Geometric representation of complex numbers and operations on them. Trigonometric form of a complex number
consider the set R2 of all possible ordered pairs (x» Y) of real numbers xxy € R. For such pairs (a, b) = (c, d) if and only if a = c and b - d. Let us introduce on this set R2 the internal laws of composition in the form of operations of addition and multiplication. We define addition by the equality £faa the operation is associative and commutative; it has (according to Definition 4.5) a neutral element (0, 0), and, by Definition 4.6, for each pair (a, 6) one can specify a symmetric (opposite) element (-a, -6). Indeed, V(a, 6) £ R2 Moreover, or The field of complex numbers. We define multiplication by equality It is easy to verify that the operation introduced in this way is associative, commutative, and distributive with respect to addition. This operation has a neutral element, which is the pair (1, 0), since So, with respect to the introduced operations of addition and multiplication, the set R2 is an Abelian ring with unit (see Table 4.1). u* Between the set of pairs (x, 0) € R2 and the set of real numbers x G R it is easy to establish a one-to-one correspondence (x, 0) x) from which it follows that, The field of complex numbers. those. addition and multiplication of such pairs are performed in the same way as for real numbers. Let us replace pairs of the form (x, 0) with real numbers, i.e. instead of (x, 0) we will simply write x, in particular, instead of (1, 0), we will simply write 1. The pair (0, 1) occupies a special place in the set R2. According to (4.3), it has the properties and has received a special notation i, and Then, in view of (4.2) and (4.3), any pair (x, y) ∈ R2 can be represented as the field of complex numbers. Denote z. The element z is called the complex conjugate of the element z. Taking into account (4.3) z-z = x2 -by2. If z does not match the neutral element (0, 0), i.e. if x and y are not equal to 0 at the same time (they denote 2^0), then x2 + + y2 φ 0. Then the inverse (symmetric, opposite with respect to the operation of multiplication - see 4.1) to the element z \u003d x + iy will be such an element r "1, that zz~l = 1 or zzz~l =z, i.e. (x2 + y2)z~l = x - y Hence -1_ X 2 Y \ Therefore, any element of gf O has an inverse to svb with respect to the operation of multiplication , and the set R2 with the operations of addition and multiplication united on it in accordance with (4.1) and (4.3) is thus a field (see Table 4.1) It is called the field (or set) of complex numbers and is denoted C. B By virtue of the above one-to-one correspondence (r, 0) € R2 ++ x € R to the fraction of complex numbers is an extension of the field of real numbers. Any element r in C is called a complex number, and its representation in the form z = x + iy> where x, y £ R and i2 = -l, is represented by an algebraic form of a complex number. In this case, £ is called the real part of the complex number and denoted by Re z, and y is called the imaginary part and denoted by Imz (t is called the imaginary unit). Note that the imaginary part of a complex number is a real number. The name for y is not entirely successful, but as a tribute to historical tradition, it has remained to this day. The term "complex number"44 was introduced in 1803 by the French mathematician JI. Carnot (1753-1823), but K. Gauss began to use this term systematically from 1828 to replace the less successful “imaginary number”44. In Russian mathematical literature of the XIX century. used the term "composite number"44. Already in R. Descartes, the real and imaginary parts of a complex number are opposed. Later, the first letters of the French words reele (real) and imagimaire (imaginary) became the designations of these parts, although many mathematicians considered the essence of imaginary quantities unclear and even mysterious and mystical. So, I. Newton did not include them in the concept of number, and G. Leibniz belongs to the Phrase: “Imaginary numbers are a wonderful and wonderful refuge of the divine spirit, almost an amphibian of being with non-being44. Since the set R2 of all possible pairs of real numbers can be identified with points on the plane, each complex number z =? x + iy corresponds to the point y) (Fig. 4.1), which allows us to talk about the geometric form of the representation of a complex number. When complex numbers are identified with points of the plane, it is called the complex plane, or the plane of complex numbers. Real numbers are placed on the x-axis, i.e. numbers z, for which lmz = y = 0, and on the Oy axis - numbers z = iy, called purely imaginary, for which Re r = x = 0. Poeto-Fig. 4.1, the coordinate axes in the complex plane are called the real and imaginary, respectively. The points of the plane corresponding to the complex conjugate elements z and z (complex conjugate numbers) are symmetrical about the real axis, and the points representing z and -z are symmetrical about the origin. Distance Field of complex numbers. point M(x, y), depicting a complex number z = x + iy on the plane, from the origin is