Добавил:

Upload Опубликованный материал нарушает ваши авторские права? Сообщите нам.

Вуз:

Одесская национальная академия связи им. А.С. Попова

Предмет:

[НЕСОРТИРОВАННОЕ]

Файл:

Reference_book_on_Higher_Mathemanics_Part_I_F

.pdf

Скачиваний:

Добавлен:

10.02.2016

Размер:

679.16 Кб

Скачать

☆

<<< < Предыдущая 1 2 34 / 74 5 6 7 > Следующая >>>

Linear Algebra

Vocabulary

addition (subtraction) of matrices – сложение (вычитание матриц) adjoint matrix– присоединенная матрица

augmented matrix – расширенная матрица capital letter – заглавная буква

check the equality – проверить равенство

cofactor of an element A - алгебраическое дополнение элементаaij

column – столбец

compatible system – совместная система

Cramer’s Rule – правило Крамера determinant - определитель

diagonal matrix – диагональная матрица dimension – размерность

elementary operations on matrices – элементарные операции над матрицами general case – общий случай

homogeneous system of equations – однородная система уравнений inverse of matrix – обратная матрица

lowercase letter (small letter) – строчная буква matrix (matrices) – матрица (матрицы) matrix form – матричный метод

matrix of order n – матрица порядка n minor – минор

multiplication of matrices – умножение матриц

multiplying of a matrix by a number – умножение матрицы на число nonsingular - невырожденный

principal (main) diagonal – главная диагональ rank of matrix – ранг матрицы

row – строка

square matrix – квадратная матрица supplement – дописать, добавить

system of n linear equation in n unknowns – система n линейных уравнение с n

неизвестными the expanding determinant by – разложение определителя по transformation Z through X - преобразование Z через X transposed matrix – транспонированная матрица

triangular matrix – треугольная матрица unique solution – единственное решение unit-matrix – единичная матрица

Matrices, determinants, linear systems

Titles, definitions, theorems

Algebraic form

• A rectangular array of numbers is called

L a

a matrix of [m× n] (read m by n)

dimension.

A =

a21

a22

L a2n

• The element in the i-th row and j-th

column of a matrix can be represented as

(

)

j = 1, n.

aij : A = aij

, i = 1, m,

am1

am2

L amn

Transposed matrix

L a

L L L

= L L L

am1

amn

a1n

amn

L a

Square matrix of order n

A =

a21

a22

L a2n

L L L

an2

an1

L ann

The diagonal containing

Principal (main) diagonal of a square

a11, a22 ,K, an −1n −1, ann

matrix

Diagonal matrix

a21

L L L

L ann

L 0

Unit-matrix E or I

L 0

I =

L L L L

L 1

A = (aij ), B = (bij ), i =

, j =

then

Let

1, m

1, n

Equality of matrices

A = B

A = B aij = bij , i = 1, m , j = 1, n

Sum of matrices of the same dimension

C = (cij

), where cij

= aij

+ bij ,

A+B = C

i = 1, m , j = 1, n

Difference of matrices of the same

D = (d ij ), where d ij

= aij

− bij ,

dimension A - B = D

i = 1, m , j = 1, n

Product of a matrix A by a number λ :

λA

(

)

, i = 1, m , j = 1, n

λA = λaij

Determinants:

• det a11

• det a11 = a11

= a

− a

• det

•

a21

a22

a21

a22

a21

a a

• det a

a22

a23

a21

a22

a23

•

a21

a22

a23

a21 a22 =

a32

a31

a32

a33

a31

a32

a33

a31 a32

a32

a33

= a11a22 a33 + a12 a23 a31 + a13 a21a32 −

− a31a22 a13 − a32 a23 a11 − a33 a21a12

A minor M ij

of an element aij

of a square matrix A is a determinant obtained from a

given matrix deleting the i-th row and j-th column.

A quantity

(−1) i + j M

ij is called a cofactor

of an element

ij .

The sum of the products of elements

= a

+ a

+ ... + a

A is

i1 11

any row (column) of a determinant and

called the expanding determinant by the i-

their cofactors is equal to one and the same

th row.

number. This number is a value of the

given determinant.

The multiplication of a matrix A[m× n] by

a matrix B[n× p] is a matrix C[m× p] whose

cij = ∑aik bkj , i =

, j =

1, m

1, p

element cij

is the product of

the i-th row

k =1

of A and the j-th column of B:

A square matrix B is said to be an inverse matrix of A if

AB = BA = I and it is denoted by the symbol

A−1 . So we have AA−1 = A−1 A = I

Transposed

matrix

cofactors

the

L A

corresponding elements of the given

matrix A is called the adjoint of A

adj A = A = L

An1

L Ann

Be sure that

≠ 0.

2. Construct the matrix of corresponding

Method of finding the inverse matrix

cofactors and transpose it (A).

