Higher_Mathematics_Part_1

UDC 513.123:517.2=111(075.8)

ББК В10я7

Н 65

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Authors:

V.P. Denisiuk, L. I. Grishina, O. V. Karupu, T. A. Oleshko, V. V. Pakhnenko, V. K. Repeta

Reviewers:

O. P. Besklinska — candidate of science (physics and mathematics), associate professor (Kyiv National Linguistic University)

P.T. Kachanov — candidate of science (engineering), associate professor (State University of Information and Communication Technologies)

English language adviser

O. Yu. Kravchuk, associate professor

Approved by the Methodical

and Editorial Board of the National Aviation University (Minutes № 8/09 of 17.12.2009)

The manual fully corresponds to the educational program of higher mathematics for engineering universities. It contains basic theoretical information, examples of problems solution, class, home and self-test assignments.

The manual is intended for the first course students of all specialities.

product (vector product) — векторний добуток, triple product (mixed product) — мішаний добуток, collinear vectors — колінеарні вектори, coplanar vectors —

компланарні вектори, orthogonal vectors — ортогональні вектора

Main tasks. 1. Calculation of determinants. 2. Solution of Linear algebraic equations system. 3. Matrix equation solution. 4. Investigation of the linear algebraic equations systems. 5. Operations with vectors. 6. Mutual disposition of vectors. 7. Dot, cross and triple product usage in solving geometrical problems.

A STUDENT MUST BE READY TO DO

THE FOLLOWING ASSIGNMENTS

1. Concepts, definitions, formulations:

•Determinants of the 2nd, the 3rd and the n-th orders.

•Matrices. Linear operations with matrices. Multiplication of matrices.

•Inverse matrix.

•Minors, rank of matrix.

•Definite, indefinite, consistent, inconsistent SLAE.

•Matrix form of SLAE.

•Gauss’ method of SLAE solution.

•Kronecker-Capellі theorem usage in SLAE investigation.

•Eigenvalues and eigenvectors of matrix.

•Geometrical vector. Vector addition and subtraction operations, multiplication by scalar.

•Linear dependence and independence of vectors.

•Basis on a plane and in space.

•Cartesian coordinate system (CCS).

•Dot product of two vectors.

•Cross product of two vectors.

•Triple product.

2. Proofs and conclusions

•Properties of determinants (2nd and 3rd orders).

•Matrix addition and multiplication properties.

•Existence of an inverse matrix.

•Inverse matrix method of SLAE solution.

•Cramer’s Theorem.

•Kronecker-Capellі Theorem.

•Projection of vector on axis.

•Representation of a vector in terms of base vectors.

•Properties of a dot product; calculation by coordinates.

•Properties of a cross product; calculation by coordinates.

•Properties of a triple product; calculation by coordinates.

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3. Assignments

•Calculate the determinants of order 2, 3 and n, to be able to lay out a determinant by the elements of any row or column, to reduce determinant to the triangle form.

•Find the matrix sum, difference and product.

•Find the matrix rank.

•Find an inverse matrix.

•Solve the square systems by Cramer’s method, through inverse matrix.

•Solve the arbitrary SLAE by Gauss’ method.

•Analyse SLAE on the consistence (compatibility) according to KroneckerCapellі Theorem.

•Find the eigenvalues and eigenvectors of matrix.

•Find the vector coordinates, it’s length, unit vector. Find the angle between vectors.

•Find the vector sum, difference, dot and cross products.

•Calculate the area of the triangle, volume of pyramid.

•Be able to represent the vector in terms of base vectors .

•Be able to use the condition of two vectors perpendicularity.

Micromodule 1

BASIC THEORETICAL INFORMATION. DETERMINANTS

Determinants of the 2nd and the 3rd orders. Properties of determinants. Minor and cofactor. General definition of a determinant of the n-th order. Calculations of determinants.

Literature: [1, chapter 1], [4, part 2, p.2.3], [6, chapter 1 §1], [7, chapter 2 §7], [10, chapter 1 §2] [11, chapter 1 §1].

1.1. Determinants of the 2nd and the 3rd orders

Definition 1.1. A number

is the determinant of the 2nd order.

