## Linear AlgebraThe original version of this book, handed out to my students in weekly in stallments, had a certain rugged charm. Now that it is dressed up as a Springer UTM volume, I feel very much like Alfred Dolittle at Eliza's wedding. I hope the reader will still sense the presence of a young lecturer, enthusiastically urging his audience to enjoy linear algebra. The book is structured in various ways. For example, you will find a test in each chapter; you may consider the material up to the test as basic and the material following the test as supplemental. In principle, it should be possible to go from the test directly to the basic material of the next chapter. Since I had a mixed audience of mathematics and physics students, I tried to give each group some special attention, which in the book results in certain sections being marked· "for physicists" or "for mathematicians. " Another structural feature of the text is its division into laconic main text, put in boxes, and more talkative unboxed side text. If you follow just the main text, jumping from box to box, you will find that it makes coherent reading, a real "book within the book," presenting all that I want to teach. |

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### Contents

Sets and Maps | x |

12 Maps | 4 |

13 Test | 9 |

14 Remarks on the Literature | 10 |

15 Exercises | 11 |

Vector Spaces | 13 |

22 Complex Numbers and Complex Vector Spaces | 18 |

23 Vector Subspaces | 22 |

66 Test | 111 |

67 Determinant of an Endomorphism | 113 |

68 The Leibniz Formula | 114 |

69 Historical Aside | 116 |

Systems of Linear Equations | 118 |

72 Cramers Rule | 120 |

73 Gaussian Elimination | 122 |

74 Test | 124 |

24 Test | 23 |

25 Fields | 25 |

26 What Are Vectors? A section for physicists | 28 |

27 Complex Numbers 400 Years Ago Historical aside | 37 |

28 Remarks on the Literature | 38 |

Dimension | 41 |

32 The Concept of Dimension | 44 |

33 Test | 48 |

34 Proof of the Basis Extension Theorem and the Exchange Lemma | 49 |

35 The Vector Product A section for physicists | 52 |

36 The Steinitz Exchange Theorem Historical aside | 57 |

37 Exercises | 58 |

Linear Maps | 60 |

42 Matrices | 66 |

43 Test | 71 |

44 Quotient Spaces | 73 |

45 Rotations and Reflections of the Plane A section for physicists | 76 |

46 Historical Aside | 80 |

Matrix Calculus | 83 |

52 The Rank of a Matrix | 87 |

53 Elementary Transformations | 88 |

54 Test | 91 |

55 How Does One Invert a Matrix? A section for mathematicians | 92 |

56 Rotations and Reflections Continued | 95 |

57 Historical Aside | 98 |

58 Exercises | 99 |

Determinants | 101 |

62 Determination of Determinants | 105 |

63 The Determinant of the Transposed Matrix | 106 |

64 Determinantal Formula for the Inverse Matrix | 108 |

65 Determinants and Matrix Products | 110 |

75 More on Systems of Linear Equations | 126 |

76 Captured on Camera A section for physicists | 128 |

77 Historical Aside | 131 |

79 Exercises | 132 |

Euclidean Vector Spaces | 134 |

82 Orthogonal Vectors | 137 |

83 Orthogonal Maps | 141 |

84 Groups | 142 |

85 Test | 144 |

86 Remark on the Literature | 146 |

Eigenvalues | 149 |

92 The Characteristic Polynomial | 152 |

93 Test | 154 |

94 Polynomials | 156 |

95 Exercises | 159 |

The Principal Axes Transformation | 160 |

102 Symmetric Matrices | 161 |

103 The Principal Axes Transformation for SelfAdjoint Endomorphisms | 164 |

104 Test | 166 |

105 Exercises Exercises for mathematicians | 168 |

Classification of Matrices | 169 |

112 The Rank Theorem | 172 |

113 The Jordan Normal Form | 174 |

114 More on the Principal Axes Transformation | 176 |

116 Test | 181 |

117 Exercises | 183 |

Answers to the Tests | 185 |

References | 198 |

Index | 199 |

### Common terms and phrases

abelian group axioms basis extension theorem bijective bilinear calculation called Chapter coefficients column rank complex numbers concept defined Definition denote determinant diagonal diagonalizable diagram dimension formula dimensional eigenspace eigenvalue eigenvectors elementary row transformations elements endomorphism Euclidean vector space example exists finite-dimensional Gaussian elimination hence injective inner product invertible Jordan normal form lemma linear algebra linear combination linear equations linear map linearly independent M(m x n,F M(n x n mathematical matrix n x n matrices n-tuples notation obtain orthogonal orthonormal basis orthonormal system physical vector physical vector space principal axes transformation PROOF properties quadratic form real numbers real vector space Remark rotation scalar domain scalar multiplication Section solution solvable space over F subset surjective Sylvester symmetric matrix system of equations systems of linear th row u x v uniquely unit vectors vector product write zero