ФЕДЕРАЛЬНОЕ АГЕНТСТВО ПО ОБРАЗОВАНИЮ 
           Государственное образовательное учреждение высшего профессионального образования 
                «ТОМСКИЙ ПОЛИТЕХНИЧЕСКИЙ УНИВЕРСИТЕТ» 
                               
                               
                               
                               
                               
                               
                               
                          V.V. Konev 
                               
                               
                               
                               
                               
            LINEAR ALGEBRA, VECTOR ALGEBRA AND  
                    ANALYTICAL GEOMETRY 
                               
                            TextBook 
                               
                               
                 Рекомендовано в качестве учебного пособия  
                    Редакционно-издательским советом 
                  Томского политехнического университета 
                               
                               
                               
                               
                               
                               
                               
                               
                               
                               
                               
                               
                               
                          Издательство 
                   Томского политехнического университета 
                             2009 
         
        UDС 517 
         
        V.V. Konev. Linear Algebra, Vector Algebra and Analytical Geometry. 
        Textbook. Tomsk: TPU Press, 2009, 114 pp. 
           This textbook consists of 3 parts devoted to the mathematical 
        methods of Linear Algebra and Analytical Geometry based on the vector 
        analysis technique. The basic concepts are explained by examples and 
        illustrated by figures. 
           The textbook is helpful for students who want to understand and be 
        able to use matrix operations, solve systems of linear equations, analyze 
        relative positions of figures, transform coordinate systems, and so on.  
           The textbook is designed to English speaking students. 
         
         
         
         
         
         
         
         
           Reviewed by: V.A. Kilin, Professor of the Higher Mathematics 
           Department, TPU, D.Sc. 
            
         
         
         
         
         
         
         
         
         
         
         
         
         
         
        © Konev V.V. 2001-2009
                     
        © Tomsk Polytechnic University, 2001-2009 
                       PREFACE 
      This textbook is intended for students who have already studied basic 
      mathematics and need to study the methods of higher mathematics. It covers 
      three content areas: Linear Algebra, Vector Algebra and Analytical Geometry. 
      Each part contains basic mathematical conceptions and explains new 
      mathematical terms. Many useful examples and exercises are presented in the 
      textbook. explained and illustrated by examples and exercises. 
      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 Analytical Geometry. 
      The author welcomes reader’s suggestions for improvement of future editions of 
      this textbook. 
       
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                            CONTENTS 
         
         
        Preface  ……………………..…………………………………………    3 
        Contents ………………………………………………………………..   4 
        LINEAR ALGEBRA 
        Chapter 1. MATRICES 
          1.1.  Basic Definitions ……………………………………………. .   7 
          1.2.  Matrix Operations …………………………………………….    8 
          1.3.  Types of Matrices …………………………………………. …  12 
          1.4.  Kronecker Delta Symbol………………………………………  15 
          1.5.  Properties of Matrix Operations………………………………  16 
         
        Chapter 2. DETERMINANTS 
          2.1.  Permutations and Transpositions………………………………  20 
          2.2.  Determinant General Definition ……………………………...   23 
          2.3.  Properties of Determinants …………………………………...   25 
          2.4.  Determinant Calculation………………………………………  31 
         
        Chapter 3. INVERSE MATRICES 
          3.1.  Three Lemmas ………………………………………………..  36 
          3.2.  Theorem of Inverse Matrix …………………………………...  38 
           3.2.1. Examples ……….………………………………………….   39 
          3.3.  Calculation of Inverse Matrices by Elementary 
              Transformations ………………………………………………  42 
         
        Chapter 4. SYSTEMS OF LINEAR EQUATIONS 
          4.1.  Matrix Rank …………………………………………………..  43 
          4.2.  Basic Concepts ……………………………………………….  45 
          4.3.  Gaussian Elimination …………………………………………  46 
           4.3.1. Examples …………………………………………………..   47 
          4.4.  Homogeneous Systems of Linear Equations…………………    50 
           4.4.1. Examples ………………………………………………….   51 
          4.5.  Cramer’s Rule ………………………………………………..   54 
          4.6.  Cramer’s General Rule ………………………………………   57 
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