called the modulus of the complex number and is denoted \z\ or r. The angle that forms the radius vector of the point M with the positive direction of the Ox axis is called argument of a complex number and denote Argz or (p (see Fig. 4.1). The angle is measured as in trigonometry: the positive direction of the angle change is considered to be the counterclockwise direction. It is clear that Arg z is not uniquely defined, but up to a term that is a multiple of 2n. z - 0, the value of Args is not defined. The point corresponding to this number (the origin) is characterized only by the condition \z\ = r = 0. Thus, to each complex number z on the complex plane there corresponds the radius vector of the point M(x, y), which can be set it by polar coordinates: the polar radius r ^ 0, equal to the modulus of the complex number, and the polar angle coinciding with the principal value of the argument of this complex number.According to the definitions of trigonometric functions and their inverses known from the school course of trigonometry (see point z on the complex plane we have x=rcosy>= X Taking into account the restrictions imposed on the principal value of the argument of the complex number, we obtain if x > 0, if x 0, if x = 0 and y. From (4.6) it follows that pr the notation + tsiny> is avomeric), (4.8) It is called the trigonometric form of the representation of a complex number. For the transition from the algebraic form of representation to the trigonometric form, use (4.5) and (4.7) ”and for the reverse transition - (4.6). Note that two non-zero complex numbers are equal if and only if their moduli are equal and the arguments differ by terms that are multiples of 2n. According to (4.1), the sum of the complex numbers z \ and r2 will be a complex number and their difference - From these formulas it follows that the addition (or subtraction) of complex numbers is similar to the addition (or subtraction) of vectors in the complex plane according to the parallelogram rule (Fig. 4.2) ( while the corresponding coordinates of the vectors are added or subtracted). Therefore, for the moduli of complex numbers, the triangle inequalities a are valid in the form (the length of any side of a triangle is not greater than the sum of the lengths of its two other sides). However, this is where the analogy between complex numbers and vectors ends. The sum or difference of complex numbers can be a real number (for example, the sum of complex conjugate numbers r-f z = = 2x, x = Rez e R). According to (4.3), the product of the complex numbers z\ and z2 is a complex number. the quotient Z1/22 for V*2 φ 0 is understood to be a complex number -r satisfying the equality z^z = z\. After multiplying both parts of this equality by 22, we get. Raising a complex number z to the power n ∈ N is multiplying z by itself n times, taking into account the fact that for k 6 N is the field of complex numbers. The trigonometric notation (4.8) makes it possible to simplify multiplication, division, and exponentiation of complex numbers. So, for z\ \u003d r\ (cos (p\ + isiny?i) and Z2 \u003d Г2 (co + -f isin no (4.3) it can be established that On the complex plane (Fig. 4.3) multiplication corresponds to the rotation of the segment OM by angle (counterclockwise at 0) and a change in its length by r2 = \z2\ times; exponentiation n £ N as multiplying z by itself n times, semi-nay In honor of the English mathematician A. de Moivre (1667-1754), this relation is called the Moivre formula for raising a complex number to a positive integer power. /n, q € Q, m € Z, n6N, is related to raising this number to the power 1/n, or, as they say, extracting the nth root of a complex number. degree, i.e. = w, if wn = z. Let) Then from (4.13) we have and, taking into account the equality of complex numbers, we obtain From expression (4.14), we call derived from the Moivre formula for extracting the root of a positive integer power from a complex number) it follows that among the possible values of y/z, n values corresponding to k = 0, n - 1 will be different. All n different values for $fz have the same modulus, and their arguments differ by angles that are multiples of 2jr/n. The values correspond to the points of the complex plane at the vertices of a regular n-gon inscribed in a circle of radius 1/f centered at the origin. In this case, the radius vector of one of the vertices forms an angle (p/n) with the Ox-axis. From (4.13) and (4.14) follows the formula for raising the complex number z /0 to a rational power g € Q. Beli g = m/n, where m € Z and n € N, taking into account (4.7), we obtain (Therefore, in trigonometric form. According to (4.11) and (4.12) we find: Using (4.13), we raise z\ to the power n = 4, applying (4.14), we extract from z2 the root of degree n = 3 Results of calculations are shown in Fig. 4.4 Three values of the root of the third degree from zi correspond to the vertices of a regular triangle ABC inscribed in a circle of radius and the polar angles of these vertices \u003d i * / 18, 4\u003e v \u003d 13m / 18 and \u003d 25m / 18 (or \u003d - 11^/18).
