Exercises and Problems in Linear Algebra
                              John M. Erdman
                          Portland State University
                            Version July 13, 2014
                           c
                           
2010 John M. Erdman
           E-mail address: erdman@pdx.edu
                                                   Contents
              PREFACE                                                                            vii
              Part 1.  MATRICESANDLINEAREQUATIONS                                                 1
              Chapter 1. SYSTEMSOFLINEAREQUATIONS                                                 3
                1.1. Background                                                                   3
                1.2. Exercises                                                                    4
                1.3. Problems                                                                     7
                1.4. Answers to Odd-Numbered Exercises                                            8
              Chapter 2. ARITHMETICOFMATRICES                                                     9
                2.1. Background                                                                   9
                2.2. Exercises                                                                   10
                2.3. Problems                                                                    12
                2.4. Answers to Odd-Numbered Exercises                                           14
              Chapter 3. ELEMENTARYMATRICES;DETERMINANTS                                         15
                3.1. Background                                                                  15
                3.2. Exercises                                                                   17
                3.3. Problems                                                                    22
                3.4. Answers to Odd-Numbered Exercises                                           23
              Chapter 4. VECTORGEOMETRYINRn                                                      25
                4.1. Background                                                                  25
                4.2. Exercises                                                                   26
                4.3. Problems                                                                    28
                4.4. Answers to Odd-Numbered Exercises                                           29
              Part 2.  VECTORSPACES                                                              31
              Chapter 5. VECTORSPACES                                                            33
                5.1. Background                                                                  33
                5.2. Exercises                                                                   34
                5.3. Problems                                                                    37
                5.4. Answers to Odd-Numbered Exercises                                           38
              Chapter 6. SUBSPACES                                                               39
                6.1. Background                                                                  39
                6.2. Exercises                                                                   40
                6.3. Problems                                                                    44
                6.4. Answers to Odd-Numbered Exercises                                           45
              Chapter 7. LINEAR INDEPENDENCE                                                     47
                7.1. Background                                                                  47
                7.2. Exercises                                                                   49
                                                        iii
                  iv                                               CONTENTS
                    7.3.   Problems                                                                                           51
                    7.4.   Answers to Odd-Numbered Exercises                                                                  53
                  Chapter 8.    BASIS FOR A VECTOR SPACE                                                                      55
                    8.1.   Background                                                                                         55
                    8.2.   Exercises                                                                                          56
                    8.3.   Problems                                                                                           57
                    8.4.   Answers to Odd-Numbered Exercises                                                                  58
                  Part 3.    LINEAR MAPS BETWEENVECTORSPACES                                                                  59
                  Chapter 9.    LINEARITY                                                                                     61
                    9.1.   Background                                                                                         61
                    9.2.   Exercises                                                                                          63
                    9.3.   Problems                                                                                           67
                    9.4.   Answers to Odd-Numbered Exercises                                                                  70
                  Chapter 10.     LINEAR MAPS BETWEENEUCLIDEANSPACES                                                          71
                    10.1.   Background                                                                                        71
                    10.2.   Exercises                                                                                         72
                    10.3.   Problems                                                                                          74
                    10.4.   Answers to Odd-Numbered Exercises                                                                 75
                  Chapter 11.     PROJECTIONOPERATORS                                                                         77
                    11.1.   Background                                                                                        77
                    11.2.   Exercises                                                                                         78
                    11.3.   Problems                                                                                          79
                    11.4.   Answers to Odd-Numbered Exercises                                                                 80
                  Part 4.    SPECTRALTHEORYOFVECTORSPACES                                                                     81
                  Chapter 12.     EIGENVALUES AND EIGENVECTORS                                                                83
                    12.1.   Background                                                                                        83
                    12.2.   Exercises                                                                                         84
                    12.3.   Problems                                                                                          85
                    12.4.   Answers to Odd-Numbered Exercises                                                                 86
                  Chapter 13.     DIAGONALIZATION OF MATRICES                                                                 87
                    13.1.   Background                                                                                        87
                    13.2.   Exercises                                                                                         89
                    13.3.   Problems                                                                                          91
                    13.4.   Answers to Odd-Numbered Exercises                                                                 92
                  Chapter 14.     SPECTRALTHEOREMFORVECTORSPACES                                                              93
                    14.1.   Background                                                                                        93
                    14.2.   Exercises                                                                                         94
                    14.3.   Answers to Odd-Numbered Exercises                                                                 96
                  Chapter 15.     SOMEAPPLICATIONSOFTHESPECTRALTHEOREM                                                        97
                    15.1.   Background                                                                                        97
                    15.2.   Exercises                                                                                         98
                    15.3.   Problems                                                                                         102
                    15.4.   Answers to Odd-Numbered Exercises                                                                103
                  Chapter 16.     EVERYOPERATORISDIAGONALIZABLEPLUSNILPOTENT                                                 105