Divide the matrix

A by

A−1 =

A .

det A

a x + a x

+ K + a

= b

The system of n linear equation with n

11 1

unknowns

a21x1 + a22 x2 + K + a2n xn = b2

.......................................

x + a

+ K + a

= b

n1 1

AX = B , where

L a

Matrix form of a system:

AX =B

K a

A =

L L L

an1

an2 K ann

X =

B =

Methods of solution of linear systems

Matrix method solution of a square

X = A−1 B

linear system

Cramer’s Rule :If the determinant of

x =

, x

, K, x

the system does not equal zero then this

system has a unique solution .

where

i is

the

determinant

formed by replacing the ith column of the

matrix A with the column of constant

terms

b1, b2 , b3 ,K, bn , i = 1, n

System of linear equations in the general

•

Matrix of a system:

Case

L a

a11 x1 + a12 x2 + K + a1n xn = b1

L a

+ a22 x + K + a2n x2 = b2

A =

a21 x1

L L

.......................................

(1)

L a

am2 x2

amn xn

• Augmented matrix:

am1 x1

L a

a21

a22

L a2n

B =

am1

am2

L amn

Let an arbitrary matrix A dimension of which is [m× n] be given

Let us strike out k rows and k columns in this matrix. Then elements aij found at the

intersection of these rows and columns form the matrix of order k. Determinant of this matrix is said to be minor of the k order of the matrix A.

The highest order of the minor of matrix A different from zero is called the rank of this matrix and denoted r(A).

Matrix A is equivalent to matrix B if their ranks are equal: A ~B if r(A) = r(B).

A system (1) is compatible if and only if the rank of matrix A is equal to the rank of matrix B: r(A) = r(B)

• Gauss Method. The goal of this method is to reduce an augmented matrix to triangular form. After that it is able to answer the next questions:

1.Is the given linear system compatible or not?

2.How many solutions has this system?

3.If the system is compatible you can find its solution.

• Homogeneous System of Equations is a linear system of equations for which the constant terms are zeroes:

a11x1 + a12 x2 + K + a1n xn = 0a21x1 + a22 x2 + K + a2n xn = 0

LLLLLLLLLLL

am1x1 + am2 x2 + L + amn xn = 0

Homogeneous system is compatible as the solution (0,0,...,0) is always the solution of this system. If a homogeneous system has non-zero solutions you can find them using the Gauss method

Example. Solve the system of linear equation using Gauss method.

2x + x				+ x		+ 2x		= 8,
	1		2		3		4
x1		- x2		+ 3x3 + x4				= 10,
	x1	+ x2				+ x4 = 5.
	x1	+ x2				+ x4 = 5.

Solution.
Reduce the augmented matrix to triangular form:
	2 1 1								1 1 0 1		5
	2 1 1		2		8				1 1 0 1		5

B = 1 − 1 3			1			10		~	2 1 1 2			8 R2 − 2R1 ~
							R3 → R1			1 − 1 3 1			R2 − R1
	1 1 0		1		5		R3 → R1			1 − 1 3 1	10		R2 − R1
	1 1 0			1						1 1 0 1		5
	1 1 0			1		5				1 1 0 1		5
~								~
~	0 − 1 1			0		− 2		~		0 − 1 1 0		− 2
		− 2 3						− 2R2
	0	− 2 3		0		5	R3	− 2R2		0 0 1 0		9

As we can see r(A) = r(B) = 3 . But n = 4 > r = 3. It means that the general solution depends on n − r arbitrary constants. Here is one constant in this case. Let x4 = C , then the equivalent system is

	x + x			= 5 − C			x	3	= 9,
	1		2					3
	− x2 + x3 = −2					− x2 + 9 = −2 x2 = 11,
			x3 = 9						+ 11 = 5 − C x1 = −6 − C.
			x3 = 9				x1		+ 11 = 5 − C x1 = −6 − C.
Check the solution: AX = B .
		2 1 1 2				− 6 − C				-12-2C + 11 + 9 + 2C		8
		2 1 1 2								-12-2C + 11 + 9 + 2C		8
							11
AX = 1 − 1 3 1							9			= − 6 − C − 11 + 27 + C = 10 = B .
							9				− 6 - C + 11 + C

		1 1 0 1										5
							C
			− 6 − C

				11
Answer: X =				9	.
				9

				C
				C

VECTORS

Vocabulary accordingly – соответственно

be observed - наблюдается clockwise – по часовой стрелке collinear - коллинеарный coplanar - компланарный

counterclockwise – против часовой стрелки directed – направленный

direction cosines – направляющие косинусы

expansion of the vector a through the base – разложение вектора по базису

initial point - начальная точка lefthanded triple – левая тройка

middle point of a segment – точка, делящая отрезок пополам ordered triple –упорядоченная тройка

ort of a vector (unit vector) – единичный вектор orthogonal – ортогональный

right -handed triple – правая тройка rotation – поворот

scalar – скаляр

scalar product (dot product) – скалярное произведение terminal (end) point – конечная точка

test of collinearity of vectors – признак коллинеарности векторов test of coplanarity of vectors – признак компланарности векторов test of orthogonality of vectors – признак ортогональности векторов tetrahedron – тетраэдр

torque – момент (силы) triangle – треугольник

triple scalar product – смешанное произведение vector – вектор

vector product (cross product) – векторное произведение volume of the parallelepiped – объем параллелепипеда

• Vector is a line segment. a = AB , A is the initial point. B is the terminal (end)

point.