Field axioms. The field of complex numbers. Trigonometric notation of 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 same complex numbers usually denoted by a "bold" or thickened letter
Complex numbers are displayed on 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
Operations with complex numbers
To add two complex numbers, add their real and imaginary parts.
Subtraction of complex numbers
The action is similar to addition, the only peculiarity is that the subtrahend must be taken in brackets, and then, as a standard, open these brackets with a sign change
Multiplication of complex numbers
open brackets according to the rule of multiplication of 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 of the properties of 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 (the existence of an opposite element);
5) a(b + c) = ab + ac ();
6) (a + b)c = ac + bc (distributivity of multiplication with respect to addition);
7) (ab)c = a(bc) (multiplication associativity);
8) ab = ba (commutativity of multiplication);
9) a∙1 = 1∙a = a (existence of a neutral element by multiplication);
10) for any a≠ 0 b, what ab = ba = 1 (existence of inverse element);
11) 0 ≠ 1 (no name).
The set of objects of arbitrary nature, on which the operations of addition and multiplication are defined, which have 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 complex number modulus, a - complex number argument.
The modulus of a complex number is the distance from the origin of coordinates to the corresponding point of the complex plane. Simply put, modulus is the length radius vector, which is marked 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 valid for any meanings "a" and "be".
The argument of a complex number called injection between positive 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 usually denoted by: or
Let and φ = arg z. Then, by the definition of the argument, we have:
Ring of matrices over the field of real numbers. Basic operations on matrices. Operation properties.
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 place of each element is uniquely determined by the number of the row and column at the intersection of which it is located. Matrix elements are denoted a ij , where i is the row number and j is the column number.
Definition. If the number of columns of the matrix 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 a mn = a nm, then the matrix is called symmetrical.
Example. 
- symmetric matrix
Definition.
Square view matrix 
called diagonal matrix.
Multiplying a Matrix by a Number
Multiplying a Matrix by a Number(notation: ) is to construct a matrix whose elements are obtained by multiplying each element of the matrix by this number, that is, each element of the matrix is equal to
Properties of multiplication of 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 to
Matrix addition properties:
1. commutativity: A+B = B+A;
2.associativity: (A+B)+C =A+(B+C);
3.addition with a zero matrix: A + Θ = A;
4.existence of the opposite matrix: A+(-A)=Θ;
All properties of linear operations repeat the axioms of a linear space, and therefore the following theorem is true:
The set of all matrices of the same size m x n with elements from the field P(fields of all real or complex numbers) forms a linear space over the field P (each such matrix is a vector of this space). However, primarily to avoid terminological confusion, matrices in common contexts are avoided without the need (which is not in the most common standard applications) and clear specification of the use of the term to call vectors.
Matrix multiplication
Matrix multiplication(notation: , rarely with the multiplication sign) - there is an operation to calculate 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 the 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 with a matrix. If the matrix has dimension , - , then the dimension of their product is .
Matrix multiplication properties:
1.associativity (AB)C = A(BC);
2.non-commutativity (generally): AB BA;
3. The product is commutative in the case of multiplication with an 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 transposition.
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 degenerate 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.
Notation: r(A), R(A), Rang A.
Comment. Obviously, the value of the rank of a matrix cannot exceed the smallest 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 . For this, we can assume that the reversible minor M order r is in the first r matrix rows A . It follows from this that the first r matrix rows A are linearly independent and the set of minor rows M linearly independent. Let be a -- length string r , composed of elements i -th row of the matrix , which are located in the same columns as the minor M . Since the minor strings M form the basis of k r , then 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 the result is a string containing a non-null element in the column with the number t , then consider the minor M 1 order r+1 matrices A , adding to the rows of the minor the th row of the matrix A and to the minor columns -th column of the matrix A (they say that minor M 1 received edging minor M via i -th line and t -th column of the matrix A ). By our choice t , this minor is invertible (it suffices to subtract from the last row of this minor the linear combination of the first r rows, and then expand its determinant over the last row to make sure that this determinant, up to a non-zero scalar factor, matches the determinant of the minor 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 row is a linear combination of the first r matrix rows A . We have shown that the first r rows form the base of the matrix rowset A , that is, the lowercase 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 it makes no sense to distinguish between 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 (the notation r(A) -- matrix rank A ). We also note that it follows from the proof of the rank theorem that the rank of a matrix coincides with the dimension of any invertible minor of the matrix such that all minors surrounding 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.
