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Introduction

This textbook is intended for students who have already studied basic mathematics and need
to study the methods of advanced mathematics. It covers three content areas: Linear Algebra,
Vector Algebra and Analytic Geometry.

Many useful examples and exercises are presented in the textbook.

Each part contains basic mathematical conceptions and explains new mathematical terms.
The Linear Algebra topics include matrix operations, determinants and systems of linear
equations.

In the section “Vector Algebra”, a main attention is paid to the geometrical applications
of vector operations. The vector approach is considered to be basic for discussion of classic
problems of Analytic Geometry.

The authors welcome reader’s suggestions for improvement of future editions of this text-
book.

Lecture 1. Elements of Determinants’ Theory

1.1. Determinants and their properties

Let’s have a look at the square table, which consists of four digits:

( )
a 11 a 12
. (1.1)
a 21 a 22

In future we will call these tables matrices. Every digit has its place in the table, which is
determined by double indexations. The first index indicates the row number while the second
- the column number, which contains the given digit. The table (1.1) consists of two rows and
two columns.

Definition 1.1. A digit which equals to the difference of multiples of the digits,
located on the main and side diagonals (main diagonal is the one connecting the
first digit with the last one, and side diagonal - is the other diagonal) is called the
determinant of the second order. ✓

The determinant is denoted by the symbol:

a 11 a 12
∆ = .

a 21 a 22
So, according to the Definition 1.1,

∆ = a 11 · a 22 − a 21 · a 22 . (1.2)

Digits a 11 , a 12 , a 21 , a 22 are called elements of the determinant.

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