• Let i , j, k be the unit and orthogonal vectors giving the direction of x-axis, y-

k . Then vector

axis and

z-axis accordingly:

= 1,

− are the projections of the vector

= a x i + a y

j + a z k , where a x , a y , a z

onto the vectors i, j and k accordingly and are called the coordinates of the vector.

• a = (a x , a y , a z ) – the coordinate form of the vector a ;

• a = axi + a y j + az k – the vector form of the vector a , or the expansion of the

through the base i , j, k.

vector

№

Theorems, definitions

Formulas for Calculation


Let the coordinates of the points А and В and vectors					a	, b ,			c	be given:
A(x A , y A , z A ), B(xB , y B , z B ),		= (a x , a y , a z ),		= (bx , b y , bz ),				=		(c x , c y , c z ) then
			b
	a						c

Vector’s coordinates through the
1 coordinates of initial and end points	AB = (x B − x A ; y B − y A , z B − z A )

2The length of a vector

= a x2 + a 2y + a z2

Equal vectors

(

) have equal

= b a x

= bx , a y

= by , a z = bz

corresponding coordinates

Sum and difference of vectors

(

)

a ± b =

± bx ; a y

± b y ; a z ± bz

a x

A product of a vector

by a scalar

(number) λ

= (λa x ; λa y ; λa z )

A scalar product (dot product) of vectors

and b is the number equal to the product

(

)

of the moduli of these vectors and the

= a x bx + a y by

+ a z bz

a , b

cosine of the angle φ between them:

(

cos ϕ

cos ϕ =

(

)

Cosine of the angle φ between vectors

ϕ = (

, ^ b )

The scalar product

is the work.

(

)

A = F , S

9The ordered triple of vectors is called a right (left)-handed triple if the shortest rotation of the first vector to the second one is observed from the end point of the third vector in the counterclockwise (clockwise)

Vector product (cross product) of two

vectors a and b is the vector S such that

| = |

| | b | sin φ , where φ is the

angle between a and b ;

S =

a x

a y

a z

, S b ( the vector S is orthogonal

to both of the vectors

and b );

,b , S is the right-handed triple of

vectors.

The area of the parallelogram constructed on

S =

[

]

the vectors

and b

[

]

The area of a triangle constructed on the

vectors

and b

The vector product is the torque of the

, F ]= L 0

force

The triple scalar product:

a x

a y

a z

( a , b , c )=

( a , b , c ) = ( [a , b ], c ) ( a , [b , c ])

c x

c y

c z

The volume of the parallelepiped

V =

(

)

determined by vectors a, b, c is equal to the

module of the triple scalar product of these

vectors

(

)

The volume of a tetrahedron determined by

tet.

vectors a, b, c

a y

• a

Tests of collinearity of vectors a

•

b a = λ b

]= 0

•

18			(	)=
					0

	Test of orthogonality of vectors a b	a b	a , b

Test of coplanarity

b and

are coplanar if and

only if (

)=0

20 Coordinates of a point M dividing a segment

x A + λx B

1 + λ

y A

+ λy B

АВ in the given ratio λ :

= λ

y M

1 + λ

z A + λz B

1 + λ

x A + x B

Coordinates of a middle point of a

y A + y B

segment АВ ( AC = CB)

z A + z B

cosα =

a x

Direction cosines

a y

(α = a^ i, β = a^ j, γ = a^ k )

cos β =

cos γ =

a z

0 = (cosα; cos β ; cos γ )

Ort of a vector

(the unit vector): a 0

<<< < Предыдущая 1 2 34 / 74 5 6 7 > Следующая >>>

Соседние файлы в предмете [НЕСОРТИРОВАННОЕ]

#
10.02.20161.79 Mб136prorobot.ru-19-0214.doc
#
10.02.20161.35 Mб183P_PART_all.docx
#
23.11.20181.48 Mб10RAZDYeL_1.doc
#
23.11.20181.1 Mб10RAZDYeL_2.doc
#
23.11.2018443.39 Кб4RAZDYeL_3.doc
#
10.02.2016679.16 Кб5Reference_book_on_Higher_Mathemanics_Part_I_F.pdf
#
10.02.2016112.51 Кб46REGIONAL_NA_EKONOMIKA.docx
#
05.05.20191.58 Mб2RPS_Kovalevskiy.doc
#
26.08.2019539.14 Кб4Sotsiologia.doc
#
13.08.2019892.93 Кб9Strategichesky_menedzhment.doc
#
08.11.2019591.36 Кб0Switches.doc