Need
Let the system be consistent. 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 from the system of its rows (columns) or a row (column) that is a linear combination of other rows (columns) follows that .
Adequacy
Let be . 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 members 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 compatible system will be defined (its solution is unique) if the rank of the system is equal to the number of all its variables.
Basis minor theorem.
Theorem. In an arbitrary matrix A, each column (row) is a linear combination of 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 other columns. The same is true for strings. This statement follows from the property of linear dependence with the determinant equal to zero.
7. SLU solution. Cramer's 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 coefficients of the system. 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 matrix of the system is not equal to 0.
Indeed, if any equation of the system is a linear combination of the others, then if the elements of any row are added to the elements of another, 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 matrix of the system is not equal to zero, it has a unique solution and this solution is found by 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 members 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 applying the properties of matrix multiplication.
Let the system of equations be given:
Compose 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 \u003d A -1 × B
To apply this method, it is necessary to find the inverse matrix, which may be associated with computational difficulties in solving high-order systems.
Definition. The system of m equations with n unknowns is generally 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 the system has no solution, then it is called incompatible.
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*= 
is called the augmented matrix of the system
Definition. If b 1 , b 2 , …,b m = 0, then the system is called homogeneous. homogeneous system is always consistent.
Elementary transformations of systems.
The elementary transformations are:
1) The addition to both parts of one equation of the corresponding parts of the other, multiplied by the same number, not equal to zero.
2) Permutation of equations in places.
3) Removal from the system of 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 successive elimination of variables, when, with the help of elementary transformations, the system of equations is reduced to an equivalent system of a triangular form, from which all other variables are found sequentially, starting from the last (by number) variables
Let the original system look like this
The matrix is called the main matrix of the system, - the column of free members.
Then, according to the property of elementary transformations over rows, the main matrix of this system can be reduced to a stepped form (the same transformations must be applied to the column of free members):
Then the variables are called main variables. All others are called free.
If at least one number , where , then the system under consideration is inconsistent, i.e. she has no solution.
Let for any .
We transfer the free variables beyond the equal signs and divide each of the equations of the system by its coefficient at the leftmost one ( , where is the line number):
If we assign all possible values to the free variables of system (2) and solve the new system with respect to the main unknowns from the bottom up (that is, from the lower equation to the upper one), then we will get all the solutions of this SLAE. Since this system was obtained by elementary transformations over the original system (1), then by the equivalence theorem under elementary transformations, systems (1) and (2) are equivalent, that is, the sets of their solutions coincide.
Consequences:
1: If in a joint system all variables are principal, then such a system is definite.
2: If the number of variables in the system exceeds the number of equations, then such a system is either indeterminate or inconsistent.
Algorithm
The algorithm for solving SLAE by the Gaussian method is divided into two stages.
At the first stage, the so-called direct move is carried out, when, by means of elementary transformations over rows, the system is brought to a stepped or triangular form, or it is established that the system is inconsistent. Namely, among the elements of the first column of the matrix, a non-zero one is chosen, it is moved to the uppermost position by permuting the rows, and the first row obtained after the permutation is subtracted from the remaining rows, 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 the indicated transformations have been made, the first row and the first column are mentally crossed out and continue until a zero-size matrix remains. If at some of the iterations among the elements of the first column there was not found a non-zero one, 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 construct a fundamental system of solutions, or, if all variables are basic, then numerically express 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 there) and substituted into the previous equations, and so on, going up the “steps”. Each line corresponds to exactly one basic variable, so at each step, except for the last (topmost), the situation exactly repeats the case of the last line.
Vectors. Basic concepts. Scalar product, its properties.
Vector is called a directed segment (an ordered pair of points). Also applies to vectors. null a vector whose start and end are the same.
