, through elementary transformations over the rows, the system is brought to a stepped or triangular shape, or it is established that the system is incompatible. Namely, among the elements of the first column of the matrix, select a non-zero one, move it to the uppermost position by rearranging the rows, and subtract the resulting first row from the remaining rows after the rearrangement, multiplying it by a value equal to the ratio of the first element of each of these rows to the first element of the first row, zeroing thus the column below it. After these transformations have been completed, the first row and first column are mentally crossed out and continued until a zero-size matrix remains. If at any iteration there is no non-zero element among the elements of the first column, then go to the next column and perform a similar operation.

At the second stage, the so-called reverse move is carried out, the essence of which is to express all the resulting basic variables in terms of non-basic ones and build a fundamental system of solutions, or, if all the variables are basic, then express numerically the only solution to the system of linear equations. This procedure begins with the last equation, from which the corresponding basic variable is expressed (and there is only one) and substituted into the previous equations, and so on, going up the “steps”. Each line corresponds to exactly one basis variable, so at every step except the last (topmost), the situation exactly repeats the case of the last line.

Vectors. Basic concepts. Dot product, its properties.

Vector called a directed segment (an ordered pair of points). Vectors also include null a vector whose beginning and end coincide.

Length (module) vector is the distance between the beginning and end of the vector.

The vectors are called collinear, if they are located on the same or parallel lines. The null vector is collinear to any vector.

The vectors are called coplanar, if there is a plane to which they are parallel.

Collinear vectors are always coplanar, but not all coplanar vectors are collinear.

The vectors are called equal, if they are collinear, identically directed and have the same modules.

All vectors can be brought to a common origin, i.e. construct vectors that are respectively equal to the data and have a common origin. From the definition of equality of vectors it follows that any vector has infinitely many vectors equal to it.

Linear operations over vectors is called addition and multiplication by a number.

The sum of vectors is the vector -

Work - , and is collinear.

The vector is codirectional with the vector ( ) if a > 0.

The vector is oppositely directed with the vector ( ¯ ), if a< 0.

Properties of vectors.

1) + = + - commutativity.

2) + ( + ) = ( + )+

5) (a×b) = a(b) – associativity

6) (a+b) = a + b - distributivity

7) a( + ) = a + a

1) Basis in space any 3 non-coplanar vectors taken in a certain order are called.

2) Basis on a plane any 2 non-collinear vectors taken in a certain order are called.

3)Basis Any non-zero vector on a line is called.

If is a basis in the space and , then the numbers a, b and g are called components or coordinates vectors in this basis.

In this regard, we can write the following properties:

equal vectors have identical coordinates,

when a vector is multiplied by a number, its components are also multiplied by this number,

When adding vectors, their corresponding components are added.

;
;

Linear dependence of vectors.

Definition. Vectors are called linearly dependent, if such a linear combination exists, with a i not equal to zero at the same time, i.e. .

If only when a i = 0 is satisfied, then the vectors are called linearly independent.

Property 1. If there is a zero vector among the vectors, then these vectors are linearly dependent.

Property 2. If one or more vectors are added to a system of linearly dependent vectors, then the resulting system will also be linearly dependent.

Property 3. A system of vectors is linearly dependent if and only if one of the vectors is decomposed into a linear combination of the remaining vectors.

Property 4. Any 2 collinear vectors are linearly dependent and, conversely, any 2 linearly dependent vectors are collinear.

Property 5. Any 3 coplanar vectors are linearly dependent and, conversely, any 3 linearly dependent vectors are coplanar.

Property 6. Any 4 vectors are linearly dependent.

Vector length in coordinates is defined as the distance between the start and end points of a vector. If two points are given in space A(x 1, y 1, z 1), B(x 2, y 2, z 2), then.

If the point M(x, y, z) divides the segment AB in the ratio l/m, then the coordinates of this point are determined as:

In a special case, the coordinates midpoint of the segment are found like:

x = (x 1 + x 2)/2; y = (y 1 + y 2)/2; z = (z 1 + z 2)/2.

Linear operations on vectors in coordinates.

Rotating coordinate axes

Under turning Coordinate axes mean a coordinate transformation in which both axes are rotated by the same angle, but the origin and scale remain unchanged.

Let the new system O 1 x 1 y 1 be obtained by rotating the Oxy system by an angle α.

Let M be an arbitrary point on the plane, (x;y) its coordinates in the old system and (x";y") - in the new system.

Let us introduce two polar coordinate systems with a common pole O and polar axes Ox and Οx 1 (the scale is the same). The polar radius r is the same in both systems, and the polar angles are respectively equal to α + j and φ, where φ is the polar angle in the new polar system.

According to the formulas for the transition from polar to rectangular coordinates, we have

But rcosj = x" and rsinφ = y". That's why

The resulting formulas are called axis rotation formulas . They allow you to determine the old coordinates (x; y) of an arbitrary point M through the new coordinates (x"; y") of the same point M, and vice versa.

If a new coordinate system O 1 x 1 y 1 is obtained from the old Oxy by parallel transfer of coordinate axes and subsequent rotation of the axes by angle α (see Fig. 30), then by introducing an auxiliary system it is easy to obtain the formulas

expressing the old x and y coordinates of an arbitrary point in terms of its new x" and y" coordinates.

Ellipse

An ellipse is a set of points on a plane, the sum of the distances from each

which is constant up to two given points. These points are called foci and

are designated F1 And F2, the distance between them 2s, and the sum of the distances from each point to

focuses – 2a(by condition 2a>2c). Let us construct a Cartesian coordinate system so that F1 And F2 were on the x-axis, and the origin coincided with the middle of the segment F1F2. Let us derive the equation of the ellipse. To do this, consider an arbitrary point M(x, y) ellipse. A-priory: | F1M |+| F2M |=2a. F1M =(x+c; y);F2M =(x-c; y).

|F1M|=(x+ c)2 + y 2 ; |F2M| = (x- c)2 + y 2

(x+ c)2 + y 2 + (x- c)2 + y 2 =2a(5)

x2+2cx+c2+y2=4a2-4a(x- c)2 + y 2 +x2-2cx+c2+y2

4cx-4a2=4a(x- c)2 + y 2

a2-cx=a(x- c)2 + y 2

a4-2a2cx+c2x2=a2(x-c)2+a2y2

a4-2a2cx+c2x2=a2x2-2a2cx+a2c2+a2y2

x2(a2-c2)+a2y2=a2(a2-c2)

because 2a>2c(the sum of two sides of a triangle is greater than the third side), then a2-c2>0.

Let a2-c2=b2

Points with coordinates (a, 0), (−a, 0), (b, 0) and (−b, 0) are called the vertices of the ellipse, the value a is the semi-major axis of the ellipse, and the value b is its semi-minor axis. The points F1(c, 0) and F2(−c, 0) are called foci

ellipse, and the focus F1 is called right, and the focus F2 is called left. If point M belongs to an ellipse, then the distances |F1M| and |F2M| are called focal radii and are denoted by r1 and r2, respectively. The quantity e =c/a is called the eccentricity of the ellipse. Lines with equations x =a/e

and x = −a/e are called directrixes of the ellipse (for e = 0 there are no directrixes of the ellipse).

General plane equation

Consider a general first-degree equation with three variables x, y and z:

Assuming that at least one of the coefficients A, B or C is not equal to zero, for example, we rewrite equation (12.4) in the form