Math 2: Linear Algebra
                           Problems, Solutions and Tips
                   FORTHEELECTRONICSANDTELECOMMUNICATIONSTUDENTS
                             Chosen, selected and prepared by:
                                  Andrzej Mackiewicz
                                          ´
                              Technical University of Poznan
                                                  ´
            2
                     Contents
                     1 Complex Numbers (Exercises)                                  7
                     2 Systems of Linear Equations (Exercises)                     17
                        2.1 PracticeProblems......................... 17
                     3 Row Reduction and Echelon Forms (Exercises)                 23
                        3.1 Practiceproblems.......................... 23
                        3.2 SolvingSeveralSystemsSimultaneously ............. 26
                     4 Vector equations (Exercises)                                31
                        4.1 Practiceproblems.......................... 31
                        4.2 Exercises .............................. 35
                     5 The Matrix Equation Ax=b (Exercises)                        39
                        5.1 PracticeProblems ......................... 39
                        5.2 Exercises .............................. 43
                     6  Solutions Sets of Linear Systems (Exercises)               47
                        6.1 PracticeProblems ......................... 47
                        6.2 Exercises .............................. 52
                     7 Linear Independence (Exercises)                             55
                        7.1 PracticeProblems ......................... 55
                        7.2 Exercises .............................. 58
                     8 Introduction to Linear Transformations (Exercises)          61
                        8.1 PracticeProblems ......................... 61
                        8.2 Exercises .............................. 66
                     9 The Matrix of a Linear Transformation (Exercises)           69
                        9.1 PracticeProblems ......................... 69
                        9.2 Exercises .............................. 72
                     10 Matrix Operations (Exercises)                              73
                          4Contents
                              10.1 Diagonal Matrices . . . . . . . . . . . . . . . . . . . . . . . . .   73
                              10.2 Matrix addition and scalar multiplication . . . . . . . . . . . .     73
                              10.3Matrixmultiplication........................ 74
                              10.4Whydoitthisway......................... 78
                              10.5Matrixalgebra ........................... 79
                              10.6Exercises .............................. 83
                          11 The Inverse of a Matrix (Exercises)                                         87
                              11.1PracticeProblems ......................... 87
                                    11.1.1Propertiesoftheinverse.................. 90
                                    11.1.2 Inverses and Powers of Diagonal Matrices . . . . . . . .      92
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                                    11.1.3 An Algorithm for finding         ............... 92
                              11.2Exercises .............................. 94
                          12 Characterizations of Invertible Matrices (Exercises)                        97
                              12.1PracticeProblems ......................... 97
                              12.2Exercises .............................. 99
                          13 Introduction to Determinants (Exercises)                                  105
                              13.1PracticeProblems .........................105
                              13.2ApplicationtoEngineering ....................109
                              13.3Exercises ..............................110
                          14 Eigenvectors and Eigenvalues (Exercises)                                  113
                              14.1PracticeProblems .........................113
                              14.2Exercises ..............................115
                          15 The Characteristic Equation (Exercises)                                   117
                              15.1PracticeProblems .........................117
                              15.2Exercises ..............................119
                              Bibliography                                                             123