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 zero vector is collinear to any vector.
The vectors are called coplanar if there exists 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, have the same direction, and have the same absolute value.
Any vectors can be reduced to a common origin, i.e. construct vectors correspondingly equal to the data and having a common origin. From the definition of vector equality 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 - 
, while being collinear.
The vector is codirectional with the vector ( ) if a > 0.
The vector is opposite to the vector ( ¯ ) if a< 0.
Vector properties.
1) + = + - commutativity.
2) + ( + ) = ( + )+
5) (a×b) = a(b) – associativity
6) (a + b) = a + b - distributivity
7) a( + ) = a + a
1) Basis in space are called any 3 non-coplanar vectors, taken in a certain order.
2) Basis on the plane are any 2 non-collinear vectors taken in a certain order.
3)Basis any non-zero vector is called on the line.
If a 
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 the same coordinates,
when a vector is multiplied by a number, its components are also multiplied by that number,
when vectors are added, their corresponding components are added.
; 
;
Linear dependence of vectors.
Definition.
Vectors 
called linearly dependent, if there is such a linear combination , if a i is 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 null vector among the vectors, then these vectors are linearly dependent.
Property 2. If one or more vectors are added to the 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 other 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 defined as the distance between the start and end points of the vector. If two points are given in the 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 defined as:
In a particular case, the coordinates middle of the segment are located like:
x \u003d (x 1 + x 2) / 2; y = (y 1 + y 2)/2; z = (z 1 + z 2)/2.
Linear operations on vectors in coordinates.
Rotation of coordinate axes
Under turn coordinate axes understand such a coordinate transformation in which both axes are rotated by the same angle, while the origin and scale remain unchanged.
Let a new system O 1 x 1 y 1 be obtained by rotating the Oxy system through an angle α.
Let Μ be an arbitrary point of the plane, (x; y) - its coordinates in the old system and (x"; y") - in the new system.
We 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 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". So
The resulting formulas are called axis rotation formulas . They make it possible to determine the old coordinates (x; y) of an arbitrary point M in terms of the new coordinates (x"; y") of the same point M, and vice versa.
If the new coordinate system O 1 x 1 y 1 is obtained from the old Oxy by parallel transfer of the coordinate axes and subsequent rotation of the axes by an 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 in a plane, the sum of the distances from each of
up to two given points is constant. 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
tricks - 2a(by condition 2a>2c). We 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's 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)
as 2a>2c(the sum of two sides of a triangle is greater than the third side), then a2-c2>0.
Let be a2-c2=b2
The points with coordinates (a, 0), (−a, 0), (b, 0) and (−b, 0) are called the vertices of the ellipse, the value a is the major semi-axis of the ellipse, and the value b is its minor semi-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 the point M belongs to the ellipse, then the distances |F1M| and |F2M| are called focal radii and are denoted by r1 and r2, respectively. The value e \u003d c / a is called the eccentricity of the ellipse. Straight 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 equation of the plane
Consider a general equation of the first degree 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
Definitions . Let be a, b are real numbers, i is some character. A complex number is a record 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 forms a field. Addition properties
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 z1=a+bi, z2=c+di, z3=e+fi, then (z 1 z 2)z 3=z 1 (z 2 z 3).
7) Distributivity: if z1=a+bi, z2=c+di, z3=e+fi, then z 1 (z 2+z3)=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 be z=a+bi. Real numbers a, called valid, a b - imaginary parts complex number z. Notations are used: a=Rez, b=imz.
If a b=0 , then z=a+ 0i=a is 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 number property i, as well as the properties of the operations proved in Theorem 1, one can perform operations with complex numbers according to the usual rules, replacing i 2 on the - 1 .
Comment. The relations £, ³ (“less than”, “greater than”) for complex numbers are not defined.
2 Trigonometric notation .
The notation z = a+bi is called algebraic notation of a complex number . Consider a plane with a chosen Cartesian coordinate system. Let's represent the number z point with coordinates (a,b). Then the 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 the term 2kp; value for which - p< j £ p , is 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 notation of a complex number.
Conjugate numbers . A complex number is called a conjugate of a number.z = a + bi . It's clear that . Properties : .