Such determinants appear to be true, for example, for the solution of such linear systems as

a11 x1 + a12 x2 = b1 ,a21 x1 + a22 x2 = b2 .

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Let’s find x . We multiply both parts of the first equation by a22 , and of the 2nd equation by ( −a12 ), whereupon, by adding the left-hand and right-hand sides of the given equations, we get the equality

a22 (a11 x1 + a12 x2 ) − a12 (a21 x1 + a22 x2 ) = a22b1 − a12b2 ,

which, after simplifications, we obtain

(a22 a11 − a12 a21 )x1 = a22b1 − a12b2 .

In the case of condition a22 a11 − a12 a21 ≠ 0 , we get

So, the solutions of the given system are expressed through the determinants of the 2nd order.

Definition 1.2. A number

a11 a12 a13

= a21 a22 a23 = a11a22 a33 + a12 a23a31 + a13a21a32 − a31 a32 a33

−a13 a22 a31 − a11a23 a32 − a12 a21a33 .

is called the determinant of the 3rd order.

The order of determinant equals to the number of its rows or columns.

The numbers aij are elements of a determinant thus the first index i indicates a

number of row, and the 2nd index j is a number of column, in which the element lies. Elements a11 , a22 in the determinant of the 2nd order and a11 , a22 , a33 of the

determinant of the 3rd order, all together, form a principal (leading) diagonal of a determinant and elements a12 , a21 and a13 , a22 , a31 form a secondary diagonal.

For the calculation of determinant of the 2nd order it is necessary to subtract the product of all elements of a secondary diagonal from the product of elements of a principal diagonal.

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The determinant of the 3rd order is calculated by the rule of triangle. First three multipliers with a sign plus are the product of all elements of a principal diagonal and in the tops of two triangles in which one side is parallel to the principal diagonal. Next three multipliers with a sign minus are the product of elements of a secondary diagonal and in the tops of two triangles in which one side is parallel to secondary diagonal.

To remember this formula it is advisable to use the scheme represented in Fig. 1.1.

Fig. 1.1

Remark. It is important to know the following formula. If a determinant is upper (lower) triangular form, then this determinant is equal to the product of all elements of the principal diagonal, for example

1.2. Principal properties of determinants

We shall formulate the properties for the third-order determinants, although they are true for determinant of any order.

1. The value of a determinant does not vary if its rows and columns are interchanged.

Property 1 establishes the equivalence of the rows and columns of a determinant.

2. Interchanging of two rows or of two columns of a determinant is equivalent to its multiplication by –1.

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For example,

3.If all elements of a row (or a column) of a determinant are zero, then the determinant itself is zero.

4.If a determinant has two identical rows (or columns), then it is equal to zero.

5.The product of all elements of one column or one row of a determinant by any number k is equivalent to the multiplication of the determinant by this number.

For example,

6.If the elements of two rows (or columns) of a determinant are proportional then the determinant is zero.

7.If every element of the i-th row (or j-th column) of a determinant is the sum of two terms, then the determinant can be represented as the sum of two determinants.

For example,

8. If the elements of a row (or a column) of a determinant are summed up with the elements of another row (column) multiplied by any common factor k, then the value of the determinant remains unchanged.

For example,

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a11 a12 a13 = a21 a22 a23 .

a31 a32 a33

All properties of determinants of order 3 are true for the determinants of any order.

1.3. Expansion of a determinant by the elements of the row or the column

Let’s consider a determinant of order 3

a11 a12 a13

= a21 a22 a23 . a31 a32 a33

Definition 1.3. The Minor Мij of an element aij of a determinant of order n

is a determinant of order (n – 1) obtained from the given determinant by striking out a row i and a column j in which the element lies.

For instance, the minor of the element a23 of the determinant ∆ is the

Definition 1.4. The Cofactor, or algebraic adjunct, Aij of an element aij of a determinant is the minor Мij of that element multiplied by (−1)i+ j :

Aij = (−1)i+ j Mij .

Laplas’ Theorem 1.1. A determinant is equal to the sum of the products of all elements of a row or of a column by their cofactors.

For example, for the determinant of the 3rd order such equalities are executed:

column).

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