## Linear Algebra: A Pure Mathematical ApproachLinear algebra is one of the most important branches of mathematics - important because of its many applications to other areas of mathematics, and important because it contains a wealth of ideas and results which are basic to pure mathematics. This book gives an introduction to linear algebra, and develops and proves its fundamental properties and theorems taking a pure mathematical approach - linear algebra contains some fine pure mathematics. Main topics: - vector spaces and algebras, dimension, linear maps, direct sums, and (briefly) exact sequences - matrices and their connections with linear maps, determinants (properties proved using some elementary group theory), and linear equations - Cayley-Hamilton and Jordan theorems leading to the spectrum of a linear map - this provides a geometric-type description of these maps - Hermitian and inner product spaces introducing some metric properties (distance, perpendicularity etc.) into the theory, also unitary and orthogonal maps and matrices - applications to finite fields, mathematical coding theory, finite matrix groups, the geometry of quadratic forms, quaternions and Cayley numbers, and some basic group representation theory A large number of examples, exercises and problems are provided. Answers and/or sketch solutions to all of the problems are given in an appendix. Some of these are theoretical and some numerical, both types are important. No particular computer algebra package is discussed but a number of the exercises are intended to be solved using one of these packages chosen by the reader.The approach is pure-mathematical, and the intended readership is undergraduate mathematicians, also anyone who requires a more than basic understanding of the subject. This book will be most useful for a "second course" in linear algebra, that is for students that have seen some elementary matrix algebra. But as all terms are defined from scratch, the book can be used for a "first course" for more advanced students. |

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### Contents

Algebraic Preamble | 1 |

Permutation Groups | 6 |

Problems 1 | 9 |

Vector Spaces and Linear Maps | 13 |

Bases and Dimension | 17 |

Linear Maps | 22 |

Direct Sums | 28 |

Addendum Modules | 32 |

Problems 5 | 123 |

Hermitian and Inner Product Spaces | 125 |

Hermitian and Inner Products and Norms | 126 |

Unitary and Selfadjoint Maps | 139 |

Orthogonal and Symmetric Maps | 149 |

Problems 6 | 153 |

Selected Topics | 159 |

Normed Algebras Quaternions and Cayley Numbers | 169 |

Problems 2 | 33 |

Matrices Determinants and Linear Equations | 39 |

Determinants | 51 |

Systems of Linear Equations | 62 |

Problems 3 | 67 |

CayleyHamilton Theorem and Jordan Form | 75 |

The CayleyHamilton and Spectral Theorems | 81 |

Jordan Form | 92 |

Problems 4 | 103 |

Interlude on Finite Fields | 109 |

Applications | 116 |

Introduction to the Representation of Finite Groups | 178 |

Problems 7 | 190 |

Set Theory | 195 |

Problems A | 202 |

Answers and Solutions to the Problems | 205 |

Bibliography | 239 |

241 | |

243 | |

244 | |

245 | |

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### Common terms and phrases

2-cycles associated axioms bijection called Cayley-Hamilton Theorem Chapter characteristic polynomial coefficient column commutative consider Corollary denoted determinant diagonal entries direct sum eigenvalues eigenvectors element elementary operations End(V equals equivalent example exists factor finite field finite groups finite-dimensional function G-module G-submodule given in Problem gives Hence Hermitian product identity map Inequality injective inner product space inverse irreducible isomorphism Jordan Form Lemma linear algebra linear equations linear map linearly independent matrix multiplication minimum polynomial module n x n matrix non-singular non-zero normed algebra Note orthogonal orthonormal basis permutation positive definite positive integer prove quadratic form Rank Theorem reader should check real numbers result follows ring satisfying scalar multiplication Secondly self-adjoint similar solution spectral standard basis subgroup subset subspace superdiagonal Suppose surjective symmetric theory unique unitary vector space zero