Comment. The sum and product of conjugate numbers are real numbers:
Def. The system of complex numbers is the min-th 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>called sys-th comp-th numbers, if you issue the following conditions (axioms):
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.
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
St. va ℂ 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. The fundamental theorem of algebra: The field ℂ of numbers is algebraically closed, that is, any pl. degrees over the field ℂ of numbers has at least one set. root
The next from the main. theorems alg.: Any plural posit. degrees over the field of complex numbers can be decomposed into a product ... of the first degree with a positive coefficient.
Next: any square ur-e has 2 roots: 1) D>0 2-a diff. action root 2)D=0 2-a real. coincident-x root 3)D<0 2-а компл-х корня.
4. Axioms. 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 the upper grades, 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 the 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 principles 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 ideas about polynomials and numbers laid down in the basic mathematics course, in a sense, completing the development of the concept of number in high 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 planes. Or vice versa, 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 addition, multiplication, subtraction, division, raising to a power, extracting a root from a complex number on complex numbers; translate complex numbers from algebraic form into trigonometric, have an idea about 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 beginning of mathematical analysis", the topic "Complex numbers" is introduced in the 11th grade. The 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 in the 11th grade - the integral and differential equations, exponential, logarithmic and power functions, polynomials. 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 a consideration of the issue of solving quadratic equations, equations of the third and fourth degree and, as a result, 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 rigorous definition of the concept of a complex number, properties of the operations of addition and multiplication, subtraction and division is given. The next subsection deals with conjugate complex numbers and some of their properties. Next, we consider the question of extracting square roots from complex numbers and solving quadratic equations with complex coefficients. The following paragraph deals with: geometric representation of complex numbers; polar coordinate system and trigonometric form of complex numbers; multiplication, exponentiation and division of complex numbers in trigonometric form; de Moivre's formula, the application of complex numbers to the proof of trigonometric identities; extracting a root from a complex number; the 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", the topic "Complex numbers is considered in the 11th grade after studying all topics, i.e. at the end of the 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, contains many concepts, definitions, theorems. The paragraph "Algebraic form and geometric interpretation of complex numbers" contains three sections: the algebraic form of a complex number; conjugate complex numbers; geometric interpretation of a complex number. The paragraph "Trigonometric form of a complex number" contains definitions and concepts necessary for introducing the concept of a trigonometric form of a complex number, as well as an algorithm for switching from an algebraic form of notation to a trigonometric form of a complex number. In the last paragraph “Roots of polynomials. The exponential form of complex numbers” contains three sections: roots from 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 the minimum knowledge about them. The textbook has a small number of exercises and does not address the issue of raising a complex number to a power and De Moivre's 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 grade 10 immediately after studying the topics "Real numbers" and "Trigonometry". This placement is not accidental: both the numerical circle and the trigonometry formulas are actively used in the study of the trigonometric form of a complex number, the Moivre formula, when extracting square and cubic roots from a complex number. The topic "Complex numbers" is presented in the 6th chapter and is divided into 5 sections: complex numbers and arithmetic operations on them; complex numbers and 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 a number and the impossibility of performing certain operations in real numbers. The textbook contains 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, they pass 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 consider the trigonometric form of writing 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, the solution of quadratic equations. And in the last paragraph, the Moivre formula is introduced and an algorithm for extracting the cube root from a complex number is derived.
Also in the textbook under consideration, 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.
complex number z called expression , where a and in- real numbers, i is an imaginary unit or a special sign.
The following agreements are followed:
1) with the expression a + bi, arithmetic operations can be performed 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, takes place 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 as a=a+ 0i. In particular, 0=0+0i, but then if a+bi=0, then a+bi=0+0i, hence a=b=0.
Thus, a complex number a+bi=0 if and only if a=0 and b=0.
The laws of transformation of complex numbers follow from the conventions:
(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, in turn, is a complex number.
Number a called real part of a complex number z(denoted) in is 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 they have the same real and imaginary parts.
Two complex numbers are called. conjugated if they have substances. parts coincide, and imaginary ones differ in signs. , then the conjugate to it .
The sum of conjugate numbers is the number of substances, and the difference is a purely imaginary number. On the set of complex numbers, the operations of multiplication and addition of numbers are naturally defined. Namely, if and are two complex numbers, then the sum is: ; work: .
