Linear Algebra

In the spirit of the author s Basic Language of Mathematics, this companion volume is a careful exposition of the concepts and processes of Linear Algebra. It stresses cautious and clear explanations, avoiding reliance on co ordinates as much as possible, and with special, but not exclusive, attention to the finite dimensional situation. It is intended to also serve as a conceptual and technical background for use in geometry and analysis as well as other applications. Readership: Undergraduate and graduate students in mathematics; Mathematicians in algebra. CONTENTS 8 PREFACE 6 Chapter 1. LINEAR SPACES AND LINEAR MAPPINGS 10 11. Linear spaces 10 12. Linear mappings 15 13. Subspaces 20 14. Linear partitions 29 15. Supplements 33 Chapter 2. PROPERTIES OF LINEAR MAPPINGS 38 21. Linear invertibility 38 22. Cancellability and factorization 43 Chapter 3. LINEAR PRODUCTS AND COPRODUCTS 48 31. Linear products 48 32. Linear coproducts 53 33. Decompositions 58 Chapter 4. FAMILIES IN LINEAR SPACES 66 41. Linear combination 66 42. Linear independence 69 43. Linear independence, spanning, bases 74 Chapter 5. FINITE DIMENSION 82 51. Matrices 82 52. Finite-dimensional spaces 86 53. Finite dimension and linear mappings 92 54. Finite- and infinite-dimensional spaces 99 Chapter 6. DUALITY 102 61. Dual spaces and transposes 102 62. Annihilators 106 63. Higher-order dual spaces and transposes 115 64. Biorthogonal families 119 Chapter 7. DUALITY AND FINITE DIMENSION 122 71. Duality for finite-dimensional spaces 122 72. Dual bases and matrices 126 73. The trace form 130 74. The canonical identifications 136 INDEXES 138 Index of terms 138 Index of symbols 140