We now define the operations of subtraction and division.
Note that the product of two complex numbers is the number of substances.
(because i=-1). This number is called module square numbers. Thus, if a number , then its modulus is a real number.
Unlike real numbers, for complex numbers the concept of "more", "less" is not introduced.
Geometric representation of complex numbers. Real numbers are represented by points on the number line:
Here is the point A means number -3, dot B is the number 2, and O- zero. In contrast, complex numbers are represented by points on the coordinate plane. For this, 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 called the length of the vector OP, depicting a complex number on the coordinate ( comprehensive) plane. Complex number modulus a + bi denoted by | 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 one is also used, called trigonometric.
Let the complex number z=a+bi be represented by the vector ОА with coordinates (a,b). Let's designate the length of the OA vector as r: r=|OA|, and the angle that it forms with the positive direction of the Ox axis through 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 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 denoted by 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:
It's famous Moivre formula.
8 .Vector space. Examples and simple properties of vector spaces. Linear dependence and independence of the system of vectors. Basis and rank of a finite system of vectors
Vector space - mathematical concept that generalizes the concept of the totality 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 given. Applied to any vectors x, y, z and any numbers α, β these rules satisfy the 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+βx(distributive property with respect to a numerical factor);
8) α (X+at)=αх+αy(distributive property with respect to the vector factor).
A vector (or linear) space is a set R, consisting of elements of any nature (called vectors), which defines the operations of adding elements and multiplying elements by real numbers that satisfy conditions 1-8.
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 null vector in a vector space.
2. In a vector space, any vector has a unique opposite to it.
4. .
Doc-in
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 , p.t.d.
First we prove that the product of a zero scalar and any vector is equal to a zero vector.
Let be . Then, applying the vector space axioms, we get:
With respect to addition, a vector space is an Abelian group, and the cancellation law holds in any group. Applying the law of reduction, it follows from the last equality 0 * x \u003d 0
We now prove assertion 4). Let be an arbitrary vector. Then
This immediately implies that the vector (-1)x is the opposite of the vector x.
Let now x=0. Then, applying the vector space axioms, we get:
Let's assume that . Since , where K is a field, there exists . Let's multiply the equality on the left by: , which implies either 1*x=0 or x=0
Linear dependence and independence of the system of vectors. A set of vectors is called a vector system.
A system of vectors is called linearly dependent if there are numbers , 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: for linearly dependent, and for linearly independent.
2. Any part of a system of vectors is called a subsystem.
Properties of linearly dependent and linearly independent vectors:
1. If the system of vectors includes a zero vector, then it is linearly dependent.
2. If there are two equal vectors in a system of vectors, then it is linearly dependent.
3. If there are two proportional vectors in the system of 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 the 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 in vectors , and, moreover, in a unique way, i.e. expansion coefficients are 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 is. in this equality. Indeed, if , then. This means that a non-trivial linear combination of vectors is equal to the zero vector, which contradicts the linear independence of the system . Therefore, and then, i.e. 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 expansion coefficients are respectively equal to each other (for example, ).
Then from equality we get .
Therefore, the linear combination of vectors is equal to the null vector. Since not all of its coefficients are equal to zero (at least ), this combination is non-trivial, which contradicts the condition of linear independence of the vectors . The resulting contradiction confirms the uniqueness of the decomposition.
Rank and basis of the system of vectors. The rank of a system of vectors is the maximum number of linearly independent vectors of the system.
The basis of the system of vectors is the maximum linearly independent subsystem of the given system of vectors.
Theorem. Any system vector can be represented as a linear combination of system basis vectors. (Any vector of the system can be decomposed into basis vectors.) The expansion coefficients are uniquely determined for a given vector and a given basis.
Doc-in:
Let the system have a basis .
1 case. Vector - from the basis. Therefore, it is equal to one of the basis vectors, let's say . Then = .
2nd case. The vector is not from the basis. Then r>k.
Consider a system of vectors . This system is linearly dependent, since is a basis, i.e. maximum linearly independent subsystem. Therefore, 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 a vector in terms of a basis is unique. Assume the opposite: there are two expansions of the vector in terms of the basis.
Subtracting these equalities, we get
Taking into account the linear independence of the basis vectors, we obtain
Therefore, the expansion of a vector in terms of a 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.
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